[stephen wiggins] global bifurcations and chaos a(bookfi.org)

Applied Mathematical SciencesVolume 73

EditorsF. John J. E. Marsden L. Sirovich

AdvisorsM. Ghil J.K. Hale J. KellerK. Kirchgassner B.J. MatkowskyJ.T. Stuart A. Weinstein

Applied Mathematical Sciences

1. John: Partial Differential Equations, 4th ed.2. Sirovich: Techniques of Asymptotic Analysis.3. Ha/e: Theory of Functional Differential Equations, 2nd ed.4. Percus: Combinatorial Methods.5. vrm Mises/Friedrichs: Fluid Dynamics.6. Freiberger/Grenander: A Short Course in Computational Probability and Statistics.7. Pipkin: Lectures on Viscoelasticity Theory.9. Friedrichs: Spectral Theory of Operators in Hilbert Space-

If. Wolovich: Linear Multivariable Systems.12. Berkovitz: Optimal Control Theory.13. BlumanlCole: Similarity Methods for Differential Equations.14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions.15. Braun: Differential Equations and Their Applications, 3rd ed.16. Lefscheiz: Applications of Algebraic Topology.17. Co//atz/Wetterling: Optimization Problems.18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol 1.20. Driver: Ordinary and Delay Differential Equations.21. Courant/Friedrichs: Supersonic Flow and Shock Waves.22. Rouche/Habets/Lalor: Stability Theory by Liapunov's Direct Method.23. Lainperti: Stochastic Processes: A Survey of the Mathematical Theory.24. Grenander: Pattern Analysis: Lectures in Pattern Theory. Vol. It.25. Davies: Integral Transforms and Their Applications, 2nd ed.26. Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems27. de Boor: A Practical Guide to Splines.28. Keilson: Markov Chain Models-Rarity and Exponentiality.29. de Veubeke: A Course in Elasticity.30. Snianrki: Geometric Quantization and Quantum Mechanics.31. Reid: Sturmian Theory for Ordinary Differential Equations.32. Meis/Markowitz? Numerical Solution of Partial Differential Equations.33. Grenander: Regular Structures: Lectures in Pattern Theory. Vol. Ill.34. Kevorkian/Cole: Perturbation methods in Applied Mathematics.35. Carr: Applications of Centre Manifold Theory.36. Bengtsson/Ghil/Ka/len: Dynamic Meteorology: Data Assimilation Methods.37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.38. Lic/uenberg/Liebertnan: Regular and Stochastic Motion.39. Pier ini/Stampaechia/Vidossieh: Ordinary Differential Equations in R".40. Nor/or/Sell: Linear Operator Theory in Engineering and Science.41. Sparrow: The Lorenz Equations: Bifurcations. Chaos, and Strange Attractors.42. Guckenheimer/Ho/mes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.43. Ockendon/Tavler: Inviscid Fluid Flows.44. Poav: Semigroups of Linear Operators and Applications to Partial Differential Equations.45. Glashoi/Gustafcon: Linear Optimization and Approximation: An Introduction to the Theoretical Analysis

and Numerical Treatment of Semi-Infinite Programs.46. Wilcox: Scattering Theory for Diffraction Gratings.47. Hale el al.: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory.48. Murray: Asymptotic Analysis.49. Lad'v:henskava: The Boundary-Value Problems of Mathematical Physics.50. Wilcox: Sound Propagation in Stratified Fluids.51. Goluhitskv/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol 1.52. Chipot: Variational Inequalities and Flow in Porous Media.53. Majda: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables.54. Wasvnv: Linear Turning Point Theory.

(continued following index)

Stephen Wiggins

Global Bifurcationsand ChaosAnalytical Methods

With 200 Illustrations

Springer-VerlagNew York Berlin HeidelbergLondon Paris Tokyo

Stephen Wiggins

Applied Mechanics 104-44California Institute of TechnologyPasadena, CA 91125USA

EditorsF. John J.E. Marsden L. SirovichCourant Institute of Department of Division of

Mathematical Sciences Mathematics Applied MathematicsNew York University University of California Brown UniversityNew York, NY 10012 Berkeley, CA 94720 Providence, RI 02912

USA USA USA

Mathematics Subject Classification (1980): 34xx, 58xx, 70

Library of Congress Cataloging-in-Publication DataWiggins, Stephen.

Global bifurcations and chaos : analytical methods / Stephen Wiggins.p. cm.-(Applied mathematical sciences ; v. 73)

Bibliography: p.Includes index.

ISBN 0-387-96775-3

1. Bifurcation theory. 2. Chaotic behavior in systems.3. Differential equations-Numerical solutions. 1. Title.II. Series: Applied mathematical sciences (Springer-Verlag New YorkInc.) ; v. 73.QA1.A647 vol. 73[QA372]510 s-dcl9[514'.74] 88-19959

© 1988 by Springer-Verlag New York Inc.All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer-Verlag, 175 Fifth Avenue, New York, NY 10010, USA),except for brief excerpts in connection with reviews or scholarly analysis. Use in connection withany form of information storage and retrieval, electronic adaptation, computer software, or by similaror dissimilar methodology now known or hereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc. in this publication, even if theformer are not especially identified, is not to be taken as a sign that such names, as understood bythe Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Camera-ready copy provided by the author.Printed and bound by R.R. Donnelley and Sons, Harrisonburg, Virginia.Printed in the United States of America.

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ISBN 0-387-96775-3 Springer-Verlag New York Berlin HeidelbergISBN 3-540-96775-3 Springer-Verlag Berlin Heidelberg New York

To Meredith

PREFACE

The study of chaotic phenomena in deterministic nonlinear dynamical systems

has attracted much attention over the last fifteen years. For the applied scientist,

this study poses three fundamental questions. First, and most simply, what is meant

by the term "chaos"? Second, by what mechanisms does chaos occur, and third,how can one predict when it will occur in a specific dynamical system? This book

begins the development of a program that will answer these questions.

I have attempted to make the book as self-contained as possible, and thus have

included some introductory material in Chapter One. The reader will find muchnew material in the remaining chapters. In particular, in Chapter Two, the tech-niques of Conley and Moser (Moser [1973]) and Afraimovich, Bykov, and Silnikov

[1983] for proving that an invertible map has a hyperbolic, chaotic invariant Cantor

set are generalized to arbitrary (finite) dimensions and to subshifts of finite type.

Similar techniques are developed for the nonhyperbolic case. These nonhyperbolic

techniques allow one to demonstrate the existence of a chaotic invariant set having

the structure of the Cartesian product of a Cantor set with a surface or a "Cantorset of surfaces". In Chapter Three the nonhyperbolic techniques are applied to the

study of the orbit structure near orbits homoclinic to normally hyperbolic invariant

tori.

In Chapter Four,I develop a class of global perturbation techniques that enable

one to detect orbits homoclinic or heteroclinic to hyperbolic fixed points, hyperbolic

periodic orbits, and normally hyperbolic invariant tori in a large class of systems.

The methods developed in Chapter Four are similar in spirit to a technique origi-nally developed by Melnikov [1963] for periodically forced, two-dimensional systems;

viii Preface

however, they are much more general in that they are applicable to arbitrary (butfinite) dimensional systems and allow for slowly varying parameters and quasiperi-

odic excitation. This general theory will hopefully be of interest to the appliedscientist, since it allows one to give a criterion for chaotic dynamics in terms of the

system parameters. Moreover, the methods apply in arbitrary dimensions, where

much work remains to be done in chaos and nonlinear dynamics.

In this book I do not deal with the question of the existence of strange attrac-

tors. Indeed, this remains a major outstanding problem in the subject. However,this book does provide useful techniques for studying strange attractors, in thatthe first step in proving that a system possesses a chaotic attracting set is to prove

that it possesses chaotic dynamics and then to show that the dynamics are con-tained in an attracting set that has no stable "regular" motions. One cannot denythat chaotic Cantor sets can radically influence the dynamics of a system; however,

the extent and nature of this influence needs to be studied. This will require thedevelopment of new ideas and techniques.

Over the past two years many people have offered much encouragement and

help in this project, and I take great pleasure in thanking them now.

Phil Holmes and Jerry Marsden gave me the initial encouragement to getstarted and criticized several early versions of the manuscript.

Steve Schecter provided extremely detailed criticisms of early versions of themanuscript which prevented many errors.

Steve Shaw read and commented on all of the manuscript.

Pat Sethna listened patiently to my explanations of various parts of the bookand helped me considerably in clarifying my thoughts and presentation style.

John Allen and Roger Samelson called my attention to a crucial error in some

earlier work.

Darryl Holm, Daniel David, and Mike Tratnik listened to several lengthy ex-

planations of material in Chapters Three and Four and pointed out several errors

in the manuscript.

Much of the material in Chapters Three and Four was first tried out in graduate

applied math courses at Caltech. I am grateful to the students in those courses for

enduring many obscure lectures and offering useful suggestions.

During the past two years Donna Gabai and Jan Patterson worked tirelesslyon the layout and typing of this manuscript. They unselfishly gave of their time

Preface ix

(often evenings and weekends) so that various deadlines could be met. Their skill

and help made the completion of this book immensely easier.

I would also like to acknowledge the artists who drew the figures for this book

and pleasantly tolerated my many requests for revisions. The figures for ChapterOne were done by Betty Wood, and those for Chapter Four by Cecilia Lin. Peggy

Firth, Pat Marble, and Bob Turring of the Caltech Graphic Arts Facilities and JoePierro, Haydee Pierro, Melissa Loftis, Gary Hatt, Marcos Prado, Bill Contado, Abe

Won, and Stacy Quinet of Imperial Drafting Inc. drew the figures for ChaptersTwo and Three.

Finally, Meredith Allen gave indispensable advice and editorial assistancethroughout this project.

CONTENTS

Preface vii

CHAPTER 1.

Introduction: Background for Ordinary Differential Equations andDynamical Systems 1

1.1. The Structure of Solutions of Ordinary Differential Equations 1

1.1a. Existence and Uniqueness of Solutions 2

1.1b. Dependence on Initial Conditions and Parameters 3

1.1c. Continuation of Solutions 4

1.1d. Autonomous Systems 8

1.1e. Nonautonomous Systems 9

1.1 f. Phase Flows 11

1.1g. Phase Space 12

1.1h. Maps 14

1.1 i. Special Solutions 15

1.1j. Stability 16

1.1k. Asymptotic Behavior 20

1.2. Conjugacies 22

1.3. Invariant Manifolds 26

1.4. Transversality, Structural Stability, and Genericity 56

1.5. Bifurcations 62

1.6. Poincare Maps 67

CHAPTER 2.

Chaos: Its Descriptions and Conditions for Existence 75

2.1. The Smale Horseshoe 76

2.1a. Definition of the Smale Horseshoe Map 76

2.1b. Construction of the Invariant Set 79

xii Contents

2.1c. Symbolic Dynamics 86

2.1d. The Dynamics on the Invariant Set 90

2.1e. Chaos 93

2.2. Symbolic Dynamics 94

2.2a. The Structure of the Space of Symbol Sequences 96

2.2b. The Shift Map 100

2.2c. The Subshift of Finite Type 101

2.2d. The Case of N = o0 106

2.3. Criteria for Chaos: The Hyperbolic Case 108

2.3a. The Geometry of Chaos 108

2.3b. The Main Theorem 118

2.3c. Sector Bundles 128

2.3d. More Alternate Conditions for Verifying Al and A2 134

2.3e. Hyperbolic Sets 145

2.3f. The Case of an Infinite Number of Horizontal Slabs 149

2.4. Criteria for Chaos: The Nonhyperbolic Case 150

2.4a. The Geometry of Chaos 151

2.4b. The Main Theorem 159

2.4c. Sector Bundles 161

CHAPTER 3.

Homoclinic and Heteroclinic Motions 171

3.1. Examples and Definitions 171

3.2. Orbits Homoclinic to Hyperbolic Fixed Points of Ordinary Differential

Equations 182

3.2a. The Technique of Analysis 183

3.2b. Planar Systems 199

3.2c. Third Order Systems 207

i) Orbits Homoclinic to a Saddle Point with Purely Real Eigenvalues 208

ii) Orbits Homoclinic to a Saddle-Focus 227

3.2.d. Fourth Order Systems 258

i) A Complex Conjugate Pair and Two Real Eigenvalues 261

ii) Silnikov's Example in 1R4 267

Contents xiii

3.2e. Orbits Homoclinic Fixed Points of 4-Dimensional Autonomous

Hamiltonian Systems 275

i) The Saddle-Focus 276

ii) The Saddle with Purely Real Eigenvalues 286

iii) Devaney's Example: Transverse Homoclinic Orbits in an Integrable

Systems 298

3.2f. Higher Dimensional Results 298

3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of Ordinary Differential

Equations 300

i) A Heteroclinic Cycle in 1R3 301

ii) A Heteroclinic Cycle in 1R4 306

3.4. Orbits Homoclinic to Periodic Orbits and Invariant Tori 313

CHAPTER 4.

Global Perturbation Methods for Detecting Chaotic Dynamics 334

4.1. The Three Basic Systems and Their Geometrical Structure 335

4.1a. System I 339

i) The Geometric Structure of the Unperturbed Phase Space 340

ii) Homoclinic Coordinates 350

iii) The Geometric Structure of the Perturbed Phase Space 352

iv) The Splitting of the Manifolds 359

4.1b. System II 370





4.1c. System III 380





v) Horseshoes and Arnold Diffusion 394

4.1d. Derivation of the Melnikov Vector 396

i) The Time Dependent Melnikov Vector 402

xiv Contents

ii) An Ordinary Differential Equation for the Melnikov Vector 404

iii) Solution of the Ordinary Differential Equation 406

iv) The Choice of SpE and Sp E 414

v) Elimination of to 416

4.1e. Reduction to a Poincare Map 417

4.2. Examples 418

4.2a. Periodically Forced Single Degree of Freedom Systems 418

i) The Pendulum: Parametrically Forced at O (E) Amplitude,

0(1) Frequency 419

ii) The Pendulum: Parametrically Forced at 0(l) Amplitude,0 (c) Frequency 426

4.2.b. Slowly Varying Oscillators 429

i) The Duffing Oscillator with Weak Feedback Control 430

ii) The Whirling Pendulum 440

4.2c. Perturbations of Completely Integrable, Two Degree of Freedom

Hamiltonian System 452

i) A Coupled Pendulum and Harmonic Oscillator 452

ii) A Strongly Coupled Two Degree of Freedom System 455

4.2d. Perturbation of a Completely Integrable Three Degree of FreedomSystem: Arnold Diffusion 458

4.2e. Quasiperiodically Forced Single Degree of Freedom Systems 460

i) The Duffing Oscillator: Forced at 0 (e) Amplitude with N 0 (1)

Frequencies 461

ii) The Pendulum: Parametrically Forced at 0(e) Amplitude, 0(1)Frequency and 0 (1) Amplitude, 0 (e) Frequency 468

4.3. Final Remarks 470

i) Heteroclinic Orbits 470

ii) Additional Applications of Melnikov's Method 470

iii) Exponentially Small Melnikov Functions 471

References 477

Index 489

CHAPTER1Introduction: Background for OrdinaryDifferential Equations and Dynamical Systems

The purpose of this first chapter is to review and develop the necessary conceptsfrom the theory of ordinary differential equations and dynamical systems which we

will need for the remainder of the book. We will begin with some results fromclassical ordinary differential equations theory such as existence and uniqueness of

solutions, dependence of solutions on initial conditions and parameters, and various

concepts of stability. We will then discuss more modern ideas such as generic-ity, structural stability, bifurcations, and Poincare maps. Standard references forthe theory of ordinary differential equations are Coddington and Levinson [1955],

Hale [1980], and Hartman [1964]. We will take a more global, geometric point of

view of the theory; some references which share this viewpoint are Arnold [1973],

Guckenheimer and Holmes [1983], Hirsch and Smale [1974], and Palis and deMelo

[1982].

1.1. The Structure of Solutions of Ordinary DifferentialEquations

In this book we will regard an ordinary differential equation as a system of equations

having the following form

x = f (x, t) , (x, t) E R" X R1 (1.1.1)

where f : U -+ lR' with U an open set in R° X R1 and x - dx/dt. The spaceof dependent variables is often referred to as the phase or state space of the system

(1.1.1). By a solution of (1.1.1) we will mean a map

0:I-->]R.n (1.1.2)

2 1. Introduction: Background for O.D.E.s and Dynamical Systems

where I is some interval in IR such that

q(t) = f (0(t),t) . (1.1.3)

Thus, geometrically (1.1.1) can be viewed as defining a vector at every point in U,

and a solution of (1.1.1) is a curve in IRn whose tangent or velocity vector at each

point is given by f (x, t) evaluated at the specific point. For this reason (1.1.1) is

often referred to as a vector field.

Now, the existence of solutions of (1.1.1) is certainly not obvious and evidently

must rely in some way on the properties of f, so now we want to give some classical

results concerning existence of solutions of (1.1.1) and their properties.

1.1a. Existence and Uniqueness of Solutions

Suppose that f is Cr in U (note: by Cr, r > 1, we mean that f has r derivativeswhich are continuous at each point of U; Co means that f is continuous at eachpoint of U) and for some fl, E2 >0 let I1 = { t E Y2. I to-E, <t < t0 + E1 } andI2 = { t E 1R I to - E2 < t < to + E2 }; then we have the following theorem.

Theorem 1.1.1. Let (x0, to) be a point in U. Then for El sufficiently small thereexists a solution of (1.1.1), 01 : 11 -. Rn, satisfying qS1(t0) = x0. Moreover, iff is Cr in U, r > 1, and 02 : 12 -4 1Rn is also a solution of (1.1.1) satisfying

02 (t0) = x0, then q51(t) _ 02 (t) for all t E I 3 = { t E R I t o - E , 3

to + E3 } where c3 = min{E1, E2}.

PROOF: See Arnold [1973] or Hale [19801. El

We make the following remarks concerning Theorem 1.1.1.

1) For a solution of (1.1.1) to exist, only continuity of f is required; however, in

this case the solution passing through a given point in U may not be unique(see Hale [1980] for an example). If f is at least C1 in U, then there is onlyone solution passing through any given point of U (note: for uniqueness ofsolutions one actually only needs f to be Lipschitz in the x variable uniformly

in t, see Hale [1980] for the proof). The degree of differentiability of the vector

field will not be a major concern to us in this book since all of the exampleswe consider will be infinitely differentiable.


2) The differentiability of solutions with respect to t was not explicitly considered

in the theorem, although evidently they must be at least Cr since f is Cr. This

result will be stated shortly.

3) Notation: In denoting the solutions of (1.1.1) it may be useful to note thedependence on initial conditions explicitly. For 0, a solution of (1.1.1) passing

through the point x = x0 at t = to, the notation would be

0(t, t0, x0) with 0(t0, t0, x0) = x0 . (1.1.4)

In some cases, the initial time is always understood to be a specific value (often

to = 0); in this case, the explicit dependence on the initial time is omitted and

the solution is written as

0(t, x0) with k(to, x0) = x0 . (1.1.5)

1.1b. Dependence on Initial Conditions and Parameters

In the computation of stability properties of solutions and in the construction ofPoincare maps (see Section 1.6) the differentiability of solutions with respect toinitial conditions is very important.

Theorem 1.1.2. If f (x, t) is Cr in U, then the solution of 1.1.1, ¢(t, to, xo)

(x0, to) E U, is a Cr function of t, t0 and xo.

PROOF: See Arnold [1973] or Hale [19801.

Theorem 1.1.2 justifies the procedure of computing the Taylor series expansion

of a solution of (1.1.1) about a given initial condition. This enables one to determine

the nature of solutions near a particular solution. Often the linear term in such an

expansion is sufficient for determining many of the local properties near a particular

solution (e.g., stability). The following theorem gives an equation which the first

derivative of the solution with respect to xo must obey.

Theorem 1.1.3. Suppose f (x, t) is Cr, r > 1, in U and let 4'(t, t0, x0), (x0, to) EU, be a solution of (1.1.1). Then the n x n matrix Dxo4' is the solution of thefollowing linear ordinary differential equation

2 = Dx f (4'(t), t) Z, Z(t0) = id, (1.1.6)


where Z is an n x n matrix and id denotes the n x n identity matrix.

PROOF: See Arnold [1973], Hale [1980], or Irwin [1980].

Equation (1.1.6) is often referred to as the first variational equation. We remark

that it is possible to find linear ordinary differential equations which the higher order

derivatives of solutions with respect to the initial conditions must obey; however,

we will not need these in this book.

Now suppose that equation (1.1.1) depends on parameters

x = f (x, t; E) , (x, t, E) E R' x R1 x RP (1.1.7)

where f : U -+ Rfz with U an open set in R'Z x Rl x RP. We have the followingtheorem.

Theorem 1.1.4. Suppose f (x, t; E) is Cr in U. Then the solution of (1.1.7),q(t, to, xo, E) (xo, to, e) E U, is a Cr function of E.

PROOF: See Arnold [1973] or Hale [1980].

In many applications it is useful to seek Taylor series expansions in E of solu-

tions of (1.1.7) (e.g., in perturbation theory and bifurcation theory). Analogous toTheorem 1.1.3, the following theorem gives an ordinary differential equation which

the first derivative of a solution of (1.1.7) with respect to c must obey.

Theorem 1.1.5. Suppose f (x, t, c) is Cr, r > 1, in U and let ¢(t, to, xo, E),

(xo, to, E) E U, be a solution of (1.1.7). Then the n x p matrix DE¢ satisfies thefollowing linear ordinary differential equation

Z = Dxf (q(t), t; E) Z + DE f (4(t), t; E) , z(to) = 0, (1.1.8)

where Z is a n x p matrix and 0 represents the n x p matrix of zeros.

PROOF: See Hale [1980].

1.1c. Continuation of Solutions

Theorem 1.1.1 gave sufficient conditions for the existence of solutions of (1.1.1) but

only on a sufficiently small time interval. We will now give a theorem which justifies

the extension of this time interval, but first we need the following definition.

1.1. The Structure of Solutions of Ordir ary Differential Equations 5

Definition I.I.I. Let 01 be a solution of (1.1.1) defined on the interval Il, andlet 02 be a solution of (1.1.1) defined on the interval 12. We say that 02 is acontinuation of ¢1 if Il C 12 and ¢1 = ¢2 on Il. A solution is noncontinuableif no such continuation exists; in this case, Il is called the maximal interval ofexistence of 01.

We now state the following theorem concerning continuation of solutions.

Theorem 1.1.6. Suppose f (x, t) is Cr in U and 0(t, to, x0), (x0, to) E U, is

a solution of (1.1.1), then there is a continuation of 0 to a maximal interval ofexistence. Furthermore, if (tl,t2) is a maximal interval of existence for 0, then

(q(t),t) tends to the boundary of U as t - tl and t -> t2.

PROOF: See Hale [1980].

Terminology

At this point we want to introduce some common terminology that applies to so-lutions of ordinary differential equations. Recall that a solution of (1.1.1) is a map0 : I - R' where I is some interval in R. Geometrically, the image of I under 0 isa curve in JR', and this geometrical picture gives rise to the following terminology.

1) A solution ¢(t, to, xo) of (1.1.1) may also be called the trajectory, phase curve

or motion through the point x0.2) The graph of the solution 0(t, to, x0), i.e.,

{ (x, t) E 1Rn X 1R1 [ x = ¢(t, to, xo), t E I }

is called an integral curve.3) Suppose we have a solution q5(t, to, x0); then the set of points in R' through

which. this solution passes as t varies through I is called the orbit through

denoted O(x0) and written as follows.

O(x0) = {x E R' I x = cb(t, to, xo), t E I) .

We remark that it follows from this definition that, for any T E I,

0 (0(T,to,xo)) = 0(xo)

The following example should serve to illustrate the terminology.

x0,


EXAMPLE 1.1.1. Consider the following equation

i+x=0. (1.1.9)

This is just the equation for a simple harmonic oscillator having frequency one.Writing (1.1.9) as a system we obtain

(1.1.10)

Equation (1.1.10) has the form of equation (1.1.1) with phase space R2. Thesolution of (1.1.10) passing through the point (1,0) at t = 0 is given by O (t) _(cost, - sin t).

1) The trajectory, phase curve or motion through the point (1,0) is illustrated inFigure 1.1.1.

r

Figure 1.1.1. Trajectory through the Point (1,0).

2) The integral curve of the solution fi(t) = (cost, -sin t) is illustrated in Figure

1.1.2.r

3) The orbit through the point (1,0) is given by { (x, y) E R2 I x2 + y2 = 1 } and

is illustrated in Figure 1.1.3. l

We remark that, although the solution through (1,0) passes through the same set of

points in R2 as the orbit through (1,0), and thus both appear to be the same object

when viewed as a locus of points in R2, we stress that they are indeed differentobjects. A solution must pass through a specific point at a specified time and an

1.1. The Structure of Solutions of Ordir_arv Differential Equations 7

Figure 1.1.2. Integral Curve of 4(t) = (cost, -sin t).

Figure 1.1.3. Orbit through (1,0).

orbit can be thought of as a one parameter family of solutions corresponding toa curve of possible initial conditions for different solutions at a specific time. Inthe qualitative theory of ordinary differential equations it is not unusual to use the

terms orbit and solution interchangeably and, usually, no harm comes from this.

There is a difference in the nature of solutions depending upon whether ornot the vector field depends explicitly on the independent variable (note: we willhenceforth always refer to the independent variable as time). If the vector field is


independent of time it is called autonomous, and if it depends explicitly on time itis called nonautonomous.

1.1d. Autonomous Systems

An autonomous system of ordinary differential equations has the following form

i= f(x), xERn (1.1.11)

where f : U --+ IR.1z with U an open set in Rn. We assume that f is C", r > 1,and let q5(t) be a solution of (1.1.11).

Lemma 1.1.7. If q(t) is a solution of (1.1.11), then so is 0(t + r) for any realnumber T.

PROOF: If j(t) is a solution of (1.1.11), then by definition

dq(t)= f (#(t)) . (1.1.12)

dt

So we have

dO(t+r)

or

dt

dO(t)

t=to dt t=to+r

dO(t+r)dt

= f (cb(to + r)) = f Wt + r))t=to

(1.1.13)

= f (qS(t + r))t=to t=to

(1.1.14)

Now (1.1.14) is true for any to, r E IR, so 4i(t + r) is also a solution of (1.1.11). 0

We remark that Lemma 1.1.7 provides us with an important fact which willprove useful in Chapter 4. Namely, if we have a solution 0(t) of an autonomousequation, then we immediately have a parametric representation of the orbit of this

solution of the form q(t + r), where r E I is regarded as the parameter. Thus,we can view t as fixed, and varying r carries us through the orbit of 4(t).

Two important properties of solutions of autonomous equations are given inthe following lemmas.

Lemma 1.1.8. Suppose that f is Cr in U, r > 1, and 01 (t), 42(t) are solutionsof (1.1.11) defined on Il and 12, respectively, with q51(tl) = #2(t2) = p. Then01(t - (t2 - t1)) = 02(t) on their common interval of definition.

PROOF: Let ry(t) = 01 (t - (t2 - t1)); then by Lemma 1.1.7 -y(t) is also a solutionof (1.1.11). Now notice that 'y(t2) = ¢1(tl) = p = 02(t2). Thus, -y(t) and 02(t)

1.1. The Structure of Solutions of Ordirary Di£erertial Equations 9

are solutions of (1.1.11) which satisfy the same initial condition (taking the initial

time as t = t2); thus, by uniqueness of solutions (Theorem 1.1.1), 'y(t) (and hence

¢1(t - (t2 - tl)) ) and 02(t) must coincide on their common interval of definition.

0

Lemma 1.1.9. Suppose that f is Cr in U, r > 1, and q5(t) is a solution of(1.1.11) defined on I. Suppose there exist two points tl, t2 E I, t1 < t2, suchthat 0(tl) = 0(t2). Then 0(t) exists for all t E R and is periodic in t with periodT = t2 - t l , i.e., q5(t) _ ¢(t +T) for all t E R.

PROOF: Let V)(t) = ql(t+tl), by Lemma 1.1.7 fi(t) is a solution of (1.1.11). Thenwe have

0(t+T) = 0(t1 + t + T) = 0(t+t2) . (1.1.15)

Now, since ql(tl) = 02), we must have 4(t + t1) = q(t + t2) by uniqueness ofsolutions (Theorem 1.1.1). Therefore,

?'(t + T) = q(t + t2) = j(t + t1) = z/i(t) . (1.1.16)

Therefore, ?P(t) is periodic in time with period T and j(t) is likewise periodic intime with period T, and since every t E R can be written in the form t = nT + r,0 < r < T, 0(t) exists for all time.

Lemmas 1.1.8 and 1.1.9 tell us that solutions (and hence all orbits) of au-tonomous equations cannot intersect themselves or each other in isolated pointswithout coinciding on their common intervals of definition. These facts can be ex-

tremely useful in determining certain global properties of the orbit structure of an

ordinary differential equation (e.g., this fact is largely responsible for the Poincare-

Bendixson theorem, see Hale [1980] or Palis and deMelo [1982]).

1.1e. Nonautonomous Systems

A nonautonomous system of ordinary differential equations has the following form

i = f (x, t) , (x, t) E R76 X Rl (1.1.17)

where f : U -4R' with U an open set in Rrt x R1. Lemma 1.1.7 does not followfor nonautonomous systems. Consider the following example.


EXAMPLE 1.1.2. Consider the following nonautonomous ordinary differential equa-

tion

i=et. (1.1.18)

The solution of equation (1.1.18) is obviously 4(t) = et, and it is clear thatq(t + r) = et+T is not a solution of (1.1.18) for r 0.

Example 1.1.2 shows that time translations of solutions of nonautonomousequations are not likewise solutions of the equations. This was the crucial property

which led to the proofs of Lemmas 1.1.8 and 1.1.9 so we conclude that it is possible

for solutions of nonautonomous ordinary differential equations to intersect them-selves and each other. This can lead to a very complicated geometrical structure of

the solutions of nonautonomous ordinary differential equations.

Often the geometry of the solution structure of nonautonomous ordinary dif-ferential equations is clarified by enlarging the phase space by redefining time as a

new dependent variable. This is done as follows: by writing (1.1.17) as

dx _ f (x, t)(1.1.19)

dt 1

and using the chain rule, we can introduce a new independent variable, s, so that

(1.1.19) becomesdx

x, = f (x, t)dsdt t =1.ds

(1.1.20)

If we define y = (x, t) and g(y) = (f (y),1), we see that (1.1.20) has the form ofan autonomous ordinary differential equation with phase space Rn X JR1.

y'=g(y), yE]Rnx]R.l. (1.1.21)

Of course, knowledge of the solutions of (1.1.21) implies knowledge of the solutions of

(1.1.17) and vice versa. For example, if 0(t) is a solution of (1.1.17) passing through

x = x0 at t = t0, i.e., 0(to) = x0, then 0(s) = (q5(s + t0), t(s) = s + to) is a

solution of (1.1..* passing through y = yo - (x0,to) at s = 0. This apparentlytrivial trick is a great aid in the construction of Poincare maps, as we shall seein Section 1.6, and it also justifies the consideration of autonomous systems exclu-

sively. Henceforth we will state concepts only for autonomous ordinary differential

equations. For the most part we will consider autonomous ordinary differential

1.1. The Structure of Solutions of Ordir. ry Differential Equations 11

equations in this book; the nonautonomous equations which we consider will have

either periodic or quasiperiodic time dependence, and in each case we will reduce the

study of such systems to the study of an associated Poincare map (see Section 1.6).

1.1f. Phase Flows

Consider the following autonomous ordinary differential equation

x= f(x), xE]EV (1.1.22)

where f is Cr, r > 1, on some open set U C IRn. Let 0(t, to, x0) be a solutionof (1.1.22) defined on the interval I. We will henceforth take to = 0 and drop theexplicit dependence on to from the solution of (1.1.22); i.e., we have ¢(t, xo) with

0((),-o) = xo.

Lemma 1.1.10. i) 0(t, x0) is C'.

ii) 0(0, x0) = x0 .

iii) 0(t + s, x0) = q5 (t, 4(s, xo)) , t + s E I.

PROOF: i) follows from Theorem 1.1.2 and ii) is by definition. The proof of iii)

goes as follows: let y(t) = c(t + s, x0), then y(t) solves (1.1.22) with y(0) =c(s, xo). Also, we have 0(t, 0(s, x0)) satisfies (1.1.22) with ¢(0, ¢(s, xo)) =c(s, xo). So y(t) - 0(t + s, xo) and 0(t, 0(s, xo)) are both solutions of (1.1.22)

satisfying the same initial condition at t = 0; hence by Lemma 1.1.8 q5(t+s,xo) =q5(t, O(s, xo)) on their common interval of definition.

Since O(t, x0) is Cr, viewing t as fixed we see that 0(t, x0) - q5t(xo) defines a

Cr map of U into ]Rfz. Thus Ot(xo) is a one parameter family of maps of U -- R.By property iii) of Lemma 1.1.10 we see that this Cr one parameter family of maps

is invertible with Cr inverse. A Cr invertible map having a Cr inverse is called aCr diffeomorphism (if r = 0, the term homeomorphism is used). So we see that the

solutions of an autonomous ordinary differential equation generate a one parameter

family of diffeomorphisms of the phase space onto itself. This one parameter family

of diffeomorphisms is called a phase flow.


1.1g. Phase Space

As mentioned earlier, the phase space of a system is the space of dependent variables

which was taken to be some open set in 1R". However, in some applications the

space of dependent variables naturally arises as a surface such as a cylinder or torus

or, more generally, a differentiable manifold (see Section 1.3 for the definition of a

differentiable manifold). We consider several common examples.

Circle: Consider the ordinary differential equation

9=w, 9E(0,27r]

where w > 0 is constant. The phase space of this equation is the interval(0,27r] with 0 and 27r identified. Thus, the phase space has the structure of acircle having length 2a which we denote as S1 (note: the superscript one refers tothe dimension of the phase space).

Cylinder: Mathematically the cylinder is denoted by R1 x S1. Consider the

following equation which describes the dynamics of a free undamped pendulum

e=v

v = -sin0.(1.1.23)

The angular velocity, v, can take on any value in ]R. but, since the motion is rota-

tional, the position, 0, is periodic with period 27r. Hence, the phase space of thependulum is the cylinder ]R1 x S1. Figure 1.1.4a shows the orbits of the pendulum

on ]R1 x S1 and Figure 1.1.4b gives an alternate representation of the cylinder.

Torus: Heuristically, we think of a torus as the surface of a donut; if we considerthe surface plus its interior, we speak of the solid torus. Mathematically, the twodimensional torus is denoted by T2 = S1 x S1, i.e., the Cartesian product of twocircles.

Consider the following ordinary differential equation

01, 02 E (0,27r] (1.1.24)

where wl and w2 are positive constants. Since 0 and B2 are angular variables, thephase space of (1.1.24) is S1 x S1 = T2. If we draw the torus as the surface of


a) b)

Figure 1.1.4. Phase Space of the Pendulum on a) R1 x S1. b) Rl X R1.

a donut in R3 the orbits of (1.1.24) spiral around the surface and close (i.e., they

are periodic) when wl/w2 is a rational number; alternatively, they densely fill the

surface when wl/w2 is an irrational number, see Arnold [1973[ for a detailed proof

of these statements.

Another useful way of representing the two torus is to first cut the torus along

01 = 0 (resulting in a tube), next cut along 02 = 0 (resulting in a sheet), andfinally flatten out the sheet into a square. Thus, we can make a torus from a square

by identifying the two vertical sides of the square and the two horizontal sides of the

square. Mathematically, this representation of the torus is written R2/7Z2 (read"r two mod z two") which means that, given any two points x, y E R2, we consider

x and y to be the same point if x = y + 27rn where n is some integer two vector.

These ideas and notations also go through in n dimensions. The n torus iswritten as Tn = S1 x x S' (note: T1 - S1) or, equivalently, Tn can be

n factorsthought of as R'/2Z', i.e., as a n dimensional cube with opposite sides identified.

Sphere: The n sphere of radius R is denoted by Sn and is defined as

Sn={xERn+11 IxI=R} . (1.1.25)


See the introduction to Chapter 3 for an example of an ordinary differential equation

whose orbits lie on a sphere.

We remark that the systems we consider in this book will have as phase spaces

1R.n, T', Sn, or some Cartesian product of 1R.n, Tn, or Sf2.

1.Ih. Maps

In this book we will mostly be concerned with the orbit structure of ordinary dif-

ferential equations. However, in certain situations much insight can be gained by

constructing a discrete time system or map from the solutions of an ordinary dif-ferential equation. In Section 1.6 we will consider this procedure in great detail;

however, at this point, we merely wish to define the term map and discuss somevarious properties of maps and their dynamics.

A Cr map of some open set U C ]R.n into 1R' is denoted as follows

f:U-')Rnx- f(x), xEU

(1.1.26)

with f Cr in U. We will be interested in the dynamics of f. By this we meanthe nature of the iterates of points in U under f. For a point x E U, equivalentnotations for the nth iterate of x under f are

f (f (... (f (x))) ...) = f of o ... of (X) = fn(x) . (1.1.27)

n times n times

By the orbit of x under f we will mean the following bi-infinite sequence if f isinvertible

{..., f-n(x),...,f-1(x),x,f(x),...,fn(x).... } , (1.1.28)

and the following infinite sequence if f is noninvertible.

{x, f (x), ... , fn(x) ....} . (1.1.29)

This brings up an important difference between orbits of ordinary differential equa-

tions and orbits of maps. Namely, orbits of ordinary differential equations are curves

and orbits of maps are discrete sets of points. In Chapter 3 we will see that thisdifference is significant.


1.1i. Special Solutions

At this time we want to consider various special solutions and orbits which are often

important in applications.

1) Fixed point, equilibrium point, stationary point, rest point, singular point, orcritical point. These are all synonyms for a point p in the phase space of anordinary differential equation that is also a solution of the equation, i.e., for

the equation i = f (x) we have

0 = f (p) (1.1.30)

or a point in the phase space of a map x F-+ f (x) such that

p = f (p) . (1.1.31)

For a map, p may also be called a period one point. In this book we willexclusively use the term fixed point when referring to such solutions.

2) Periodic Motions. A periodic solution, O(t), of an ordinary differential equation

is a solution which is periodic in time, i.e., ¢(t) = 0(t + T) for some fixed

positive constant T. T is called the period of 4(t). A periodic orbit of anordinary differential equation is the orbit of any point through which a periodic

solution passes.

For maps, a period k point, p, is a point such that f k(p) = p. The orbit of aperiod k point is a sequence of k distinct points

{p, f(p), .. , fk-1(p) }

and the orbit is called a periodic orbit of period k.

3) Quasiperiodic Motions

Definition 1.1.2. A function

h: Rl_>Rm

t ,-- h(t)

is called quasiperiodic if it can be represented in the form

(1.1.32)

h(t) = H(wlt,...,wnt) (1.1.33)


where H(x1,... , xn) is a function of period 27r in x1, ... , xn. The realnumbers w1, ... , wn are called the basic frequencies. We shall denote byCr (w1i ... , wn) the class of h(t) for which H(x1,... , xn) is r times continu-ously differentiable.

EXAMPLE 1.1.3. h(t) = 11 cos w1t + 12 sinw2t is a quasiperiodic function.

(Note: There exists a more general class of functions called almost periodic

functions which can be viewed as quasiperiodic functions having an infinitenumber of basic frequencies. These will not be considered in this book, see

Hale [1980] for a discussion and rigorous definitions.)

A quasiperiodic solution 0(t), of an ordinary differential equation is a solution

which is quasiperiodic in time. A quasiperiodic orbit is the orbit of any point

through which (k(t) passes. A quasiperiodic orbit may be interpreted geometri-cally as lying on an n dimensional torus. This can be seen as follows. Consider

the equation

y = H(xl,...,xn) .

Then, if m > n and DzH has rank n for all x = (x1, ... , x,), then

this equation can be viewed as an embedding of an n-torus in m space withX1,. .. , xn serving as coordinates on the torus. Now, viewing h(t) as a solution

of an ordinary differential equation, since xi = wit, i = 1,.. . , n, h(t) can beviewed as tracing a curve on the n-torus as t varies.

In recent years quasiperiodic orbits of maps have received much attention,mainly in the context of maps of the circle and annulus. These will not bestudied in this book but see Katok [1983] for an overview and recent references.

4) Homoclinic and Heteroclinic Motions. These will be defined and studied ingreat detail in Chapter 3.

1.1i. Stability

The general theory of stability is a very large subject to which many books have

been devoted. However, in this section we will only consider those aspects of the

theory which have particular relevance to the subjects covered in this book, namely,

the stability of specific solutions of ordinary differential equations and its determi-

nation and the stability of periodic orbits of maps and its determination. We refer

the reader to Rouche, Habets, and Laloy [1977], Yoshizawa [1966], LaSalle [1976],


Abraham and Marsden [19781, and references therein for a more complete discussion

of stability.

Consider the ordinary differential equation

i= f(x), xEIRt (1.1.34)

where f : U --* ]R" with U an open set in 1EV and f is C', r > 1. Let q(t)be asolution of (1.1.34).

Definition 1.1.3. t(t) is said to be Liapunov stable, or stable if given f > 0 wecan find a b = 6(E) > 0 such that for any other solution iO(t) of (1.1.34) withkb(to) - 0(to)I < b then we have lTP(t) - fi(t) l < E for t E [to, oo).

If 0(t) is not stable then it is said to be unstable.

Definition 1.1.4. 0(t) is said to be asymptotically stable if it is Liapunov stableand there exists b > 0 such that if 10(to) - i/'(to) j < 6, then lim 10(t) - 0(t) = 0.

t-9oo

We remark that, for autonomous systems, S and b are independent of to, seeHale [19801.

Heuristically, these definitions say that solutions starting near a Liapunov sta-

ble solution remain nearby thereafter, and solutions starting near an asymptotically

stable solution approach the solution as t -p oo, see Figure 1.1.5.

t = to

a)

t = to

b)

Figure 1.1.5. a) Liapunov Stability. b) Asymptotic Stability.


Now that we have defined stability of solutions we need to address its determi-

nation for specific problems. One method for determining the stability of a specific

solution is the direct method of Liapunov, and for this we refer the reader to thereferences given at the beginning of this section. Another method for determining

stability is linearization, which we will discuss in some detail.

Let us make the coordinate transformation x = y + q5(t) for (1.1.34) andTaylor expand f (y + q'(t)) about y = 0. Then we get the equation

y = Df (c5(t)) y + 0 ([y[2) (1.1.35)

Now the y = 0 solution of (1.1.35) corresponds to the x = 0(t) solution of(1.1.34). So, if the y = 0 solution of (1.1.35) is stable, then the x = 4(t) solutionof (1.1.34) will likewise be stable. Now, (1.1.35) is no less difficult to solve than(1.1.34), so for y small we assume that the 0 (1y12) terms can be neglected, andwe arrive at the linear equation

y = Df (qS(t)) y (1.1.36)

Now we would like to do two things: 1) determine the stability of the y = 0 so-lution of (1.1.36), and 2) conclude that stability (or instability) of the y = 0 so-lution of (1.1.36) corresponds to stability (or instability) for the x = qS(t) solution

of (1.1.34). In general, the determination of the stability of the y = 0 solution of(1.1.36) is a formidable problem (e.g., see the discussion of Hill's equation in Hale

[1980]) since, although the equation is linear, the coefficients are time dependent

and there are no general methods for solving such equations. If 4(t) has a partic-ularly simple dependence on time, then some results are available. For example, if

0(t) is constant in time, i.e., a fixed point, the Df (4(t)) is a constant matrix and

the solution of (1.1.36) may immediately be written down and, if qS(t) is periodic in

time, then Floquet Theory will apply (Hale [1980]). We will only be interested in

the case x = q(t) = constant for which we state the following result.

Theorem 1.1.11. Suppose x = qf(t) = x0 = constant is a solution of (1.1.34),and D f (xo) has no eigenvalues with zero real part. Then asymptotic stability (orinstability) of the y = 0 solution of (1.1.36) corresponds to asymptotic stability(or instability) of the x = x0 solution of (1.1.34).

PROOF: This follows from the Hartman-Grobman theorem, see Hartman [1964].

11


Next we want to state some similar results for maps. Let

xr-> f(x), xE1R' (1.1.37)

be a Cr map, r > 1 with f defined on some open set U C R'. Given an orbit off, we leave as as exercise for the reader the task of writing down discrete versions

of Definitions 1.1.3 and 1.1.4. Here we will concentrate on the stability of periodic

orbits of (1.1.37).

Let p be a period k point of f, i.e., the orbit of p under f is given by

O(p) = {p, P1 - f(p), P2 = f2(p), ..., pk - fk(p) = p} . (1.1.38)

We ask whether or not 0(p) is stable. Notice that p1,...,pk are each fixed points forfk(x) and that, by the chain rule, Dfk(x)=Df(fk-1(x)) Df(fk-2(x)) ...Df(x).Therefore, stability of 0(p) is reduced to the question of the stability of a fixedpoint p j for any j = 1, . . . , k of f k (x). The question of stability for fixed points of

maps has an answer analogous to that given for fixed points of ordinary differential

equations described in Theorem 1.1.11. Consider the map

xH fk(x), x E ]R" (1.1.39)

which has fixed points of p j, j = 1,. .. , k. Following an argument similar to thatgiven for ordinary differential equations consider the associated linear map

yf->Dfk(pj)y, yE1R', foranyj=l,...,k (1.1.40)

which has a fixed point at y = 0. We have the following result.

Theorem 1.1.12. Suppose p is a period k point for (1.1.37) and Df k(p) has no

eigenvalues of modulus one. Then asymptotic stability (or instability) of the fixed

point y = 0 of (1.1.40) corresponds to asymptotic stability (or instability) of 0(p).

PROOF: This is a consequence of the discrete version of the Hartman-GrobmanTheorem, see Hartman [1964].

We remark that, in general, any theorem pertaining to fixed points of mapshas an analogous statement for periodic orbits of maps which can be obtained byreplacing the map by its kth iterate where k is the period of the orbit. For more


information on stability of maps see Bernoussou [1977] or Guckenheimer and Holmes

[1983].

1.1k. Asymptotic Behavior

In this section we want to develop some concepts necessary for describing the asymp-

totic or observable behavior of dynamical systems. We will do this simultaneously

for ordinary differential equations and maps.

Consider the following ordinary differential equation

i= f(x), xEU (1.1.41)

and map

xHg(x), xEU (1.1.42)

where in each case f : U -4 ]Rn and g : U -> lR'L are Cr diffeomorphisms, r > 1,on some open set U C 1R'. We assume that (1.1.41) generates a flow for all time,and we denote this flow by Ot(-).

Definition 1.1.5. A set S C U is said to be invariant under ct(-) (resp. g) if

Ot(S)CS ( resp. gn(S)CS) for all tElR (resp. nE7L).

S is called an invariant set. If the above statement is true for all t E 1R+ (resp.n E ZZ+), then S is called a positive invariant set, and if true for all t E R.- (resp.n E M-), then S is called a negative invariant set.

Recurrent behavior is contained in the nonwandering set of a flow or map.

Definition 1.1.6. A point p E U is said to be a nonwandering point for ¢t(- )( resp. g(-) ) if for any neighborhood V of p there exists some nonzero T E R(resp. N E 7L) such that QST(V) n V # 0 (resp. gN(V) n V 0). The collectionof all nonwandering points for Ot( -) ( resp. g(-)) is called the nonwandering set

for qt(.) ( resp. g(-) )

EXAMPLE 1.1.4. Fixed points as well as all the points on periodic orbits arenonwandering points for both flows and maps.

EXAMPLE 1.1.5. Consider the equation

B1 = W1 (01,02)ES1xS1-T2. (1.1.43)

B2 = W2

1.2. Conjugacies 21

The flow generated by this equation is

ON = (01(t), 92 (t)) = Pit - 910, w2t + 020) . (1.1.44)

It is easy to see that if wl/w2 is a rational number all points on T2 lie on periodic

orbits, and if wl/w2 is an irrational number then all points lie on orbits that never

close but densely cover the surface of T2. Hence, in both cases, all points of T2 are

nonwandering points.

Attracting sets are thought of as the "observable" states of dynamical systems.

Definition 1.1.7. A closed invariant set A C U is called an attracting set ifthere exists some neighborhood V of A such that for all x E V 0t(x) E V (resp.gn(x) E V) for all t > 0 (resp. n > 0) and 0t(x) - A (resp. gn(x) -> A) ast - oo (resp. n -+ oo).

Definition 1.1.8. The basin or domain of attraction of A, denoted DA, is definedas follows.

DA= U Ot(V)t<0 n<0

EXAMPLE 1.1.6. Consider the following equation

( resp. U gn(V)) .

x=y(x)y) E R1 x R1 , (1.1.45)

y=x-x3-6ywith S > 0. The phase space of (1.1.45) is shown in Figure 1.1.6.

Equation 1.1.45 has three fixed points, an unstable (saddle) fixed point atthe origin and two stable (sinks) fixed points at (±1,0). The stable fixed pointsare attractors, and the domains of attraction of the two sinks are as indicated inFigure 1.1.6. Notice the two pairs of curves which issue from the saddle point atthe origin. One pair of curves consists of points which recede from the origin inpositive time and the other pair of curves consists of points which approach theorigin in positive time; these curves are called the unstable and stable manifolds of

the origin, respectively (see Section 1.3 for a discussion of invariant manifolds), and

are examples of invariant sets. Note that the unstable manifold of the origin serves

to separate the domains of attraction of the two sinks.


Y

Figure 1.1.6. Phase Space of (1.1.45).

1.2. Conjugacies

The importance of coordinate transformations in the study of dynamical systems

cannot be overestimated. For example, in the study of systems of linear constantcoefficient ordinary differential equations, coordinate transformations allow one to

decouple the system and hence reduce the system to a set of decoupled linear first

order equations which are easily solved. In the study of completely integrable Hamil-

tonian systems, the transformation to action-angle coordinates results in a triviallysolvable system (see Arnold [19781), and these coordinates are also useful in the

study of near integrable systems. If we consider general properties of dynamical

systems, coordinate transformations provide us with a way of classifying dynamical

systems according to properties which remain unchanged after a coordinate trans-

formation. In Section 1.4 we will see that the notion of structural stability is based

on such a classification scheme. In this section we want to discuss coordinate trans-

formations or, to use the more general mathematical term, conjugacies in general,giving some results which describe properties which must be retained by a map or

vector field after a coordinate transformation of a specific differentiability class. We

will discuss conjugacies for both maps and vector fields separately, beginning with

maps.

1.2. Conjugacies 23

Let us consider two C' diffeomorphisms f : Rn --, Rn, g : Rn -+ Rn, anda Ck diffeomorphism h : Rn -> Rn

Definition 1.2.1. f and g are said to be Ck conjugate (k < r) if there exists a Ckdiffeomorphism h: R' - Rn such that g o h = h o f. If k = 0, f and g are saidto be topologically conjugate.

The conjugacy of two diffeomorphisms is often represented by the following

diagram.in Rn

hi j,h (1.2.1)

Rn Rn

The diagram is said to commute if the relation g o h = h o f holds, meaningthat you can start at a point in the upper left hand corner of the diagram and reach

the same point in the lower right hand corner of the diagram by either of the two

possible routes. We note that h need not be defined on all of Rn, but possiblyonly locally about a given point. In such cases, f and g are said to be locally Ckconjugate.

If f and g are Ck conjugate then we have the following results.

Proposition 1.2.1. If f and g are Ck conjugate, then orbits of f map to orbitsof g under h.

PROOF: Let x0 E Rn; then the orbit of x0 under f is given by

Of (-0) ={...,f-n(x0),...,f-1(xo),xo,f(xo),...,fn(xo).... } . (1.2.2)

From Definition 1.2.1, f = h-1 o g o h, so for a given n > 0 we have

f n(x0) = (h-1 o g o h) o (h-1 o 9 o h) o ... o (h-1 o g o h) (xo) (1.2.3)

n factors

= h o gn o h(x0) (1.2.4)

or

h o f n(x0) = gn o h(x0) . (1.2.5)

Also from Definition 1.2.1, f -1 = h-1 o g-1 o h so, by the same argument forn > 0 we obtain

h o f -n(x0) = g-n o h(x0) . (1.2.6)


Therefore, from (1.2.5) and (1.2.6) we see that the orbit of x0 under f is mapped

by h to the orbit of h(xo) under g.

Proposition 1.2.2. If f and g are Ck conjugate, k > 1, and x0 is a fixed pointof f . Then the eigenvalues of D f (xo) are equal to the eigenvalues of Dg (h(xo)).

PROOF: From Definition 1.2.1 f (x) = h-1 o g o h(x). Note that since x0 is a fixed

point then g(h(xo)) eo) Since h is differentiable we have

D f = Dh-1 Dg Dhap 2p Ih(zo) 20

so recalling that similar matrices have equal eigenvalues gives the result.

Next we turn to flows. Let f and g be Cr vector fields on R'.

Definition 1.2.2. f and g are said to be Ck-equivalent if there exists a Ck diffeo-morphism h, which takes orbits of the flow generated by f, q5(t, x), to orbits of the

flow generated by g, ?i(t, y), preserving orientation but not necessarily parametriza-

tion by time. If h does preserve parametrization by time, then f and g are said tobe Ck-conjugate.

We remark that, as for maps, the conjugacies do not need to be defined on all

of R.Now we examine some of the consequences of Definition 1.2.2.

Proposition 1.2.3. Suppose f and g are Ck-conjugate. Thena) fixed points off are mapped to fixed points of g,b) T-periodic orbits off map to T-periodic orbits of g.

PROOF: f, g Ck conjugate under h implies the following.

h o 0(t, x) = 0(t, h(x)) (1.2.8)

Dh = (1.2.9)

The proof of a) follows from (1.2.9) and the proof of b) follows from (1.2.8).

Proposition 1.2.4. Suppose f and g are Ck-conjugate (k > 1) and f (xo) = 0.Then D f (xo) has the same eigenvalues as Dg(h(xo)).

PROOF: We have the two vector fields, i = f(x), g(y). By differentiating

(1.2.8) with respect to t we have

f(x) = g(h(x)) . (1.2.10)Dhx

1.2. Conjugacies 25

Differentiating (1.2.10) gives

D2hyf (x) + Dh yD f Dg h(yph

y. (1.2.11)

Evaluating (1.2.11) at xo gives, J

Dh D f = Dg jn'h (1.2.12)yp z0 IIL(y01 x0

or

D = Dh_11 D hf

yo yo9 h(yoVI

yo

and, since similar matrices have equal eigenvalues, the proof is complete.

(1.2.13)

The previous two propositions dealt with Ck-conjugacies. We next examine the

consequences of Ck-equivalence under the assumption that the change in parametri-

zation by time along orbits is C1.

Proposition 1.2.5. Suppose f and g are Ck-equivalent; thena) fixed points of f are mapped to fixed points of g,

b) periodic orbits off are mapped to periodic orbits of g, but the periods neednot be equal.

PROOF: If f and g are Ck-equivalent then

h o 0(t, x) = z/i(a(x, t), h(x)) (1.2.14)

where a is an increasing function of time along orbits (note: a must be increasingin order to preserve orientations of orbits).

Differentiating (1.2.14) gives:

Dhqf = at 8a (1.2.15)

So (1.2.15) implies a). Also, b) follows automatically since Ck-diffeomorphisms map

closed curves to closed curves. (If this were not true then the inverse would not be

continuous.)

Proposition 1.2.6. Suppose f and g are Ck-equivalent (k > 1) and f (xo) = 0;then the eigenvalues of D f (xo) and the eigenvalues of Dg(h(xo)) differ by a positive

multiplicative constant.

PROOF: Proceeding as in the proof of Proposition 1.2.4 we have

Dh f(x) = at g(h(x))(1.2.16)


Differentiating (1.2.16) gives

D2h f(x) + Dh Dfx

asDg h + a2a g(h(x)) . (1.2.17)

x at h(x x axat x

Evaluating at xp gives

Dh Dfxo xo = at Dgl h(xo)Dh xo

(1.2.18)

so D fxo

and Dgh(xo )

are similar, up to the multiplicative constant as/atwhich is positive since a increases on orbits.

1.3. Invariant Manifolds

In this section we want to describe some aspects of the theory of invariant manifolds

which we will use repeatedly in Chapters 3 and 4. Roughly speaking, an invariant

manifold is a surface contained in the phase space of a dynamical system which hasthe property that orbits starting on the surface remain on the surface throughoutthe course of their dynamical evolution: i.e., an invariant manifold is a collection

of orbits which form a surface (note: later we will relax this requirement by in-troducing the idea of locally invariant manifolds). Additionally, the set of orbitswhich approach or recede from an invariant manifold M asymptotically in time un-

der certain conditions are also invariant manifolds which are called the stable and

unstable manifolds, respectively, of M. Knowledge of the invariant manifolds of adynamical system as well as the interactions of their respective stable and unstable

manifolds is absolutely crucial in order to obtain a complete understanding of the

global dynamics. In Chapters 3 and 4 this statement will become evident. Also,under certain general conditions, invariant manifolds often possess the property of

persistence under perturbations. This property is used in Chapter 4 to developglobal perturbation methods for systems where we have a global knowledge of the

invariant manifold structure.

The amount of work on the subject of invariant manifolds in the past fifty years

has been prodigious and is still continuing today at a rapid pace. It is, therefore,not possible to give a full account of the various aspects of the theory or to even give

an adequate historical survey in this section. However, we will give a chronology


of some of the major results as well as some references where further information

can be obtained. (Note: results concerning invariant manifolds may be expressed

in terms of continuous time systems (vector fields) or discrete time systems (maps).

In either case, it usually poses little difficulty to translate results for one type of

system into corresponding results for the other type, see Fenichel [1971], [1974],[1977], Hirsch, Pugh and Shub [1977] or Palis and deMelo [1982] for a discussion of

some examples.)

The first rigorous results concerning invariant manifolds are due to Hadamard

[1901] and Perron [1928], [1929], [1930]. They proved the existence of stable and

unstable manifolds of fixed points of maps and ordinary differential equations using

different techniques. Levinson [1950] constructed invariant two-tori in his studies

of coupled oscillators. This work was extended and generalized by Diliberto [1960],

[1961] with additional contributions by Kyner [1956], Hufford [1956], Marcus [1956],

Hale [1961], Kurzweil [1968], McCarthy [1955] and Kelley [1967]; at the same time,

similar work was being carried out independently by the Russian school led byBogoliubov and Mitropolsky [1961]. The existence of stable and unstable manifolds

and their persistence under perturbation for an arbitrary invariant manifold wasfirst proved by Sacker [1964]. This work was later extended and generalized byFenichel [1971], [1974], [1977] with similar work and even more extensions being

done independently by Hirsch, Pugh and Shub [1977]. Some more recent resultsinclude the work of Sacker and Sell [1978], [1974] and Sell [1978] which was used

by Sell [1979] in the study of bifurcations of n-tori (note: Sell's work representsthe first rigorous results dealing with the Ruelle-Takens-Newhouse scenario for the

transition to turbulence, see Sell [1981], [1982], and the work of Pesin [1976], [1977]

dealing with the existence of invariant manifolds under nonuniform hyperbolicityassumptions. We have not mentioned any results relating to center manifolds (see

Carr [1981] or Sijbrand [1985]) or invariant manifolds in infinite dimensional systems

(see Hale, Magalhaes and Oliva [1984] and Henry [1981] since we will not use those

ideas or results in this book.

The results from invariant manifold theory which we will describe will be taken

from Fenichel [1971] since they are most closely suited for the perturbation tech-niques which we will develop in Chapter 4. However, first we will begin with amotivational example.

EXAMPLE 1.3.1. We consider a nonlinear, autonomous ordinary differential equa-


tion defined on R',

x = f (x) , x(0) = xo , x E Rn (1.3.1)

where f : Rn -+ R'L is at least C1. We make the following assumptions on (1.3.1).

Al) (1.3.1) has a fixed point at x = 0, i.e., f (0) = 0.A2) D f (0) has n - k eigenvalues having positive real parts and k eigenvalues

having negative real parts.

Thus, (1.3.1) possesses a particularly trivial type of invariant manifold, namely

the fixed point at x = 0. Let us now study the nature of the linear system obtained

by linearizing (1.3.1) about the fixed point x = 0. We denote the linearized systemby

eERn (1.3.2)

and note that the linearized system possesses a fixed point at the origin. Let

v1, ... , vn-k denote the generalized eigenvectors corresponding to the eigenvalues

having positive real parts, and vn-k+1' 'vn denote the generalized eigenvec-tors corresponding to the eigenvalues having negative real parts. Then the linearsubspaces of Rn defined as

Eu = span {v',. vn-k}

Es = span vn-k+1 vn(1.3.3)

are invariant manifolds for the linear system (1.3.2) which are known as the unstable

and stable subspaces, respectively. Eu is the set of points such that orbits of (1.3.2)

through these points approach the origin asymptotically in negative time, and Esrepresents the set of points such that orbits of (1.3.2) through these points approach

the origin asymptotically in positive time (note: these statements are not hard toprove, and we refer the reader to Arnold [1973] or Hirsch and Smale [1974] fora thorough discussion of linear, constant coefficient systems). We represent thissituation geometrically in Figure 1.3.1.

The question we now ask is what is the behavior of the nonlinear system (1.3.1)

near the fixed point x = 0? We might expect that the linearized system shouldgive us some indication of the nature of the orbit structure near the fixed pointof the nonlinear system, since the fact that none of the eigenvalues of Df(0) have


Figure 1.3.1.

zero real part implies that near x = 0 the flow of (1.3.1) is dominated by theflow of (1.3.2). (Note: fixed points of vector fields which have the property thatthe eigenvalues of the matrix associated with the linearization of the vector fieldabout the fixed point have nonzero real parts are called hyperbolic fixed points.)Indeed, the stable manifold theorem for fixed points (see Palis and deMelo [1982])

tells us that in a neighborhood N of the fixed point x = 0 for (1.3.1), there existsa differentiable (as differentiable as the vector field (1.3.1)) n - k dimensionalsurface, Wloc(0), tangent to Eu at x = 0 and a differentiable k dimensionalsurface, Wloc(0), tangent to Es at x = 0 with the properties that orbits of pointson Wloc(0) approach x = 0 asymptotically in negative time (i.e., as t -+ -oo)and orbits of points on Wloc(0) approach x = 0 asymptotically in positive time(i.e., as t --+ +oo). Wu (0) and Wloc(0) are known as the local unstable andlocstable manifolds, respectively, of x = 0. We represent this situation geometricallyin Figure 1.3.2.

Let us denote the flow generated by (1.3.1) as then we can define global

stable and unstable manifolds of x = 0 by using points on the local manifolds asinitial conditions.

W u(0) = U Ot (W1oc(0))tI>0

W6(0) = V Ot (Wloc(0))(1.3.4)

t<0

30 1. Introduction: Background for O.D.E.a and Dynamical Systems

Figure 1.3.2. Phase Space of (1.3.1) near x = 0.

W'(0) and W8(0) are called the unstable and stable manifolds, respectively, ofx = 0. We represent the situation geometrically in Figure 1.3.3.

EurW°(0)

Figure 1.3.3. Global Stable and Unstable Manifolds of x = 0.

Now suppose we add a small autonomous perturbation, Eg(x), to (1.3.1) where

g(x) is as differentiable as f (x) and e E I C JR where I = { E E JR 10 < E < cowe,denote the perturbed system by

i = f (x) + cg(x) , x(0) = xo , x E IR" . (1.3.5)


The question we now ask is how much of the structure of (1.3.1) is preserved in the

perturbed system (1.3.5). Specifically, we will be concerned with what happens to

the fixed point at the origin and its stable and unstable manifolds.

The fate of the fixed point is easy to determine by a simple application of the

implicit function theorem (note: recall that a fixed point of (1.3.5) is a solutionof f (x) + cg(x) = 0). We will set up the problem for application of the implicitfunction theorem. Let us consider the function

G : 1R.n x l --> 1Rn (1.3.6)

(x, E) H f (x) + eg(x)

It is clear that G(0,0) = 0, and we wish to determine if there exists a solution

of G(x, e) = 0 for (x, e) close to (0,0). Now the derivative of G with respect to xevaluated at (x, e) = (0, 0) is given by

DXG(0, 0) = Dx f (0) . (1.3.7)

By our assumption on the eigenvalues of D f (0) (specifically, there are no zeroeigenvalues) it is clear that det[D5G(0, 0)] = det Dx f (0) # 0; thus, by the implicit

function theorem there exists a function of c, x(e) (with x(e) as differentiable as

G(x, e)), such thatG(x(e),e) = 0 (1.3.8)

for e sufficiently small contained in I. Thus, the fixed point is preserved in theperturbed system, although it may move slightly.

The fate of the unstable and stable manifolds of x = 0 follows from the

persistence theory for stable and unstable manifolds (see Fenichel [1971] or Hirsch,

Pugh and Shub [1977]) which we will describe in some detail later on. However, for

now we will state the consequence of this theory, which tells us that in some neigh-

borhood N containing x = 0 and x = x(e) there exist differentiable manifolds

Wloc(x(E)) and Wloc(x(e)) passing through x(e) with the properties that orbits of

points in under the perturbed flow approach x = x(e) asymptotically in

negative time and orbits of points in Wloc(x(e)) under the perturbed flow approach

x = x(E) asymptotically in positive time. Wloc(x(e)) and Wloc(x(e)) have thesame dimensions and differentiability as Wloc(0) and Wloc(0), respectively. Uti-

lizing the flow generated by the perturbed system (1.3.5) and Wloc(x(E)) and


Figure 1.3.4. Perturbed and Unperturbed Structure.

Wloc(x(E)) as initial conditions, we can define global unstable and stable manifolds

of x = x(e) in exactly the same manner as we defined them for the unperturbedsystem. See Figure 1.3.4 for a geometrical interpretation.

This simple example illustrates several points that arise in invariant manifoldtheory which we now want to emphasize.

1) For the unperturbed equation it is first necessary to locate the invariant man-

ifold. In our simple example the invariant manifold is a fixed point which can

be found by solving for the zeros of a system of coupled nonlinear algebraicrelations. Locating more general types of invariant manifolds may involve hav-

ing quite a detailed knowledge of the orbit structure of a nonlinear ordinarydifferential equation, which in general is a formidable task.

2) Once the invariant manifold of the unperturbed system is obtained, it is thennecessary to study the linear system obtained by linearizing the unperturbedsystem about the invariant manifold. This procedure, where the invariantmanifold is a fixed point, periodic orbit, or quasiperiodic orbit, is quite familiar;

if the unperturbed system is of the form

i= f(x), xERn, f EC1 (1.3.9)

with an invariant manifold q(t) being a fixed point, periodic orbit, or quasiperi-


odic orbit, then letting x(t) = 0(t) - ;(t), we obtain

or, = Df(6) e + O (I 2)(1.3.10)

since 0 = f (0) (i.e., 0 is a solution of (1.3.9)). If we retain only terms linear

in C we obtain the associated linearized system

=Df(c)t. (1.3.11)

Now if the invariant manifold is more general, such as a surface containingmany different orbits of (1.3.9), then linearizing about the invariant manifold

is not a straightforward procedure, especially if the invariant manifold is not

globally representable as a graph of a function. In this case one obtains acollection of linear equations representing the linearized vector field in different

"coordinate charts" on the invariant manifold. The techniques for describing

the vector field near a general invariant manifold are obtained from the theory

of differentiable manifolds which we will describe shortly.

3) Once the linearized system is obtained, it is then necessary to study its sta-bility. This information will allow us to determine the dimension of the stable

and unstable manifolds of the invariant manifold as well as the persistence and

smoothness properties of the structure under perturbations. In general, this is

a formidable task, since the coefficients of the linear system may have a com-

plicated time dependence. There are two approaches to the problem which are

essentially equivalent, one involves the computation of Lyapunov type numbers

or exponents (this is the a?proach we shall take) and the other a consideration

of exponential dichotomies (see Coppel [1978] and Sacker and Sell [1974]).

Before discussing the general theory of invariant manifolds, we need to givesome background material from differential geometry. More specifically, we will

need to understand the definition of a differentiable manifold, the tangent space at

a point, the tangent bundle, and the derivatives of maps defined on differentiablemanifolds. We will not develop these concepts in the most abstract or mathemati-

cally crisp manner, but rather along the lines where they occur most frequently in

applications. In applications involving the modelling of the dynamics of some phys-

ical system, we typically choose certain quantities describing various aspects of the

34 1. Introduction: Background for O.D.E.s and Dynamical Systems.

system and write down equations describing the time evolution of these quantities.

These quantities constitute the phase space of the system with invariant manifolds

arising as surfaces in the phase space. Consequently, we choose to develop the con-

cept of a differentiable manifold as a surface embedded in ]R'5 (loosely following the

exposition of Milnor [1965] and Guillemin and Pollack [1974]) and refer the reader

to any differential geometry textbook for the abstract development of the theoryof differentiable manifolds (e.g., a standard and very thorough textbook is Spivak

[1979]). Our approach will allow us to bypass certain set theoretic and topologi-cal technicalities, since our manifolds will inherit much structure from 1R', whose

topology is relatively familiar. Additionally, it is hoped that this approach will ap-peal to the intuition of the reader who has little or no experience with the subjectof differential geometry.

We begin by defining the derivative of a map defined on an arbitrary subset of]Rn

Definition 1.3.1. Consider a map f : X --- ]R' where X is an arbitrary subsetof IR9t. f is said to be C' on X if for every point x c X there exists an open setU C ]R7z containing x and a Cr map F: U -+ R'2 such that f = F on U n X.

Definition 1.3.2. A map f : X -* Y of subsets of two Euclidean spaces is called aC' difeomorphism if it is one to one and onto and if the inverse map f -1:Y -* Xis also Cr.

We are now in a position to give the definition of a differentiable manifold.

Definition 1.3.3. A subset M C 1R' is called a C' manifold of dimension m ifit possesses the following two structural characteristics.

1) There exists a countable collection of open sets V a C ]R'2, a E A where A

is some countable index set, with Ua - VI nM such that M = U Ua.aEA

2) There exists a C' diffeomorphism xa defined on each Ua which maps Ua

onto some open set in lR'n.

We make the following remarks regarding Definition 1.3.3.

1. A standard terminology is that the pair (U'; xa) is called a chart for M andthe union of all charts, i.e., U (Ua; xa), is called an atlas for M.

aEA2. The sets Ua are often called relatively open sets, i.e., open with respect to M.


3. From 2) of Definition 1.3.3 we see that the degree of differentiability of a man-

ifold is the same as the degree of differentiability of the xa. This implies acertain compatibility condition which must be satisfied on overlapping charts.

More specifically, let (Ua; xa) and (UP; x1), a, 1 E A, be two charts such that

Ua n UP 54 0, see Figure 1.3.5.

Rm

Figure 1.3.5. Coordinate Charts on a Manifold.

Then the region Ua n UP can be described by two different coordinatizations,

namely (Ua n UP; xa) and (Ua n UP; xP). We denote

xa : p E Ua n UI3 _ (xi,...,xal,) E Rm(1 3 12). .

xQ:pEUaflUP (xQ,...,A)ERm

where (xi , ... , x"") and (xQ, ... , xQ,,) represent points in the Euclidean spaceRm. Now the maps

xP o (xa)-1 : xa(Ua n UP) -> xp(Ua n UP)

(xi,...,xM 'xa o (xp)-1 : xP(U' n UP) -' xa(Un n UQ)

Q p a p Q a Q Q(xl,...,xm) > xl(xl,...xm),...,xm(xl,...,xM))

(1.3.13)

represent the change of coordinates from xP to xa coordinates and from xa toxP coordinates, respectively, and the fact that xa and xP are Cr diffeomorphisms


implies that the maps describing the change of coordinates must likewise be Crdiffeomorphisms. (Note: in the change of coordinate maps in formula (1.3.13) we

should more correctly write (z (p(xi, ... , xm)), ... , xm(p(xi , ... , xm,))) for the

image of (xi,...,x') under x,6o(xa)-1 (and similarly for the map xao(xf3)however, it is standard and somewhat intuitive to identify points in the manifold

with their images in a coordinate chart, especially when the manifold is a surface

in lR'.) In particular, for r > 1 we get the familiar requirement on changes ofcoordinates that the jacobian matrices

x0 axP axi ... axiaxi axm 1 (ax axm 1

and

axm axm axa ax,ax, ... axm axA axm

be nonsingular on xa(Ua n UP) and xfl(Ua fl UP), respectively.

(1.3.14)

Heuristically, we see that a differentiable manifold is a set which locally has the

structure of ordinary Euclidean space. We now give several examples of manifolds.

EXAMPLE 1.3.2. The Euclidean space R' is a trivial example of a Coo manifold.We take as the single coordinate chart (i; R') where i is the identity map identi-fying each "point" in R9z with its coordinates; it should be clear that i is infinitely

differentiable and hence Riz is a C°O manifold.

EXAMPLE 1.3.3. Let f : I -* R be a C' function where I C R is some openconnected set. Then the graph of f is defined as follows:

graph f = { (s, t) E R2 I t = f (s), s El). (1.3.15)

Geometrically, graph f might appear as in Figure 1.3.6.

We claim that graph f is a Cr one dimensional manifold. In order to verifythis we must show that the two requirements of Definition 1.3.3 can be satisfied.

1) Let U = R2 fl graph f ; then, by definition, U = graph f.2) We define a coordinate chart on U in the following manner

x : U -- R1

f (s)) s(1.3.16)

(s, i- f

with the inverse defined in the obvious way,x-1 : R1 . U

(1.3.17)sH (s,f(s)).


Figure 1.3.6. Graph f.

S

It is clear that x and x-1 are C' since f is C'. Thus, graph f is a C' onedimensional manifold described by a single coordinate chart. We remark that this

example should remind the reader of some of the heuristic aspects of elementarycalculus, where it is common to visualize scalar functions as curves in the plane and

to identify points on the curve with the corresponding points in the domain of the

function.

EXAMPLE 1.3.4. Consider the following set of points contained in ]R3

M={(u,v,w)E]R.3I u2+v2+w2=1} . (1.3.18)

This is just the two dimensional sphere of unit radius. We want to show that M isa Coo two dimensional manifold.


Let us define the open sets

U1={(u,v,w)E1R.3lu2+v2+w2=1, w>0}

U2= {(u,v,w) EiR3u2+v2+w2=1, w<0}

U3={(u,v,w)ER 3 u2+v2+w2=1, v>0}0 (u,v,w)ER3U2+v2+w2=1,v<0}U5 = {(u,v,w) EiR3 u2+v2+w2=1, u>0}

U6= {(u,v,w)E1R3u2+v2+w2=1, u<0}

It should be clear that these six sets are open with respect to M and that they

cover M (see Figure 1.3.7).

U

Figure 1.3.7. M = U1 UU2 U U3 U U4 U U5 UU6 .

In these six sets, points of M can be represented as follows

U1 : (u, v, 1 - u2 - v2)

U2: 1-u2-v2)U3: (u, N/rl - u2 - w2, w)

U4: (u, - i - u2 - w2,l\w)

U5: ( 1-v2-w2,v,wl

U6: (-V/-1 - v2 - w2, v, w)

(1.3.20)


We define maps of the Ua, a = 1, 2, 3, 4, 5, 6, into 1R2 as follows

x1 U1 -+ JR2

(u, v, V/-l-- u2 - v2) -4 (u,v)

x2 U2 -* JR,2

(u, v - VT--_ 2---v 2(u, v)

x3 U3 -* lR2

(u, V/1--- u2 - w2, w) --> (u, w)(1.3.21)

x4 U4 -* JR2

(u, - i - u2 - w2, w) I--) (u, w)

x5 U5 -* ]R2

( i-v2-w2,v,w)"(v,w)

X6U6 -*JR2

(- 1 - v2 - w2, v, w) --. (v, w)

with the inverse maps defined in the obvious manner (see Example 1.3.3). It should

be clear that xa and a)-1, a = 1, 2, 3, 4, 5, 6, are Coo.Let us now demonstrate the compatibility of the coordinatizations on overlap-

ping regions for a particular example. The open set in M

U1nU4={ (u, v,w) Iu2+v2+w2=1, w>0, v<0} (1.3.22)

may be given coordinates by either xi or x4. The formulas for the coordinatechanges are given as follows:

x4 o (x1) : X1 (U1 fl U4) , x4(U1 fl U4)

(u, v) H (u, 1 - u2 - v2) - (u, w)

x1 o (x4)-1 :x4(U1 n U4) -' x1(U1 n U4)

(u,w) H (u,- 1-u2-w2) - (u,v).

(1.3.23)

It is easy to see that these two coordinate change maps are mutual inverses andthat they are C.


The reader should note the similarities between this example and Example1.3.3. In the present example we were not able to represent the manifold globally

as the graph of a function; however, we divided up the manifold into regions where

we could represent it as a graph and, in these regions, the construction of thecoordinate maps is exactly the same as in Example 1.3.3. Notice that the choice of

(relatively) open sets to cover M is certainly not unique, but this does not result in

any practical difficulties (see Spivak [1979] for a discussion of "maximal" atlases).

Although in Definition 1.3.1 we defined the derivative of a map defined on a

manifold, there is a geometric object associated with a manifold called the tangent

space which plays an important role in the concept of the derivative of a function

defined on a manifold. We want to motivate its construction by first recalling the

definition of differentiability of a map defined on Euclidean space. We consider amap

f : U -V (1.3.24)

where U C R1 and V C Rk are open sets. The map f is said to be differentiableat a point x0 E U if there exists a linear map

L : R1 -' Rk (1.3.25)

such that

if(xo + h) - f(x0) - Lh[ = 0 ([h12) (1.3.26)

where 1.1 is any norm on Euclidean space. The linear map L is called the derivative

of f at x0 and consists of the l x k matrix of partial derivatives of f. The linearmap L acts on elements h E RI which can be viewed as vectors emanating fromthe point x0 E U. This previous sentence is quite important. The derivative is alinear map, but linearity of a map depends crucially on the linear structure of thespace on which it operates. If we want to define the derivative of a map intrinsicto the manifold on which it is defined, we must somehow associate a linear space

on which the derivative can operate in a way that is "natural" for the manifold.This linear space will be the tangent space at a point of the manifold at which the

derivative is computed. We begin with two preliminary definitions.

Definition 1.3.4. Let I = { t E R 1 -e < t < e } for some fixed c > 0. Then aCr curve in M is a C' map from I into M.


Definition 1.3.5. Let C : I - M be a C' curve such that C(O) = p. Then thevector tangent to C at p is

dtC(t)t=0 -- C(0) .

See Figure 1.3.8 for an illustration of the geometry.

C

-E 0 E

Figure 1.3.8. A Curve and Its Tangent Vector at a Point.

Before discussing the case of the tangent space at a point for a general manifold,

let us first discuss the case where the manifold is Rm (see Example 1.3.2 following

Definition 1.3.3).

Let x be a point in Rm; then the tangent space to Rm at x, TxRm, is definedto be R'. A more geometrical, but equivalent, definition would be that TxRm isthe collection of all vectors tangent to curves passing through x at the point x. It is

easily seen that this set is equal to Rm, since every point in Rm can be viewed asthe tangent vector to some differentiable curve, e.g., take as the curve C(t) = x+t£,

E 111', then dC tdt t=0 =, see Figure 1.3.9.

Recalling our brief discussion of the differentiation of maps defined on Rm, it

should now be clear what role the tangent space at a point plays in the definitionof the derivative at a point. Namely, TxRm is the domain of the derivative D f (x),

and locally it reflects the structure of the manifold Rm, thus allowing the linearmap Df (x) to locally reflect the character of f (x). In the case where the manifold

has the structure of a linear vector space we usually do not bother with formalizing

the notion of the tangent space at a point, since the tangent space at a point isthe space itself. However, in the case where the domain of the map has no linearstructure, then in order to discuss a local linear approximation to the map at a


Figure 1.3.9. The Tangent Space of ]R at a Point x E R'.

point, i.e., the derivative of the map, it is necessary to introduce the structure of alinear vector space for the domain of the derivative, since the definition of linearity

of a map depends crucially on the fact that the domain of the map is linear.

Now we will define the tangent space at a point for an arbitrary differentiable

manifold. Let (Ua, xn) be a chart containing the point p in the m dimensionaldifferentiable manifold M. Then xa (Ua) is an open set in R' containing the pointxa(p). From the previous discussion, the tangent space at xa(p) in IR'R', Txa(p)Rm,

is just 1R"L. To construct TpM we carry Tx.(p)Rm to p in M by using xa. Sincexa : Ua --; R' is a diffeomorphism, then (xa)-1 : xa(Ua) -+ Ua is also adiffeomorphism from ]R into ]R'. Therefore, we can compute D [(xa)-1] whichis a linear isomorphism mapping R' into Rm. The tangent space at p E M,

. R".TpM is then defined to be D [(xa)-1]x°(p)

Definition 1.3.6. Let (Ua; xa) be a chart on M with p E U. Then the tangentspace to M at the point p, denoted TpM, is defined to be D [(xa)-1]

x°`(p)R"`

The tangent space at p in M has the same geometrical interpretation as thetangent space at a point at R'; namely, it can be thought of as the collection ofvectors tangent to curves which pass through p at p. This can be seen as follows: let

(U a; x') be a chart containing the point p E M. Then, as previously discussed,Txa(p)R"L consists of the collection of vectors tangent to differentiable curvespassing through xa(p) at xa(p). Let -y(t) be such a curve with -y(O) = xa(p),


then '(O) is a vector tangent to 7(t) at xa(p). Using the chart, since xa is adiffeomorphism of Ua onto xa(Ua), then (xa)-1 (-I(t)) - C(t) is a differentiable

curve satisfying C(O) = p. Using the chain rule, the tangent vector to C(t) at p isgiven by D [(xa)-1] xa(p) 7(0) - C(O). Now j(O) is a vector in Rm; thus C(0)

is an element of TpM. So we see that the elements of TpM consist of the vectorstangent to differentiable curves passing through p at p.

Before leaving the tangent space at a point, there is one last detail to beconsidered; namely, in our construction of the tangent space at a point of a manifold

we utilized a specific chart, but the tangent space is a geometrical object whichshould be intrinsic to the manifold being representative of the manifold's localstructure. Therefore, the tangent space should be independent of the specific chart

used in its construction.

Proposition 1.3.1. The construction of TpM is independent of the specific chart.

PROOF: Let (Ua; xa), (UP; xp) be two charts with Uafl UP 0 and p E Uan UP.

Then by Definition 1.3.5 TpM can be constructed as either D [(xa)-1] Ix°(p) it-

or D [(1Q)-1] I p(p) ]R"a. We must show that

D [(xa)-1] I .- (r) ]R"` D [(x,6)-1]I x

13(P) ]R

This can be established by the following argument. Consider Figure 1.3.5, on Uafl

UP we have the relationship

(xa)-1 = (xp)-1 0 [xp 0 (xa)-1] . (1.3.27)

Differentiating (1.3.27) we get

D [(xa)-1] IX" (P) D [(x")-1] I xQ(P)D

[x,6o

(xa)-1]I x11(P) (1.3.28)

but D [x,6 o (xa)-1] is an isomorphism of 1Rm so D [xA o (xa)-1] Ixa(p) Rm =

]Rm. Therefore, we get

D [(xa)-1]I x°(P) ]Rm D [(xa)-1] xfl(P)D

[x" o (xa)-1] x- (P) Im

D [(x")-1] I p(P).]Rm .

(1.3.29)


So we see that the tangent space at a point is independent of the particular chart

chosen in a neighborhood of that point.

Now that we have defined the tangent space at a point of a manifold, we want to

demonstrate the role it plays in defining the derivative of maps between manifolds.

Let f : M'n' -+ Ns be a Cr map where M' C R', m < n, is an m dimensionalmanifold and NS C Rq, s < q, is an s dimensional manifold. From Definition1.3.1, f being C' means that, for every point p E M, there is an open set V C Rn(depending on p) and a C' map F : V -4 Rq with F = f on V n MI.

Proposition 1.3.2. Let f : M' -* Ns be as defined above; then

DFIP : TTMm -> Tf(P)NS (1.3.30)

PROOF: Let (xa; Ua) be a coordinate chart on M'n containing p and let (y13;WP)be a coordinate chart on N' containing f (p). Let V C R' be the open set aroundp in R' and, if necessary, shrink Ua so that Ua C V. Then we have

F (p) = (y,8)-1 0 [yf 0 f o (xa)-1] o xa(p) . (1.3.31)

Now we must show that DF(p) D [(xa)-1] Ix"(P) R' is contained in

D [(ya)-1]I Yp(f(P))- Rn -- Tf (P)N'

Differentiating (1.3.31) we obtain

DFI P = D [(yP)-1] y$(f(P))D [yp 0 f o(xa)-1] I xa(P)DxaIP (1.3.32)

or equivalently,

DFI pD [(xa)-1] 1,(P) D [(yQ)-1] I yR(f(P))D [yO ofo(xa)-1]

I x°(P)(1.3.33)

So we get

DFI pD [(xa)-1] I xa(P) Rm D [(y")-1] Iya(f(P))D [y" o f o (xa)-1]

xa(P)Rm

(1.3.34)

but

D [yPo f o (xa)-1]

Rm C RS (1.3.35)

1.3. Invariant Manifolds

TPMmf

7 P

Mm

and

Figure 1.3.10. TpMm and Its Image under Df 1P.

D [(y")-1] I YQ(f(p))- Itn

Tf(p)N3 -

45

(1.3.36)

So we see that DFI P TTM' C T f(p)N3. From equation (1.3.34) it should be clearthat this result is independent of the particular extension F of f to some open setin 1Rn.

Geometrically, Proposition 1.3.2 may be visualized as in Figure 1.3.10.

When we study manifolds which are invariant under the flow generated by a

vector field it will be important to have information concerning the tangent spaces at

different points on the manifold. In this regard, it is useful to consider the geometric

object formed by the disjoint union of all the tangent spaces at all possible points

of the manifold. This is called the tangent bundle.

Definition 1.3.7. The tangent bundle of a C' manifold M C 1R'L, denoted TM,is defined as

TM={(p,v) EMx1RmI vETpM}. (1.3.37)

So the tangent bundle is the set of all possible tangent vectors to M, and TM itself

has the structure of a 2m dimensional Cr-1 manifold, as we show next.

Proposition 1.3.3. Let M C R' be a C' manifold of dimension m; then thetangent bundle of M, TM C 1R2n, is a Cr-1 manifold of dimension 2m.


PROOF: We must construct an atlas for TM. Let (xa; Uo), a E A, be an atlasfor M. Then (xa, Dxa; Ua, TM I is an atlas for TM, which is Cr-1, sinceDxa is Cr-1. For the remaining details of the proof, see Guillemin and Pollack[1974].

Before proceeding to discuss invariant manifolds of ordinary differential equa-

tions we need to discuss the idea of a manifold with boundary. Note that in thedefinition of a differentiable manifold given in Definition 1.3.3 each point of themanifold has a neighborhood diffeomorphic to some open set in Rn''. This rules out

the possibility of boundary points. As we shall see in Chapter 4, manifolds withboundary arise frequently in applications, and we now want to give a definition of

a Cr manifold with boundary. We begin with a preliminary definition.

Definition 1.3.8. The closed half space, R!" C R', is defined as follows

R' ={(xl,x2,...,xm)ERmI x1 <0}.

The boundary of R' , denoted 3]R , is Rm-1.

We now give the definition of a differentiable manifold with boundary.

(1.3.38)

Definition 1.3.9. A subset M C R' is called a Cr manifold of dimension mwith boundary if it possesses the following two structural characteristics:

1) There exists a countable collection of open sets Va C R'b, a E A where A issome countable index set, with Ua - Vol fl M such that M = U Ua.

aEA2) There exists a C' diffeomorphism xa defined on each Ua which maps Ua

onto some set W n 1R where W is some open set in R'.

We make the following remarks concerning Definition 1.3.9.

1) The boundary of M, denoted 8M, is defined to be the set of points in Mwhich are mapped to 8R! under xa. It is necessary to show that this set isindependent of the particular chart that is chosen, see Guillemin and Pollack[1974] for the details.

2) The boundary of M is a Cr manifold of dimension m - 1, and M - 8M is aCr manifold of dimension m.

3) The tangent space of M at a point is defined just as in Definition 1.3.6 even ifthe point is a boundary point.


We are now at the point where we can state some general results on invariant

manifolds of ordinary differential equations. As mentioned earlier, we will follow

Fenichel's development of the theory, since he explicitly treats the case of invariant

manifolds with boundary which we will encounter in our applications.

We consider a general autonomous ordinary differential equation defined onRn

i= f(x), xE1E' (1.3.39)

where f is a Cr function of x. Let us denote the flow generated by (1.3.39) by ¢t(p),

i.e., c5t(p) denotes the solution of (1.3.39) passing through the point p E R' att = 0. We remark that 't(p) need not be defined for all t E R. or all p E R. LetM - M U 3M be a compact, connected Cr manifold with boundary contained inRn

Definition 1.3.10. a) M - M U 8M is said to be overflowing invariant under(1.3.39) if for every p E M, ct(p) E M for all t < 0 and the vector field (1.3.39)is pointing strictly outward and is nonzero on 8M. b) Al - M U 8M is saidto be inflowing invariant under (1.3.39) if for every p E M, qt(p) E M for allt > 0 and the vector field (1.3.39) is pointing strictly inward and is nonzero on3M. c) M - M U 8M is said to be invariant under (1.3.39) if for every p E M,c5t(p) E M for all t E R.

We make the following remarks concerning this definition.

1) The phrase "the vector field (1.3.39) is pointing strictly outward and is nonzero

on 8M" means that for every p E 8M, cSt(p) M for all t > 0. A similardefinition is obtained for the "... pointing strictly inward ..." by reversingtime.

2) Overflowing invariant manifolds become inflowing invariant under time reversal

and vice versa.

3) Since M is compact, IM exists for all t < 0 if M is overflowing invariant,for all t > 0 if M is inflowing invariant, and for all t E R. if M is invariant.

4) M can be an invariant manifold only if the vector field (1.3.39) is identically

zero on 8M, if 8M = 0, or if the vector field (1.3.39) is parallel to W.

The following definition will also be useful.


Definition 1.3.11. Let M C 1R" be a compact, connected C' manifold in R'.We say that M is locally invariant under (1.3.39) if for each p E M there exists a

time interval Ip = {t E 1RIt1 < t < t2 where t1 < 0, t2 > 01 such that 4t(p) E Mfor all t E I.

We remark that the overflowing and inflowing invariant manifolds of Definition

1.3.10 are examples of locally invariant manifolds.

Next we want to describe the stability characteristics of the invariant manifolds.

This will be done by describing the asymptotic behavior of vectors tangent andnormal to the invariant manifold under the action of the linearized flow.

Let M - M U aM be a C' overflowing invariant manifold contained in 1Rn.Let TJR" I M denote the tangent bundle of 1R" restricted to M, i.e., TRRn I M{ (p, v) E ]R." x 1R" Ip E M , v E TplRn }. Then, by Definition 1.3.7, TM CT]R"I M and, by Proposition 1.3.2, TM is invariant under D¢t(p), p E M, forall t < 0. TM is referred to as a negatively invariant subbundle. At each pointp E M we can use the standard metric on 1R." to choose a complementary subspace

of ]Rn, Np, such that Tp1R" = TpM + Np, where "+" denotes the usual directsum of vector spaces. If we form the union of all such decompositions of TpRnover all points p E M, we obtain a decomposition or splitting of TIR"IM, i.e.,TRn I M = TM ® N - U (TpM + Np). The sum of two subbundles is denoted

pEMby ® and is referred to as the Whitney sum (see Spivak [19791).

We denote the projection onto N corresponding to the splitting T]R") M =TM E) N by IIN, and the projection onto TM by IIT. For the Cr perturbation the-

orem for overflowing invariant manifolds, we will require the manifold to be stable,

in the sense that vectors complementary to TM grow in length as t -* -oo underthe action of the linearized flow, i.e., w0 E Np implies IIND¢t(p)wo -+ oo ast -> -oo, where 1.1 denotes the norm associated with the chosen metric on ]R",and that neighborhoods of the invariant manifold "flatten out" as t -> -oo underthe action of the linearized flow. This "flattening out" property is expressed as

sDOt(p)v0I / IINDgt(p)wo 0 as t --> -oo for every vo E TpM, wo E Np.The real number s is a measure of the degree of the flattening of the neighborhoods

of M. This situation might be visualized geometrically as in Figure 1.3.11.

For computations and proving theorems the stability properties of M are more

conveniently phrased in terms of rates of growth of vectors under the linearized


Figure 1.3.11. Action of Tangent Vectors under Dcbt(p).

flow. We define the following quantities

y(p) = inf{ a E ]R+ I at/ IINDot(p)wo1 --+0 as t -+ -oo for all wo E Np }

and if y(p) < 1 (1.3.40)

a(p) = inf{ s E RI jDq5t(p)v0j / IINDOt(p)wo s -4 0 as t --+ -oo

for all vo E TpM, wo E Np } .

The functions y(p) and a(p) are called generalized Lyapunov type numbers (Fenichel

[1971]) and have several properties which we now state.

Proposition 1.3.4. The functions -f (p) and a(p) have the following properties:i) y(p) and a(p) are constant on orbits, i.e., y(p) = -y (4t(p)), a(p) = a (qt(p)),

t<0.ii) y(p) and a(p) are bounded and achieve theirsuprema on M (although in general

they are neither continuous nor semicontinuous).

iii) y(p) and a(p) are independent of the choice of metric and of the choice of N.

PROOF: See Fenichel [1971].

We remark that a more computable form for the generalized Lyapunov typenumbers which can be derived from (1.3.40) is the following:

y (p) = li m IIND*bt(p)I1/t

t-.-oolog (D0t(p)IIT (1.3.41)

a(p) = limt-+-

oo tog IIIINDbt(p)II


where 1.11 is a matrix norm.

The C' perturbation theorem for overflowing invariant manifolds is stated as

follows:

Theorem 1.3.5 (Fenichel [1971]). Suppose M = MUaM is aCr manifold withboundary overflowing invariant under the Cr vector field i = f (x), x E Mf6 with

-y(p) < 1 and a(p) < 1/r for all p E M. Then, for any CT vector field i = g(x),x E M9z with f (x) C1-close to g(x), there is a Cr manifold with boundary Mg, Cr-

close to M and of the same dimension as M such that Mg is overflowing invariant

under i = g(x), x E Mn.

PROOF: See Fenichel [19711.

This theorem can also be applied to inflowing invariant manifolds. In that case

the generalized Lyapunov type numbers are computed using the time reversed flow

and taking the limits as t -+ +oo. Theorem 1.3.5 will then read exactly the sameexcept that the word overflowing will be replaced by inflowing.

We illustrate Theorem 1.3.5 with the following simple example.

EXAMPLE 1.3.5. Consider the following Coo planar vector field

ax

-bya, b > 0. (1.3.42)

y=It should be clear that (0,0) is a fixed point of (1.3.42) with the x axis being theunstable manifold of (0,0) and the y axis being the stable manifold of (0,0). Consider

the set

M={(x,y)ElR2I-6<x<6, y=0, for some 6>0}. (1.3.43)

It is easy to verify that M is an overflowing invariant manifold under the flow gen-

erated by (1.3.42). We now show that M satisfies the hypotheses of Theorem 1.3.3.

We haveTM = M x (R1, 0)

N = M x (0, R1)

lIN0 0

=0 1

\(101TIT= J

0 0

(1.3.44)

eat 0D¢t = (0 a-bt )


Y

I

Figure 1.3.12. The Geometry of (1.3.42)

x

see Figure 1.3.12.

The generalized Lyapunov type numbers can now be computed, and we find

that

7(p) = 7 = liratcolog

1

eat 11

= -a/b < 0.a(p) = a = limlog IIe btII

e-bt 1/t= e-b

< 1

(1.3.45)

Thus, M satisfies the hypotheses of Theorem 1.3.3 so that any Cr vector field(r > 1) which is CI-close to (1.3.42) has an overflowing invariant manifold C'-close

to M. We have therefore established the local unstable manifold theorem for a fixed

point of a planar nonlinear vector field whose linear part is given by (1.3.42). Local

stable manifolds can be shown to exist by considering inflowing invariant manifolds.

Now for overflowing invariant manifolds it makes sense to consider the unstable

manifold of the overflowing invariant manifold, and we have an existence and per-

turbation theorem for unstable manifolds of overflowing invariant manifolds. The

set-up is as follows: Let M = MUBM be overflowing invariant under (1.3.39), and

let Nu C T]RnIM be a subbundle which contains TM and is negatively invariant

under the linearized flow generated by (1.3.39). Let I C Nu be any subbundlecomplementary to TM, and let J C T]R.nI M be any subbundle complementary


Figure 1.3.13.

to Nu. Then we have the splitting T1RnIM = TM ® I ® J. Let III, IIJ, and 1ITbe the projections onto I, J, and TM, respectively, corresponding to this splitting.We define generalized Lyapunov type numbers as follows:

For any p E M

A(p) = inf{ b E 1R.+ IlIDtt(p)uol bt -0 as t --oo for all u0 E Ip }

-y(p) = inf{a E R.+ at/ lJDq5t(p)wol -> 0 as t -+ -oo for all wo E Jp}

and if y(p) < 1 define (1.3.46)

a(p) = inf{ s E 1R JDOt(p)v01 /IIJDct(p)wols

-> 0 as t --> -oo

for all vo E TpM, w0 E Jp } .

Conclusions identical to those in Proposition 1.3.4 follow for A(p), -y(p) and a(p)

defined above. More computable expressions for the generalized Lyapunov typenumbers can be derived from (1.3.46) and have the following form

A (P) =

7(p) =

limt

lim

III Dq5t(p)

11-1/t

IIJ D.kt (p)1111t

(1 3 47)t- -oo . .

log IID.Pt(p)IIT IIa(p) = t li Do log I111 JDkt(p) I I

where is some matrix norm. See Figure 1.3.13 for an illustration of the geometry.

We have the following theorem.


Theorem 1.3.6 (Fenichel [1971], [1979]). Suppose M = M U aM is a Crmanifold with boundary overflowing invariant under the Cr vector field i = f (x),

x E Rn, with N" C TRnI M a subbundle containing M negatively invariant under

the linearized flow generated by i = f (x), x E Rn. Then if A(p) < 1, j(p) < 1,and o(p) < 1/r for all p E M, the following conclusions hold:

i) (Existence) There exists a Cr manifold Wu overflowing invariant under i =f (x), x E Rn, such that W' contains M and is tangent to N' along M.

ii) (Persistence) Suppose i = g(x), x E Rn, is a Cr vector field C1 close toi = f (x), x E Rn. Then there exists a Cr manifold W9 overflowing invariantunder i = g(x), x E Rn which is Cr close to W'i and has the same dimension asW U.

PROOF: See Fenichel [1971].

We remark that Theorem 1.3.6 can be applied to inflowing invariant manifolds.

In that case, the generalized Lyapunov type numbers are computed using the time

reversed flow with the limits taken as t -> +oo, and the phrase "overflowinginvariant" in Theorem 1.3.6 is replaced by "inflowing invariant." Also, Nu and W'

are replaced by N8 and W', with N8 taken to be a positively invariant subbundleunder the linearized flow generated by i = f (x), x E 1Rn (i.e., N8 is negativelyinvariant under the time reversed linearized flow).

We illustrate this theorem with the following example.

EXAMPLE 1.3.6. Consider the vector field (1.3.42). We will regard the fixed point

(0,0) as the overflowing invariant manifold M. Then we have Nu _ (0,0) x(1R1, 0) is a negatively invariant subbundle. We now show that M and N' satisfy


the hypotheses of Theorem 1.3.6. We have

M = (0, 0)

TM =0

I (0, 0) x (R1, o)

J (0, 0) x (0, R1)III=

1 0

(0 0

J= 0 0lI

0 1

eat 0

DOt =(

0 e -bt )

The generalized Lyapunov type numbers are given by

A(0) = limt-*-oo

y(0) = lirat--00

ebt -l/t =-be < 1

= e-a< 1e-atll -'It

a(0) = 0 since TM = 0.

(1.3.48)

(1.3.49)

Thus, M and Ne satisfy the hypotheses of Theorem 1.3.6 so any Cr vector fieldC1-close to (1.3.42) contains an overflowing invariant manifold.

We remark that in Examples 1.3.5 and 1.3.6 the same conclusions can be made

concerning the vector field (1.3.42); however, different conditions which are neces-

sary in order to arrive at these conclusions are computed in each case. In Example

1.3.5 generalized Lyapunov type numbers are computed on a manifold, and in Ex-

ample 1.3.6 generalized Lyapunov type numbers are computed at a point. In fact,

Theorem 1.3.6 is proven by showing that N" under the hypotheses given in thetheorem is an overflowing invariant manifold satisfying the hypotheses of Theorem

1.3.5.

Let us now give the usual theorem (Sacker [1964], Hirsch, Pugh and Shub[1977]) for compact, boundaryless manifolds invariant under (1.3.39).

Theorem 1.3.7. Let M be a compact, boundaryless C" manifold invariant underx = f (x), x E R'i. Let N9 and N' be subbundles ofTR" lM such that Ns®N" =TR'ZIM and N9 n Nu = TM. Suppose Nu satisfies the hypotheses of Theorem


1.3.6, and N9 satisfies the hypotheses of Theorem 1.3.6 for the time reversed flow.

Then the following conclusions hold.

i) (Existence) There exist Cr manifolds W11, W3 tangent to Nu, Ns along Mwith W' overflowing invariant under z= f (x), x E R', and W 3 overflowinginvariant under i= -f (x), x E 1R°, moreover M = W1 n W1.

ii) (Persistence) Suppose i = g(x), x E R' is a Cr vector field with g(x) Clclose to f (x). Then there are C' manifolds N U, Wg Cr close to W U' Ws'respectively, and having the same respective dimensions with W9 overflowing

invariant under i = g(x), x E Rn and W9 overflowing invariant underi = -g(x), x E Rn; moreover, W9 n Wg -- Mg is a Cr manifold invariantunder i = g(x), x E R'E and Cr close to M.

We remark that compact boundaryless invariant manifolds satisfying the hy-potheses of Theorem 1.3.7 are said to be normally hyperbolic.

EXAMPLE 1.3.1 - continued. Let us now return to our original example concerning

invariant manifolds in order to demonstrate the calculations necessary for the veri-

fication of the hypotheses of the theorems. Recall that we considered the equation

x= f(x), xEia'E (1.3.50)

under the following assumptions.

Al) (1.3.50) has a fixed point at x = 0, i.e., f (0) = 0.A2) D f (0) has n - k eigenvalues having positive real parts and k eigenvalues

having negative real parts.

In this simple case, the invariant manifold M is just the fixed point x = 0 andT1R'IM = 1R't with the unstable and stable subspaces of the linear problem, Euand E'8, corresponding to the invariant subbundles N' and Ns where TJRTI M =1Rn = E' + E'8. We must show that Eu satisfies the hypotheses of Theorem 1.3.6,

and E'8 satisfies the hypotheses of Theorem 1.3.6 under the time reversed flow.

Step 1: Show that Eu satisfies the hypotheses of Theorem 1.3.6.

Let I = Eu and J = Es; then, in coordinates given by the unstable andstable subspaces, the linearized vector field written as

=Df(0)C, eER.n (1.3.51)


with

D f (0) =Al

A ) (1.3.52)2

where Al is a n - k x n - k matrix with all eigenvalues having positive real partsand A2 is a k x k matrix with all eigenvalues having negative real parts. Then we

haveRIeDf(0)t = eAlt

HJeD f (0)t = eA2t .(1.3.53)

Let Al be the real part of the eigenvalue of Al having the smallest real part, andA2 the real part of eigenvalue of A2 having the largest real part. Then it is easy to

see thatA(0) = e-ai < 1

y(0) = eat < 1 (1.3.54)

a(0) = 0 since TM = 0 .

Step 2: Show that E3 satisfies the hypotheses of Theorem 1.3.6 for the time reversed

vector field.

Under the time reversed vector field E'i and E3 are interchanged, so now,letting I = E3, J = Eu and using (1.3.52), we get

A(0) = eat < 1

-y(O) = e-A1 < 1

a(0) = 0 since TM = 0.

(1.3.55)

Thus, we can conclude that there exist manifolds W', W' as differentiableas f and tangent to Eu and E3 at x = 0. W'L is overflowing invariant under(1.3.50) having the same dimension as E'L, and W3 is overflowing invariant under

(1.3.50), with time reversed, having the same dimension as Es; moreover, for anyother vector field C1 close to (1.3.50) this structure persists.

1.4. Transversality, Structural Stability, and Genericity

The concepts of transversality, structural stability, and genericity have played animportant role in the development of dynamical systems theory, and in this section

we want to present a brief discussion of these ideas.

1.4. Transversality, Structural Stability, ar.:: Genericity 57

Transversality is a geometric notion which deals with the intersection of surfaces

or manifolds (see Section 1.3). Let M and N be differentiable (at least C1) manifolds

in Rn.

Definition 1.4.1. Let p be a point in R'; then M and N are said to be transversalat p if p M n N or, if p E M n N, then TpM +TpN = Rn where TpM andTpN denote the tangent spaces of M and N, respectively, at the point p. M andN are said to be transversal if they are transversal at every point p E Rn, seeFigure 1.4.1.

Figure 1.4.1. M and N Transverse at p.

Note that transversality of two manifolds at a point requires more than just the

two manifolds geometrically piercing each other at the point. Consider the following

example.

EXAMPLE 1.4.1. Let M be the x axis in R2, and let N be the graph of thefunction f (x) = x3, see Figure 1.4.2. Then M and N intersect at the origin inR2, but they are not transversal at the origin, since the tangent space of M isjust the x axis and the tangent space of N is the span of the vector (1,0); thus,T(O O)N = T(O,O)M and, therefore, T(O O)N +T(O,O)M: R2.

The most important characteristic of transversality is that it persists undersufficiently small perturbations. This fact will play a useful role in many of ourgeometric arguments in Chapters 3 and 4. Finally, we remark that a term oftenused synonymously for transversal is general position, i.e., two or more manifolds

which are transversal are said to be in general position.


Figure 1.4.2. Nontransversal Manifolds.

The concept of structural stability was introduced by Andronov and Pontryagin

[1931] and has played a central role in the development of dynamical systems theory.

Roughly speaking, a dynamical system (vector field or map) is said to be structurally

stable if nearby systems have qualitatively the same dynamics. This sounds like a

simple enough idea; however, first one must provide a recipe for determining when

two systems are "close" and then one must specify what is meant by saying that,qualitatively, two systems have the same dynamics. We will discuss each question

separately.

Let Cr(Rn,Rn) denote the space of Cr maps of Rn into Rn. In terms ofdynamical systems, we can think of the elements of Cr(R',Rn) as being vectorfields. We denote the subset of C'(Rn,Rn) consisting of the Cr diffeomorphismsby Diffr(Rn, Rn)

Two elements of Cr(Rn,Rn) are said to be Ck e-close (k < r), or justCk-close, if they, along with their first k derivatives, are within a as measured insome norm. There is a problem with this definition; namely, Rn is unbounded, and

the behavior at infinity needs to be brought under control (note: this explains why

most of dynamical systems theory has been developed using compact phase spaces;

however, in applications this is not sufficient, and appropriate modifications mustbe made).

There are several ways of handling this difficulty. For the purpose of ourdiscussion we will choose the usual way and assume that our maps act on compact,

1.4. Tranaversality, Structural Stability, ar.c Ger_ericity 59

boundaryless n dimensional differentiable manifolds, M, rather than all of R. The

topology induced on Cr(M,M) by this measure of distance between two elements

of Cr(M, M) is called the Ck topology, and we refer the reader to Palis and deMelo

[1982] or Hirsch [1976] for a more thorough discussion.

The question of what is meant by saying that two dynamical systems havequalitatively the same dynamics is usually answered in terms of conjugacies (see

Section 1.2). Specifically, CO conjugate maps and CO equivalent vector fields have

qualitatively the same orbit structures in the sense of the propositions given in Sec-

tion 1.2. It should also be clear from Section 1.2 why we do not use differentiable

conjugacies, e.g., from Proposition 1.2.2, two maps having a fixed point cannot be

Ck (k > 1) conjugate unless the eigenvalues associated with the linearized maps

about the respective fixed points are equal. This is a much too strong require-ment if we are only interested in distinguishing qualitative differences between the

dynamics of different dynamical systems. We remark that in recent years, as ourknowledge of the global dynamics of nonlinear systems has increased, it is beginning

to appear possible that even CO conjugacies may be too strong for distinguishingthe important dynamical features in different dynamical systems. This is evidenced

by the great difficulties encountered in trying to ascertain generic properties andstructural stability criteria for higher dimensional systems (i.e., n dimensional maps,

n > 2 and n dimensional vector fields, n > 3) using the classical concepts (i.e.,CO conjugacies). This could be caused by the fact that CO conjugacies are rela-tions between specific orbits, and much of the complicated and chaotic phenomena

that occur in higher dimensional dynamical systems arise via interactions amongst

families of orbits (note: we will see many examples of this in Chapters 3 and 4).

We are now at the point where we can formally define structural stability.

Definition 1.4.2. Consider a map f E Diffr(M,M) (resp. a Cr vector field inCr(M,M) ); then f is said to be structurally stable if there exists a neighborhood,N, of f in the Ck-topology such that f is CO conjugate (resp. CO equivalent) toevery map (resp. vector field) in N.

Now that we have defined structural stability, it would be nice if we coulddetermine the characteristics of a specific system which result in the system being

structurally stable. From the point of view of the applied scientist, this would beuseful, since one might presume that dynamical systems modelling phenomena oc-


curring in nature should possess the property of structural stability. Unfortunately,

such a characterization does not exist, although some partial results are knownwhich we will describe shortly. One approach to the characterization of structural

stability has been through the identification of typical or generic properties of dy-

namical systems, and we now discuss this idea.

Naively, one might expect a typical or generic property of a dynamical system

to be one that is common to a dense set of dynamical systems in Cr(M,M). Thisis not quite adequate, since it is possible for a set and its complement to both bedense. For example, the set of rational numbers is dense in the real line, and so is its

complement, the set of irrational numbers; however, there are many more irrational

numbers than rational numbers, and one might expect the irrationals to be moretypical than the rationals in some sense. The proper sense in which this is true iscaptured by the idea of a residual set.

Definition 1.4.3. Let X be a topological space and let U be a subset of X. U iscalled a residual set if it is the intersection of a countable number of sets each ofwhich are open and dense in X. If a residual set in X is itself dense in X then Xis called a Baire space.

We remark that Cr (M, M) equipped with the Ck topology (k < r) is a Bairespace (see Palis and deMelo [1982]). We now give the definition of a generic property.

Definition 1.4.4. A property of a map (resp. vector field) is said to be Ck genericif the set of maps (resp. vector fields) possessing that property contains a residual

subset in the Ck topology.

An example of some important generic properties are listed in the followingtheorem due to Kupka and Smale.

Theorem 1.4.1. Let N be the set of diffeomorphisms of M, where M has di-mension > 2, such that all fixed points and periodic orbits of elements of N arehyperbolic and the stable and unstable manifolds of each fixed point and periodic

orbit intersect transversely. Then N is a residual set.

For a proof of the Kupka-Smale theorem, see Palis and deMelo [1982].

In utilizing the idea of a generic property to characterize the structurally stable

systems, one first identifies some generic property. Then, since a structurally stable

system is Co conjugate (resp. equivalent for vector fields) to all nearby systems,

1.4. Transversality, Structural Stability, ar.d Gerericity 61

the structurally stable systems must have this property if the property is one thatis preserved under CO conjugacy (resp. equivalence for vector fields). Now, one

would like to go the other way with this argument; namely, it would be nice toshow that structurally stable systems are generic. For two dimensional vector fields

on compact manifolds we have the following result due to Peixoto [1962].

Theorem 1.4.2. A Cr vector field on a compact boundaryless two dimensionalmanifold M is structurally stable if and only if

1) The number of fixed points and periodic orbits is finite and each is hyperbolic.

2) There are no orbits connecting saddle points.

3) The nonwandering set consists of fixed points and periodic orbits.

Moreover, if M is orientable, then the set of such vector fields is open and dense in

CT(M) (note: this is stronger than generic).

This theorem is useful because it spells out precise conditions on the dynam-

ics of a vector field on a compact boundaryless two manifold under which it isstructurally stable. Unfortunately, we do not have a similar theorem in higher di-mensions. This is in part due to the presence of complicated recurrent motions (e.g.,

the Smale horseshoe, see Section 2.1) which are not possible for two dimensional

vector fields. Even more disappointing is the fact that structural stability is not ageneric property for n dimensional diffeomorphisms (n > 2) or n dimensional vector

fields (n > 3). This fact was first demonstrated by Smale [1966].

At this point we will conclude our brief discussion of the ideas of transversality,

structural stability, and genericity. For more information, we refer the reader toChillingworth [1976], Hirsch [1976], Arnold [1982], Nitecki [1971], Smale [1967],

and Shub [1987]. However, before ending this section, we want to make somebrief comments concerning the relevance of these ideas to the applied scientist, i.e.,

someone who has a specific dynamical system and must somehow discover what

types of dynamics are present in that system.

Genericity and structural stability as defined above have been guiding forcesbehind much of the development of dynamical systems theory. The approach often

taken has been to postulate some "reasonable" form of dynamics for a certain class

of dynamical systems and then to prove that this form of dynamics is structurally

stable and/or generic within this class. If one is persistent and occasionally success-

ful in this approach, eventually a significant catalogue of generic and structurally


stable dynamical properties is obtained. This catalogue is useful to the applied sci-

entist in that it gives him or her some idea of what dynamics to expect in a specific

dynamical system. However, this is hardly adequate. Given a specific dynamical

system, is it structurally stable and/or generic? If this question could be answered,

then very general and powerful theorems such as the Kupka-Smale theorem could

be invoked, resulting in far-reaching conclusions concerning the dynamics of the

system in question. So we would like to give computable conditions under which a

specific dynamical system is structurally stable and/or generic. For certain special

types of motions such as periodic orbits and fixed points this can be done in terms of

the eigenvalues of the linearized system. However, for more general global motions

such as homoclinic orbits and quasiperiodic orbits, this cannot be done so easily,since the nearby orbit structure may be exceedingly complicated and defy any local

description (see Chapter 3). What this boils down to is that, in order to determinewhether or not a specific dynamical system is structurally stable, one needs a fairly

complete understanding of its orbit structure or, to put it more cynically, one needs

to know the answer before asking the question. It might therefore seem that these

ideas are of little use to the applied scientist; however, this is not exactly true,since the theorems describing structural stability and generic properties do giveone a good idea of what to expect, although they cannot tell one what is precisely

happening in a specific system.

1.5. BifurcationsThe term bifurcation is broadly used to describe significant qualitative changes that

occur in the orbit structure of a dynamical system as the parameters on which the

dynamical system depends are varied. In this section, we want to describe some of

the ideas behind bifurcation theory, beginning with the general framework of the

theory and then addressing various special situations.

Let us consider the infinite dimensional space of dynamical systems, eithervector fields or diffeomorphisms. The set of structurally stable dynamical systems

forms an open set S in this infinite dimensional space. The complement of S,denoted S', is defined to be the bifurcation set. We would like to describe thestructure of the bifurcation set Sc; to begin with, we would like to show that Sc is a

codimension one submanifold or, more generally, a stratified subvariety (see Arnold


[1983]) in the infinite dimensional space of dynamical systems. In order to motivate

this we must first make a slight digression and explain the term "codimension."

Let M be an m dimensional manifold and let N be an n dimensional subman-

ifold contained in M. Then the codimension of N is defined to be m - n. Thus,the codimension of a submanifold is a measure of the avoidability of the subman-

ifold as one moves about the ambient space; in particular, the codimension of asubmanifold N is equal to the minimum dimension of a submanifold P C M that

intersects N such that the intersection is transversal. This defines codimension in

a finite dimensional setting and allows one to gain some intuition. Now we move

to the infinite dimensional setting. Let M be an infinite dimensional manifold and

let N be a submanifold contained in M. (Note: for the definition of an infinitedimensional manifold see Hirsch [1976]. Roughly speaking, an infinite dimensional

manifold is a set which is locally diffeomorphic to an infinite dimensional Banach

space. Since we only mention infinite dimensional manifolds in this section, and

mainly in a heuristic fashion, we refer the reader to the literature for the properdefinitions.) We say that N is of codimension k if every point of N is contained insome open set in M which is diffeomorphic to U x Rk where U is an open set inN. This implies that k is the smallest dimension of a submanifold P C M thatintersects N such that the intersection is transversal. Thus, the definition of codi-mension in the infinite dimensional case has the same geometrical connotations as

in the finite dimensional case. Now we return to our main discussion.

Suppose S' is a codimension one submanifold or, more generally, a stratified

subvariety. We might think of S° as a surface dividing the infinite dimensional space

of dynamical systems as depicted in Figure 1.5.1. Bifurcations (i.e., topologicallydistinct orbit structures) occur as one passes through Sc. Thus, in the infinitedimensional space of dynamical systems, one might define a bifurcation point as

being any dynamical system which is structurally unstable.

Now in this setting one might initially conclude that bifurcations seldom occur

and are unimportant, since any point p on Sc may be perturbed to S by (most)arbitrarily small perturbations. Also, from a practical point of view, dynamical sys-

tems contained in Sc are probably not very good models for physical systems, since

any model is only an approximation to reality and, therefore, we should require our

model to be structurally stable. However, suppose we have a curve ry of dynamical

systems transverse to S°, i.e., a one parameter family of dynamical systems. Then


Figure 1.5.1. The Bifurcation Surface S° Contained in S.

any sufficiently small perturbation of this curve ry of systems still results in a curve

y' transverse to S So even though any particular point on Sc may be removedfrom Sc by (most) arbitrarily small perturbations, a curve transverse to S' remains

transverse to Sc under perturbation. Thus, bifurcation may be unavoidable in aparametrized family of dynamical systems.

Although it may be possible to show that Sc is a codimension one submanifold

or stratified subvariety, the detailed structure of S' may be quite complicated, forit may be divided up into submanifolds of higher codimension corresponding tomore degenerate forms of bifurcations. Then a particular type of codimension kbifurcation in SC would be persistent in a k parameter family of dynamical systems

transverse to the codimension k submanifold.

This is essentially the program for bifurcation theory originally outlined byPoincare. In order to utilize it in practice one would proceed as follows:

1) Given a specific dynamical system, determine whether or not it is structurally

stable.

2) If it is not structurally stable, compute the codimension of the bifurcation.

3) Embed the system in a parametrized family of systems transverse to the bifur-

cation surface with the number of parameters equal to the codimension of the


bifurcation. These parametrized systems are called unfoldings or deformations

and, if they contain all possible qualitative dynamics that can occur near the

bifurcation, they are called universal unfoldings or versal deformations, see

Arnold [1982].

4) Study the dynamics of the parametrized systems.

In this way one obtains structurally stable families of systems. Moreover,

this provides a method for gaining a complete understanding of the qualitativedynamics of the space of dynamical systems with as little work as possible. Namely,

one uses the degenerate bifurcation points as "organizing centers" around whichone studies the dynamics. Since elsewhere the dynamical systems are structurally

stable, one need not worry about the details of their dynamics, since qualitativelythey will be topologically conjugate to the structurally stable dynamical systems in

a neighborhood of the bifurcation point.

Now this program for the development of bifurcation theory is far from com-plete, and the obstacles preventing its completion are exactly those discussed atthe end of Section 1.4; namely, in the space of dynamical systems, one must first

identify S and S°, and this involves a detailed knowledge of the orbit structure of

each element in the space of dynamical systems. Although the situation appearshopeless, some progress has been made along two fronts:

1) Local Bifurcations.

2) Global Bifurcations of Specific Orbits.

We will discuss each of these situations separately.

Local bifurcation theory is concerned with the bifurcation of fixed points ofvector fields and maps, or in situations where the problem can be cast into thisform, such as in the study of bifurcations of periodic motions; for vector fieldsone can construct a local Poincare map (see Section 1.6) near the periodic orbit,thus reducing the problem to one of studying the bifurcation of a fixed point ofa map, and for maps with a k periodic orbit one can consider the kth iterate ofthe map thus reducing the problem to one of studying the bifurcation of a fixedpoint of the kth iterate of the map (see Section 1.1h). Utilizing a procedure such as

the center manifold theorem (see Carr [1981] or Guckenheimer and Holmes [1983])

or the Lyapunov-Schmidt reduction (see Chow and Hale [1982]), one can usually


reduce the problem to that of studying an equation of the form

f (x, A) = 0 (1.5.1)

where x E 1R't, A E ]RP are the system parameters, and f : ]R." x RP -> Rrz is

assumed to be sufficiently smooth. The goal is to study the nature of the solutions

of (1.5.1) as A varies. In particular, it would be interesting to know for whatparameter values solutions disappear or are created. These particular parameters

are called bifurcation values. Now there exists an extensive mathematical machinery

called singularity theory (see Golubitsky and Guillemin 11973]) which deals with

such questions. Singularity theory is concerned with the local properties of smooth

functions near a zero of the function. It provides a classification of the various

cases based on codimension in a spirit similar to that described in the beginningof this section. The reason this is possible is that the codimension k submanifolds

in the space of all smooth functions having zeros can be described algebraically by

imposing conditions on derivatives of the functions. This gives us a way of classifying

the various possible bifurcations and of computing the proper unfoldings. From this

one might be led to believe that local bifurcation theory is a well-understood subject;

however, this is not the case. The problem arises in the study of degenerate local

bifurcations, specifically, in codimension k (k > 2) bifurcations of vector fields.Fundamental work of Takens [1974], Langford [1979], and Guckenheimer [1981]has shown that arbitrarily near these degenerate bifurcation points complicatedglobal dynamical phenomena such as invariant tori and Smale horseshoes may arise.

These phenomena cannot be described or detected via singularity theory techniques.

We refer the reader to Chapter 7 of Guckenheimer and Holmes [1983] for a more

thorough discussion of these issues.

Global bifurcations will be defined to be bifurcations which are not local inthe sense described above, i.e., a qualitative change in the orbit structure of anextended region of phase space. Typical examples are homoclinic and heteroclinic

bifurcations. In both of these examples the complete story is far from known,mainly because techniques for the global analysis of the orbit structure of dynamical

systems are just now beginning to be developed. In Chapter 3 we will present agreat deal of what is known at this point regarding homoclinic and heteroclinicbifurcations and comment also on the large gaps in our knowledge, and in Chapter 4

we will develop a variety of analytical techniques suitable for dealing with these


situations.

1.6. Poincare Maps

The idea of reducing the study of continuous time systems (flows) to the study of an

associated discrete time system (map) is due to Poincare [1899], who first utilized it

in his studies of the three body problem in celestial mechanics. Nowadays virtually

any discrete time system which is associated to an ordinary differential equationis referred to as a Poincare map. This technique offers several advantages in thestudy of ordinary differential equations, three of which are the following:

1) Dimensional Reduction. Construction of the Poincare map involves the elim-

ination of at least one of the variables of the problem resulting in a lowerdimensional problem to be studied.

2) Global Dynamics. In lower dimensional problems (say dimension < 4) numeri-

cally computed Poincare maps provide an insightful and striking display of the

global dynamics of a system, see Guckenheimer and Holmes [1983] and Licht-

enberg and Lieberman [1982] for examples of numerically computed Poincare

maps.

3) Conceptual Clarity. Many concepts that are somewhat cumbersome to state for

ordinary differential equations may often be succinctly stated for the associated

Poincare map. An example would be the notion of orbital stability of a periodic

orbit of an ordinary differential equation (see Hale [1980]). In terms of thePoincare map, this problem would reduce to the problem of the stability of afixed point of the map which is simply characterized in terms of the eigenvalues

of the map linearized about the fixed point (see Case 1 to follow in this section).

It would be useful to give methods for constructing the Poincare map associ-

ated with an ordinary differential equation. Unfortunately, there exist no generalmethods applicable to arbitrary ordinary differential equations, since construction

of the Poincare map of an ordinary differential equation requires some knowledge of

the geometrical structure of the phase space of the ordinary differential equation.

So constructing a Poincare map requires ingenuity specific to the problem at hand;

however, in four cases which come up frequently, the construction of a specific type

of Poincare map can in some sense be said to be canonical. The four cases are:


1) In the study of the orbit structure near a periodic orbit of an ordinary differ-ential equation.

2) In the case where the phase space of an ordinary differential equation is periodic

such as in periodically forced oscillators.

3) In the case where the phase space of an ordinary differential equation is quasi-

periodic such as in quasiperiodically forced oscillators.

4) In the study of the orbit structure near a homoclinic or heteroclinic orbit.

We will discuss Cases 1, 2 and 3 now; all of Chapter 3 is devoted to Case 4.

Case 1. Consider the following ordinary differential equation

i = f (x) , x E 1R'i (1.6.1)

where f: U -- 1R'L is Cr on some open set U C R. Let q(t, ) denote theflow generated by (1.6.1). Suppose that (1.6.1) has a periodic solution of periodT which we denote by 0(t, x0), where xp E 1R' is any point through whichthis periodic solution passes (i.e., O(t + T,.0) = cS(t, xo)). Let E be an n - 1dimensional surface transverse to the vector field at xp (note: "transverse" meansthat f(x) n(xp) # 0 where denotes the vector dot product and n(xp) is thenormal to E at x0); we refer to E as a cross-section to the vector field (1.6.1). Now

in Theorem 1.1.2 we proved that q(t, x) is Cr if f (x) is C'; thus, we can find anopen set V C E such that trajectories starting in V return to E in a time close toT. The map which associates points in V with their points of first return to E iscalled the Poincare map which we denote by P. To be more precise,

P:V --+E

X H qS(r(x), x)(1.6.2)

where r (x) is the time of first return of the point x to E. Note that by constructionwe have r(xp) = T and P(xp) = xp.

So a fixed point of P corresponds to a periodic orbit of (1.6.1) and a periodk point of P (i.e., a point x c V such that Pk(x) = x provided P'(x) E V,i = 1,... , k) corresponds to a periodic orbit of (1.6.1) which pierces E k timesbefore closing, see Figure 1.6.1.

A question that arises is how does the Poincare map change if the cross-section

E is changed. Let xp and x1 be two points on the periodic solution of (1.6.1), and


Figure 1.6.1. The Geometry of the Poincare Map.

let EO and E1 be two n - 1 dimensional surfaces at xp and z1, respectively, whichare transverse to the vector field, and suppose that E1 is chosen such that it is the

image of EO under the flow generated by (1.6.1) see Figure 1.6.2. This defines a Cr

diffeomorphism

h:Eo-+E1.

Figure 1.6.2. The Cross-sections EO and E1.

We define Poincare maps PO and P1 as in the previous construction.

(1.6.3)

Po: Vo -* EO


x0 0(r(x0),x0) , x0 E VO C EO (1.6.4)

P1:V1-'E1

xl H c5(r(xl),xl) , x1 E V1 C E1 . (1.6.5)

Then we have the following result.

Proposition 1.6.1. P0 and Pl are locally Cr conjugate.

PROOF: We need to show that

P1 o h = h o P0

from which the result is immediate since h is a Cr diffeomorphism. However, weneed to worry a bit about the domains of the maps. We have

h(Eo) = E1

P0(V0) CEO (1.6.6)

P1(V1) C E1 .

So h o P0: V0 -+ E1 is well-defined; but P1 o h need not be defined since P1 isnot defined on all of E1; however, this problem is solved if we choose E1 such that

E1 = h(VO) rather than h(E0) and the result follows.

Case 2. Consider the following ordinary differential equation

i= f(x,t), xER'L (1.6.7)

where f : U -+ R' is C' on some open set U C R" x Rl. Suppose the timedependence of (1.6.7) is periodic with fixed period T = 2 > 0, i.e., f (x, t) _f (x,t + T). We rewrite (1.6.7) in the form of an autonomous equation in n + 1dimensions (see Section 1.1e) by defining the function

B:R1-+S1,(1.6.8)

t F--, B(t) wt , mod2ir..

Using (1.6.8) the equation (1.6.7) becomes

x = f(x,0)(x01ERnxS1. (1.6.9)

B=w


We denote the flow generated by (1.6.9) by qS(t) = (x(t), B(t) = wt+Bo

We define a cross-section Ego to the vector field (1.6.9) by

(mod 21r)).

Ego={(x,o)ER- x S' 0=O E(0,2,r]}. (1.6.10)

The unit normal to Ego in Rn x S1 is given by the vector (0,1), and it is clear that

Ego is transverse to the vector field (1.6.9) for all x E R76, since (f (x, 0), w) (0, 1) _

w # 0. In this case Ego is called a global cross-section.

We define the Poincare map of Ego into itself as follows:

or

PB0 : Ego _, Ego

W Bo)Bo) H \x

(60 0+ 2ir = Bo)

x(Bo - Bo) H x(Bo - Bo + 27r)w w J

(1.6.11)

Thus, the Poincare map merely tracks initial conditions in x at a fixed phase after

successive periods of the vector field.

It should be clear that fixed points of Peo correspond to 27r/w-periodic orbitsof (1.6.9) and k-periodic points of Peo correspond to periodic orbits of (1.6.9) which

pierce Ego k times before closing.

As in Case 1, suppose we construct a different Poincare map Poi as above but

with cross-section

E91={(x,0)ERnxS11°=81E(0,21r1} . (1.6.12)

Then we have the following result.

Proposition 1.6.2. Peo and Pe, are C' conjugate.

PROOF: The proof follows a construction similar to that given in Proposition 1.6.1.

We construct a C' diffeomorphism, h of Ego into E61 by mapping points on Ego

into Eg1 under the action of the flow generated by (1.6.9). Points starting on Egohave initial time to = (Bo - 00)1w, and they reach Egl after time

t-B1 - 00

thus we have

h: Ego -4 E81

lx(Bow- e0 ),#0) F--, (x (B1 e0),B1)

(1.6.13)


Using (1.6.13) and the expressions for the Poincare maps defined on the differentcross-sections we obtain

hoP-o:E90 - - + E8'

(x(60_90)B0/ H

(x(91-6`0+27r)' 1+2x=e1)

and

l o h: Eeo E6iPg

(x\6pcv60/ 60) H

Thus from (1.6.14) and (1.6.15) we have that

ho P- =P-loh.

(1.6.14)

(1.6.15)

(1.6.16)

Case S. Consider the following ordinary differential equation

x= f(x,t),xER'b (1.6.17)

where f : U --' R' is Cr on some open set U C R' x R1. We assume that forfixed x, f (x, t) is a quasiperiodic function of time. Recalling the definition of aquasiperiodic function given in Section 1.1i, (1.6.17) can be written as

x = f(x,B1,...,9m)

91 =W1(x, 01, ... , Omm) E Rn X S1.. XS, . (1.6.18)

m factors

We denote the flow generated by (1.6.18) by 4i(t) = (x(t),w1t+610, ,Wmt+6m0) .

In analogy with Case 2, we construct a cross-section to the vector field (1.6.18)

by fixing the phase of one of the angular variables. To be more precise, the globalcross-section E93 0 is defined as

Ee,O_{(x,6...... em)ER'LxS1X...xS1I 9j=ej0E(0,27r)}. (1.6.19)


(Note: the fact that (1.6.19) is a global cross-section to (1.6.18) follows by anargument similar to that given in Case 2.) Using the flow generated by (1.6.18), the

Poincare map, Pej 0, is constructed by choosing an initial time to = (Bj 0-0j 0) 1-j

such that solutions of (1.6.14) start with Bj = Bj 0 and evolve for time t = to+W 7,

thus returning to Eej 0. To be more precise, we have

P- : Egj O -> Eej 0 (1.6.20)gjo

(x(Bj0 Bj0wl+Bm01I\\

wj wjll wj

)w ,i(ej0-Bj0+2rr)+010, ej0+27r-ej0,

w.. w (Bjp-0jo+2ir +9mo .

7 \

We remark that changing the cross-section by changing the angle corresponding to

a fixed frequency (i.e., Egj 0 and EgiO) results in two Poincare maps defined onthe respective cross-sections which are Cr conjugate. However, changing the cross-

section by changing angles which correspond to different frequencies (i.e., EeiO and

Egk0) results in two Poincare maps defined on the respective cross-sections which

are Cr conjugate only if the frequencies wj and wk are commensurate.

Before concluding our discussion on Poincare maps we want to address animportant issue of a more general nature. In Cases 1, 2, and 3 the Poincare mapswere all constructed by considering a portion of the phase space and allowing it to

evolve in time under the action of the flow generated by the vector field. The region

of the phase space and the "time of flight" were not chosen arbitrarily but in such a

manner that the dynamics of the resulting Poincare map could be directly relatedto the dynamics of the flow. In these three cases, this was accomplished by choosing

a portion of the phase space which was mapped back onto itself (or at least nearitself) after a certain amount of time. The ability to make this choice depended on

our knowledge of certain recurrent properties in the dynamics of the vector field(e.g., a periodic orbit, periodic or quasiperiodic phase space, or, as we shall see,a homoclinic or heteroclinic orbit). There is a property which is common to allsuch "flow maps" which is quite useful in making certain global arguments; namely,


Poincare maps constructed as discrete time flow maps from the flow generated by

an ordinary differential equation are orientation preserving (note: recall that amap f : U --> V, U, V open sets in IRh, is said to be orientation preserving if thedeterminant of D f (denoted det D f) is positive in U). We now want to give aproof of this fact. The set up is as follows.


i= f(x), xEIR'i (1.6.21)

where f : U - 1R'i is Cr on some open set U C R'. Let 0(t, x) denote the flowgenerated by (1.6.21), and we assume that it exists for a sufficiently long time on

some set V C U C 1R°. Consider the map

P:V -* R' (1.6.22)

x -4 q5(r, x)

where r is some fixed real number which may depend on x. Then we have thefollowing result.

Proposition 1.6.3. The determinant of DP - Dxcb(r,x) is positive on V; henceP is orientation preserving on V.

PROOF: We have 0(0, x) = x and, therefore, Dxq(0, x) = id where id denotesthe n x n identity matrix. It follows that det Dxq5(0, x) = 1.

Now from Theorem 1.1.3 Dxo(t, x) is a solution of the linear matrix equation

,z = Dx f (O(t, x)) z (1.6.23)

where we regard the x in the argument of q5 as fixed. Utilizing the formula for the

determinant of the fundamental solution matrix of a linear system based on theknowledge of the determinant at a fixed time (see Hale [1980], Chapter 3, Lemma1.5) we see that

[!trDzf((tx))dtdet Dx(r, x) = det Dz (0, x) exp0

= exp

0

[!trDzf((tx))dt]

(1.6.24)

Since (1.6.24) holds for each x E V, it follows that det DP = det Dxq(r, x) > 0 foreach x E V.

CHAPTER 2Chaos: Its Descriptionsand Conditions for Existence

In this chapter we will discuss and derive sufficient conditions for a dynamicalsystem to exhibit complicated dynamical, or chaotic, behavior. We will also discuss

a characterization of this behavior in terms of symbolic dynamics.

We begin in Section 2.1 with a discussion of the two dimensional, piecewise

linear, Smale horseshoe map, which is the prototypical chaotic dynamical system.

We will use this specific example to introduce many techniques and concepts such as

symbolic dynamics, sensitive dependence on initial conditions, and chaos that will

appear in a broader context later on. We believe that a complete understanding ofthis example is absolutely essential in order to understand the meaning of the term"chaos" as it applies to deterministic dynamical systems.

In Section 2.2 we will discuss separately the subject of symbolic dynamics, since

it will play a crucial role in most of our examples in Chapter 3. We will derive many

of the topological properties of the space of symbol sequences as well as discuss the

dynamics of the shift map and the subshift of finite type which acts on the space of

symbol sequences.

In Section 2.3 we will give sufficient conditions in order for a map to possess

an invariant set of points on which the dynamics can be described via the tech-niques of symbolic dynamics. Among other things, these conditions will require

that there be a uniform splitting of the domain of the map into strongly expanding

and contracting directions; this results in the invariant set being an invariant set of

points.

In Section 2.4 we will weaken the conditions of Section 2.3 in order to allow

the map to possess directions which exhibit neutral growth. This will result in theinvariant set not being an invariant set of points, but rather an invariant set of

76 2. Chaos: Its Descriptions and Conditions for Existence

surfaces or, more precisely, the Cartesian product of a Cantor set and a surface.

The dynamics of the map on this invariant set will still admit a description via the

techniques of symbolic dynamics.

2.1. The Smale Horseshoe

In this section we will describe a two dimensional map possessing an invariant set

having a delightfully complicated structure. Our map is a simplified version of a

map first studied by Smale [1963], [1980] and, due to the shape of the image of the

domain of the map, is called a Smale horseshoe. At this stage of our understanding

of chaotic dynamics it is safe to call the Smale horseshoe the prototypical mappossessing a chaotic invariant set (note: the phrase "chaotic invariant set" willbe precisely defined later on in the discussion). Therefore, we feel that a thorough

understanding of the Smale horseshoe is absolutely essential for understanding what

is meant by the term "chaos" as it is applied to the dynamics of specific physicalsystems. For this reason we will first endeavor to define as simple a two dimensional

map as possible that contains the necessary ingredients for possessing a complicated

and chaotic dynamical structure so that the reader may get a feel for what isgoing on in the map with a minimum of distractions. As a result, our constructionmay not appeal to those interested in applications, since it may appear ratherartificial. However, following our discussion of the simplified Smale horseshoe map,

we will give sufficient conditions for the existence of Smale horseshoe-like dynamics

in n-dimensional maps which are of a very general nature. We will begin by defining

the map and then proceed to a geometrical construction of the invariant set of the

map. We will utilize the nature of the geometrical construction in such a way as to

motivate a description of the dynamics of the map on its invariant set by symbolic

dynamics, following which we will make precise the idea of chaotic dynamics.

2.1a. Definition of the Smale horseshoe map

We will give a combination geometrical-analytical definition of the map. Consider

a map, f, from the square having sides of unit length into R2

f:D-+R2, D=S(x,y)ER2I0<x<1, 0<y<1} (2.1.1)


Lx

Contraction andexpansion

H1

Hi

Ho

Hp

Folding

f

f(Hf(HO)0 A 1-X1

Figure 2.1.1. The Action of f on D.

11

which contracts the x-direction, expands the y-direction, and twists D around,laying it back on itself as shown in Figure 2.1.1.

We will assume that f acts affinely on the "horizontal" rectangles

HO ={(x,y)ER210<x<1,0<y<1/µ} (2.1.2a)

and

H1={(x,y)ER2I0<x<1,1-1Iµ<y<1} (2.1.2b)

taking them to the "vertical" rectangles

f(HO) - Vo= {(x,y) ER210<x<A,0<y<1} (2.1.3)

and

f(H1) -V1={(x,y)ER211-A<x<1,0<y<1} (2.1.4)

with the form of f on HO and H1 given by

HO \y/ ~ \0 µl \y/(2.1.5)

H1 \y/ 0 µJ \y/ + \/A/

with 0 < A < 1/2, µ > 2 (note: the fact that, on H1, the matrix elements arenegative means that, in addition to being contracted in the x-direction by a factor A


and expanded in the y-direction by a factor /L, H1 is also rotated 180°). Addition-

ally, it follows that f -1 acts on D as shown in Figure 2.1.2, taking the "vertical"rectangles Vo and V1 to the "horizontal" rectangles Ho and H1, respectively (note:

by "vertical rectangle" we will mean a rectangle in D whose sides parallel to the y

axis each have length one, and by "horizontal rectangle" we will mean a rectangle in

D whose sides parallel to the x axis each have length one). This serves to define f;however, before proceeding to study the dynamics of f on D, there is a consequence

of the definition of f which we want to single out, since it will be very importantlater.

Y

Lx

f-1

Vp V,Vp

0 X 1-k 1Expansion and contraction

Figure 2.1.2. The Action of f -1 on D.

Folding

Lemma 2.1.1. a) Suppose V is a vertical rectangle; then f (V) n D consists ofprecisely two vertical rectangles, one in Vo and one in V1, with their widths eachbeing equal to a factor of A times the width of V. b) Suppose H is a horizontalrectangle; then f -1(H) fl D consists of precisely two horizontal rectangles, one inHo and one in H1, with their widths being a factor of 1/p times the width of H.

PROOF: We will prove case a). Note that from the definition of f the horizontaland vertical boundaries of Ho and Hl are mapped to the horizontal and verticalboundaries of V0 and V1. So let V be a vertical rectangle. Then V intersects thehorizontal boundaries of Ho and Hl; hence, f (V) fl D consists of two verticalrectangles, one in Ho and one in H1. The contraction of the width follows from the


form off on Hp and H1, which indicates that the x-direction is contracted uniformly

by a factor A on Hp and H1. Case b) is proved similarly. See Figure 2.1.3.

V

(a)

H f-t(H)--

(b)

Figure 2.1.3. (a) Geometry of Lemma 2.1.1a. (b) Geometry of Lemma 2.1.1b.

2.1b. Construction of the Invariant Set

We now will geometrically construct the set of points, A , which remain in D under

all possible iterations by f ; thus A is defined as

nf-n(D)n...nf-1(D)nDnf(D)n...nfn(D)n...00

or n fn(D) . (2.1.6)n=-oo

We will construct this set inductively, and it will be convenient to construct sepa-

rately the "halves" of A corresponding to the positive iterates and the negativeiterates, and then take their intersections to obtain A. Before proceeding with theconstruction, we need some notation in order to keep track of the iterates of f ateach step of the inductive process. Let S = {O, 1} be an index set, and let sidenote one of the two elements of S, i.e., si E S, i = 0,±1,±2,... (note: thereason for this notation will become apparent later on).


00 n=kWe will construct n f n(D) by constructing n f' (D) and then deter-

n=0 n=0mining the nature of the limit as k -+ oo.

D n f (D): By the definition of f, D n f (D) consists of the two vertical rectanglesVO and V1, which we denote as follows:

D n f(D) = UVs_1= {p E D I p E Vs_1, s_l E S} (2.1.7)s_1ES

where Vs_1 is a vertical rectangle of width A. See Figure 2.1.4.

H1

f

Hp

Figure 2.1.4. D n f (D).

D n f (D) n f 2 (D): It is easy to see that this set is obtained by acting on D nf (D) with f and taking the intersection with D, since Dn f (Dn f (D)) = Dn f (D)n

f 2(D). Thus, by Lemma 2.1.1, since Dn f (D) consists of the vertical rectangles VO

and V1 with each intersecting HO and H1 and their respective horizontal boundaries

in two components, then D n f (D) n f 2 (D) corresponds to four vertical rectangles,

two each in VO and V1, with each of width A2. We label this set as follows:

D n f(D) n f2(D) =U (f (Vs_2) n Vs_1) = UVs_ls_2 (2.1.8)a_iES a_iESi=1,2 i=1,2

={pEDIpEV3_1,f-1(p)EV3_2,s_{ES,i=1,2}.

Pictorially, this set is described in Figure 2.1.5.

D n f (D) n f 2 (D) n f 3(D): Using the same reasoning as in the previous steps, this

set consists of eight vertical rectangles, each having width A3, which we denote as


f

Figure 2.1.5. D n f (D) n f2(D).

follows:

D n f (D) n f2(D) n f3(D) = U (f (Vs_28-31 n V3_1) = U V3_13_23_3 =a_iES a_iESi=1,2,3 i=1,2,3

{p E Djp E V3-1, f-1(P) E V3_2, f-2(p) E V5_s, 8_4 E S, s= 1, 2, 3},(2.1.9)

and is represented pictorially in Figure 2.1.6.

V00

f

Figure 2.1.6. D n f (D) n f2(D) n f3(D).

If we continually repeat this procedure, we almost immediately encounter ex-treme difficulty in trying to represent this process pictorially as in Figures 2.1.4


through 2.1.6. However, using Lemma 2.1.1 and our labelling scheme developed

above, it is not hard to see that at the kth step we obtain

D n f(D) n ... n fk(D) = U (f (Vs_2...s_k) n V3_1) = U V9_1...s_ka_iES a_iES

={pEDIf i+l(p)EV3_i, s-iES, 1 =1,...,k}(2.1.10)

and that this set consists of 2k vertical rectangles, each of width Ak.

Before proceeding to discuss the limit as k --> oo, we want to make the following

important observation concerning the nature of this construction process. Note that

at the kth stage, we obtain 2k vertical rectangles, and that each vertical rectanglecan be labelled by a sequence of 0's and 1's of length k. The important point torealize is that there are 2k possible distinct sequences of 0's and 1's having length k

and that each of these is realized in our construction process; thus, the labelling of

each vertical rectangle is unique at each step. This fact follows from the geometric

definition of f and the fact that VO and V1 are disjoint.

Letting k --> oo, since a decreasing intersection of compact sets is nonempty, it

is clear that we obtain an infinite number of vertical rectangles and that the widthof each of these rectangles is zero, since lim Ak = 0 for 0 < A < 1/2. Thus, we

k--.oohave shown that

00

n f n(D) = U (f (V8_2...3_k...) n V,s_1) = U Vs_1...3_k (2.1.11)n=O a_iES a-iES

i=1,2,... i=1,2,...

_ {p E DI f-i+l(p) E V5_i, si E S, i = 1,2,...}

consists of an infinite number of vertical lines and that each line can be labelledby a unique infinite sequence of 0's and 1's (note: we will give a more detailed set

00theoretic description of n f n(D) later on).

n=0n=0

Next we will construct n f n(D) inductively.-00

D n f (D): From the definition of f , this set consists of the two horizontal rect-

angles HO and Hl and is denoted as follows:

Dnf-l(D)=U Hso={pEDIpEH30, so ES}. (2.1.12)so ES


Vp

See Figure 2.1.7.

f-i

Figure 2.1.7. D n f -1(D).

83

D n f -1(D) n f -2(D): We obtain this set from the previously constructed set,D n f -1(D), by acting on D n f -1(D) with f -1 and taking the intersectionwith D, since D n f-1 (Dn f-1(D)) =

D n f-1(D)n f-2(D). Also, by Lemma2.1.1, since Hp intersects both vertical boundaries of Vp and V1 as does H1, D n

f -1(D) n f -2 (D) consists of four horizontal rectangles, each of width l/µ2, andwe denote this set as follows:

D n f-1(D) o f-2(D) = U(f-1(Hsl) n H.,0) =UHsas1 (2.1.13)aiES aiESi=0,1 i-0,1

={pEDIpEHso, f(p) EHs1, siES, i=0,1}.

See Figure 2.1.8.

f-1

Figure 2.1.8. D n f-1(D) n f-2(D).


D n f -1(D) n f-2(D) n f -3(D): Using the same arguments as those given in theprevious steps, it is not hard to see that this set consists of eight horizontal rectangles

each having width l/µ3 and can be denoted as follows:

D n f-1(D) o f-2(D) o f-3(D) =U(f-1(Hsls2) nHso) =UHs0s1s2l eiES siES

i=0,1,2 i=0,1,2

={pEDIpEHso, f(p)EHs1,f2(p)EHs2, siES,i=0,1,2}.(2.1.14)

See Figure 2.1.9.

H100

f-1

H101

Hoot H000

Figure 2.1.9. D n f-1(D) n f-2(D) n f-3(D).

Continuing this procedure, at the kth step we obtain D n f -1(D) n . . . n f -k(D),which consists of 2k horizontal rectangles each having width 11µk and is denotedby

D n f-1(D) n ... o f-k(D) = U (f-1(Hsl sk_1) n H30) = U HsO---sk-1siES eiES

i=O,...,k- i=O,...,k-1

{pEDlf9'(p)EHsi, siES, i=0,...,k-1}.(2.1.15)

As in the case of vertical rectangles, we note the important fact that at the kthstep of the inductive process, each one of the 2k can be labelled uniquely with asequence of 0's and 1's of length k. Now, as we take the limit as k -+ oo, we


n=0arrive at n f n (D), which is an infinite set of horizontal lines, since a decreasing

00intersection of compact sets is nonempty and the width of each component of theintersection is given by lim (1µk) = 0, µ > 2. Each line is labelled by a uniquek00infinite sequence of 0's and 1's as follows:

0

n f n(D) = U (f (H91...sk...) n H30) = U Hso...sk_.. (2.1.16)n=-oo . ES siES

i=0,1.... i=0,1,...

={pEDIf'(p)EHsi, siES, i=0,1,...

Thus, we have

00 0

A = n fn(D) = [flfn(D)j n I00

n fn(D)n=-00 n=00 L n=0

(2.1.17)

00which consists of an infinite set of points, since each vertical line inn f n (D) in-

n=00

tersects each horizontal line in n f n(D) in a unique point. Furthermore,n=-oo

each point p E A can be labelled uniquely by a bi-infinite sequence of 0's and1's which is obtained by concatenating the sequences associated with the respec-

tive vertical and horizontal lines which serve to define p. Stated more precisely, lets_i . s-k . . be a particular infinite sequence of 0's and 1's; then Vs_1...3_k..

corresponds to a unique vertical line. Let s0 . sk ... likewise be a particular

infinite sequence of 0's and 1's; then Hs0...sk... corresponds to a unique horizontal

line. Now a horizontal line and vertical line intersect in a unique point p; thus, wehave a well-defined map from points p E A to bi-infinite sequences of 0's and 1's

which we call 0.0

Notice that since

(2.1.18)

VS _1...s_k...={pEDIf-t+i(p)EV3_i, i = 1, ...} (2.1.19)

={pEDIf_'(p)EH3_i, i= 1, ...} since f(Hsi)=Vsi

and

H3o...sk... _ { p E DI f1(p) E Hsi , i = 0, ... } (2.1.20)


we have

p=Vs_l...s_k...nHso sk ={pEDIft(p)EHsi, i = 0, f1, ±2,...}(2.1.21)

Therefore, we see that the unique sequence of 0's and l's which we have associated

with p contains information concerning the behavior of p under iteration by f.In particular, the skth element in the sequence associated with p indicates thatf k (p) E Hsk . Now, note that for the bi-infinite sequence of 0's and l's associated

with p, the decimal point separates the past iterates from the future iterates; thus,

the sequence of 0's and l's associated with f k(p) is obtained from the sequenceassociated with p merely by shifting the decimal point in the sequence associated

with p k places to the right if k is positive or k places to the left if k is negative,until sk is the symbol immediately to the right of the decimal point. We can define

a map of bi-infinite sequences of 0's and l's, called the shift map, a, which takesa sequence and shifts the decimal point one place to the right. Therefore, if weconsider a point p E A and its associated bi-infinite sequence of 0's and l's, qS(p),

we can take any iterate of p, f k(p), and we can immediately obtain its associatedbi-infinite sequence of 0's and l's given by ak (O(p)). So there is a direct relationship

between iterating any point p E A under f and iterating the sequence of 0's andl's associated with p under the shift map or.

Now at this point it is not clear where we are going with this analogy between

points in A and bi-infinite sequences of 0's and l's since, although the sequenceassociated with a given point p E A contains information on the entire future and

past as to whether or not it is in H0 or Hl for any given iterate, it is not hard toimagine different points, both contained in the same horizontal rectangle after any

given iteration, whose orbits are completely different. The fact that this cannothappen for our map and that the dynamics of f on A are completely modeled bythe dynamics of the shift map acting on sequences of 0's and l's is an amazing fact

which before we justify we must make a slight digression into symbolic dynamics.

2.1c. Symbolic Dynamics

Let S = {0, 1} be the set of nonnegative integers consisting of 0 and 1. Let E bethe collection of all bi-infinite sequences of elements of S, i.e., s E E implies

8 = {... s_n ... s-1.s0 ... S. } , sa E S V i. (2.1.22)


We will refer to E as the space of bi-infinite sequences of 2 symbols. We wish to

introduce some structure on E in the form of a metric, d( , ), which we do asfollows:

Consider

we define the distance between s and s

006 0 if Si = si

=where bi1ifsi si

(2.1.23)d(s, s) _i=-oo

Thus, two sequences are "close" if they agree on a long central block. (Note: thereader should check that d( , ) does indeed satisfy the properties of a metric. SeeDevaney [1986] for a proof.)

We consider a map of E into itself, which we shall call the shift map, a; it isdefined as follows:

For

C(s) = {... S_ ... sn ...} (2.1.24)

or, a(s)i = si+1. Next, we want to consider the dynamics of a on E (note: forour purposes the phrase "dynamics of a on E" refers to the orbits of points in Eunder iteration by a). It should be clear that or has precisely two fixed points,namely, the sequence whose elements are all zeros and the sequence whose elements

are all ones (notation: bi-infinite sequences which periodically repeat after some

fixed length will be denoted by the finite length sequence with an overbar, e.g.,{... 101010.101010...} is denoted by {10.10}).

In particular, it is easy to see that the orbits of sequences which periodically re-

peat are periodic under iteration by a. For example, consider the sequence {10.10}.

We have

a{10.10} _ {01.01} (2.1.25)

and

a{01.01} _ {10.10} . (2.1.26)

Thus

a2{10.10} _ {10.10}. (2.1.27)


Therefore, the orbit of {10.10} is an orbit of period two for a. So, from this

particular example, it is easy to see that for any fixed k, the orbits of a havingperiod k correspond to the orbits of sequences made up of periodically repeatingblocks of 0's and 1's with the blocks having length k. Thus, since for any fixed k

the number of sequences having a periodically repeating block of length k is finite,

we see that a has a countable infinity of periodic orbits having all possible periods.

We list the first few below.

Period 1 {1.1}

Period 2 : {01.01} - {10.10} - {01.01}

Period 3 {001. 001} {010.010} {100.100}(2 1 28)

{110.110} {101.101} - {011.011}. .

etc.

Also, a has an uncountable number of nonperiodic orbits. To show this, we needonly construct a non-periodic sequence and show that there are an uncountablenumber of such sequences. A proof of this fact goes as follows: we can easilyassociate an infinite sequence of 0's and 1's with a given bi-infinite sequence by the

following rule:

so s1 s-1 s2 s-2 .. . (2.1.29)

Now, we will take it as a known fact that the irrational numbers in the closed unitinterval [0, 1] constitute an uncountable set, and that every number in this interval

can be expressed in base 2 as a binary expansion of 0's and 1's with the irrationalnumbers corresponding to non-repeating sequences. Thus, we have a one-to-onecorrespondence between an uncountable set of points and non-repeating sequences

of 1's and 0's. As a result, the orbits of these sequences are the non-periodic orbits

of or, and there are an uncountable number of such orbits.

Another interesting fact concerning the dynamics of a on E is that there exists

an element, say s E E, whose orbit is dense in E, i.e., for any given s1 E Eand e > 0, there exists some integer n such that d (a'L(s), s') < c. This is easiestto see by constructing s directly. We do this by first constructing all possiblesequences of 0's and 1's having length 1, 2, 3, .... This process is well defined in a


set theoretic sense, since there are only a finite number of possibilities at each step

(more specifically, there are 2k distinct sequences of 0's and 1's of length k). The

first few of these sequences would be as follows:

length 1 : {0} , {1}

length 2 : {00} , {01}, {10} , {11}

length 3 : {000} , {001}, {010} , {011}, {100} , {101}, {110} , {111}

etc.

(2.1.30)

Now we can introduce an ordering on the collection of sequences of 0's and 1's inorder to keep track of the different sequences in the following way. Consider two

finite sequences of 0's and is

s={sl...sk}, s = Ig1 ... skr} . (2.1.31)

Then we say

s<s ifk<k' (2.1.32)

and if k = k's < s if Si < si , (2.1.33)

where i is the first integer such that si # si. For example, using this ordering wehave

{0} < {1} ,

{0} < {00},

{00} < {01}, etc.

(2.1.34)

This ordering gives us a systematic way of distinguishing different sequences that

have the same length. Thus, we will denote the sequences of 0's and 1's havinglength k as follows:

Si < ... <skk (2.1.35)

where the superscript refers to the length of the sequence and the subscript refers to

a particular sequence of length k which is uniquely specified by the above ordering

scheme. This will give us a systematic way of writing down our candidate for adense orbit.


Now consider the following sequence.:

s = { s8 s6 s4 s2 s4 s2 , s1 s1 s3 s1 s3 s5 s7 (2.1.36)

Thus, s contains all possible sequences of 0's and 1's of any fixed length. Now, in

order to show that the orbit of s is dense in E, we argue as follows: let s' be anarbitrary point in E and let E > 0 be given. An e-neighborhood of st consistsof all points sit E E such that d(s', sit) < e, where d is the metric given in(2.1.23). Therefore, by definition of the metric on E, there must be some integerN = N(E) such that s? = s2' , lil < N (note: a proof of this statement canbe found in Devaney [19861 or in Section 2.2). Now, by construction, the finitesequence {s' N s I _ 1 s0 . s 11 is contained somewhere in s; therefore, there

must be some integer N such that d (aN(s), s') < E, so we can conclude that theorbit of s is dense in E.

We summarize these facts concerning the dynamics of or on E in the following

theorem.

Theorem 2.1.2. The shift map a acting on the space of bi-infinite sequences of0's and 1's, E, has1) a countable infinity of periodic orbits of arbitrarily high period;

2) an uncountable infinity of non-periodic orbits; and3) a dense orbit.

2.1d. The Dynamics on the Invariant Set

Now we want to relate the dynamics of a on E, on which we have a great deal ofinformation, to the dynamics of the Smale horseshoe f on its invariant set A, ofwhich, at this point, we know little except for its complicated geometric structure.

Recall that we have shown the existence of a well-defined map 0 which associates

to each point, p E A, a bi-infinite sequence of 0's and 1's, gy(p). Furthermore, we

noted that the sequence associated with any iterate of p, say f k(p), can be foundmerely by shifting the decimal point in the sequence associated with p k places to

the right if k is positive or k places to the left if k is negative. In particular, therelation a o 0(p) = 0 o f (p) holds for every p e A. Now, if q5 were invertible and

continuous (continuity is necessary since f is continuous), the following relationship

would hold:

0-1 0 a 0 O(P) = f (p), d p E A. (2.1.37)


Thus, if the orbit of p E A under f is denoted by

{...fn(P), ..., f-1(P), P, f(P), ..., fn(P), ...}

since 0-1 o a o (k(p) = f (p), we /see that

f n(P) _ (0-1 o a o 0) o (0-1 o a o §6) ... o (0-1 o a o o(p))

=(k-1 no0(p)

91

(2.1.38)

(2.1.39)

Therefore, the orbit of p E A under f would correspond directly to the orbit of0(p) under a in E. In particular, the entire orbit structure of a on E would beidentical to the structure of f on A. So, in order to verify that this situation holdswe need to show that 0 is a homeomorphism of A and E.

Theorem 2.1.3. The map 0 : A -- E is a homeomorphism.

PROOF: We need only show that 0 is one-to-one, onto, and continuous, since conti-

nuity of the inverse will follow from the fact that one-to-one, onto, and continuous

maps from compact sets into Hausdorff spaces are homeomorphisms (see Dugundji

[1966]). We prove each condition separately.

is one-to-one : This means that given p, p' E A, if p 54 p', then 4(p) # q5(p').We give a proof by contradiction. Suppose

O(P) = O(P1) = (2.1.40)

Then, by construction of A, p and p' lie in the intersection of the vertical lineVS_l...s_n... and the horizontal line H30...3, .... However, the intersection of ahorizontal line and a vertical line consists of a unique point; therefore p = p',contradicting our original assumption. This contradiction is due to the fact that we

have assumed 0(p) = 0(p'); thus, for p# p', qS(p) # q5(p').

is onto: This means that given any bi-infinite sequence of 0's and 1's in E,say {. s-n ... 8-1-so sn ..}, there is a point p E A such that qS(p) _

00The proof goes as follows: Recall the construction of n f n(D) and

n=00n f n(D); given any infinite sequence of 0's and 1's, {.so . . sn . . .}, there

n=-oo


,XN

Figure 2.1.10. The Location of p and p'.

ccis a unique vertical line inn f' (D) corresponding to this sequence. Similarly,

n=0given any infinite sequence of 0's and 1's, { s_n s_1 .}, there is a unique

0horizontal line in n f n(D) corresponding to this sequence. Therefore, we see

n=-oothat for a given horizontal and vertical line, we can associate a unique bi-infinitesequence of 0's and 1's, { s_n s_1 .80 sn .. -} and since a horizontal and

vertical line intersect in a unique point, p, to every bi-infinite sequence of 0's and1's, there corresponds a unique point in A.

0 is continuous: This means that, given any point p E A and e > 0, we can finda 6 = 6 (e, p) such that

Ip - p'I < 6 implies d (q,(p), ¢(p )) < e (2.1.41)

where is the usual distance measurement in R2 and d(-, ) is the metric onE introduced earlier.

Let e > 0 be given; then, if we are to have d(q(p), 4(p')) < e, there must besome integer N = N(e) such that if

-0 (p) = {... s-n ... S-1-so ... an ...}(2.1.42)

'(PI) _ {... s' n ... sl 1 . s0 ... S' ...}

then si = sq, i = 0, ±1, ...,±N. Thus, by construction of A, p and p' lie inthe rectangle defined by Hso...sN n Vs_I...s_N, see Figure 2.1.10. Recall that the

width and height of this rectangle are AN and 11 N+1, respectively. Thus we have

1p - p'1 < (AN+ 1µN+1). Therefore, if we take 6 = AN+ 11µN+1, continuity isproved.


Remarks:

1) When 4' is a homeomorphism recall from Section 1.2 that the dynamical systems

f acting on A and a acting on E are said to be topologically conjugate if 0 of (p) = a o qS(p). (Note: the equation 4 o f (p) = a o q5(p) is also expressed by

saying that the following diagram "commutes.")

z

A f IA

(2.1.43)

2) The fact that A and 0 are homeomorphic allows us to make several conclusions

concerning the set theoretic nature of A. We have already shown that E isuncountable, and we state without proof that E is a closed, perfect (meaningevery point is a limit point), totally disconnected set and that these properties

carry over to A via the homeomorphism 0. A set having these propertiesis called a Cantor Set. We will give more detailed information concerningsymbolic dynamics and Cantor sets in Section 2.2.

Now we can state a theorem regarding the dynamics of f on A that is almostprecisely the same as Theorem 2.1.2, which describes the dynamics of a on E.

Theorem 2.1.4. The Smale horseshoe, f, has1) a countable infinity of periodic orbits of arbitrarily high period. These periodic

orbits are all of saddle type;

2) an uncountable infinity of non-periodic orbits; and

3) a dense orbit.

PROOF: This is an immediate consequence of the topological conjugacy of f on Awith a on E, except for the stability result. The stability result follows from theform of f on Ho and H1 given in (2.1.5).

2.1e. Chaos

Now we can make precise the statement that the dynamics of f on A is chaotic.

Let p E A with corresponding symbol sequence

ON = {... s-n ... S-1. so ... sn ...} . (2.1.44)


We want to consider points close to p and how they behave under iteration byf as compared to p. Let e > 0 be given; then we consider an c-neighborhoodof p determined by the usual topology of the plane. Also, there exists an integerN = N(c) such that the corresponding neighborhood of q5(p) includes the set of

sequences .s _ { s1 n "'s 1 . s . sn } E E such that si = si, Jil < N.

Now suppose the N + 1 entry in the sequence corresponding to qS(p) is 0 and the

N + 1 sequence corresponding to s! is 1. Thus, after N iterations, no matter howsmall e, the point p is in Ho and the point, say p', corresponding to sl under 0-1is in H1 and they are at least a distance 1 - 2A apart. Therefore, for any pointp C A, no matter how small a neighborhood of p we consider, there is at least one

point in this neighborhood such that after a finite number of iterations, p and this

point have separated by some fixed distance. A system displaying such behavior is

said to exhibit sensitive dependence on initial conditions.

Now we want to end our discussion of this simplified version of the Smalehorseshoe with some final observations.

1) If you consider carefully the main ingredients of f which led to Theorem 2.1.4,

you will see that there are two key elements.

a) The square is contracted, expanded, and folded in such a way that we can

find disjoint regions which are mapped over themselves.

b) There exists "strong" stretching and contraction in complementary direc-tions.

2) From observation 1), the fact that the image of the square appears in theshape of a horseshoe is not important. Other possible scenarios are shown inFigure 2.1.11.

Notice that, in our study of the invariant set of f, we do not consider the question

of the geometry of the points which escape from the square. We remark that thiscould be an interesting research topic, since this more global question may enable

one to determine conditions under which the horseshoe becomes an attractor.

2.2. Symbolic DynamicsIn the previous section we saw an example of a two dimensional map which pos-

sessed an invariant Cantor set. The map, restricted to its invariant set, was shown

to have a countable infinity of periodic orbits of all periods, an uncountable infinity


rrr

Figure 2.1.11. Two Other Possible Horseshoe Scenarios.

of nonperiodic orbits, and a dense orbit. Now, in general, the determination ofsuch detailed information concerning the orbit structure of a map is not possible.However, in our example we were able to show that the map restricted to its in-variant set behaved the same as the shift map acting on the space of bi-infinitesequences of 0's and 1's (more precisely, these two dynamical systems were shown

to be topologically equivalent; thus their orbit structures are identical). The shiftmap was no less complicated than our original map but, due to its structure, many

of the features concerning its dynamics (e.g., the nature and number of its peri-odic orbits) were more or less obvious. The technique of characterizing the orbitstructure of a dynamical system via infinite sequences of "symbols" (in our case 0's

and 1's) is known as symbolic dynamics. The technique is not new and appearsto have originally been applied by Hadamard 11898] in the study of geodesics onsurfaces of negative curvature and Birkhoff [1927], [1935] in his studies of dynamical

systems. The first exposition of symbolic dynamics as an independent subject was

given by Morse and Hedlund [1938]. Applications of this idea to differential equa-

tions can be found in Levinson's work on the forced Van der Pol equation (Levinson

[1949]), from which came Smale's inspiration for his construction of the horseshoe

map (Smale [1963] and [1980]), and also in the work of Alekseev [1968], [1969], who

gives a systematic account of the technique and applies it to problems arising from

celestial mechanics. These references by no means represent a complete account of


the history of symbolic dynamics or of its applications and we refer the reader to the

bibliographies of the above listed references or Moser [1973] for a more complete list

of references on the subject and its applications. In recent times (say from about1965 to the present) there has been a flood of applications of the technique, and we

will refer to many of these throughout the remainder of the book.

Symbolic dynamics will play a key role in explaining the dynamical phenomena

which we encounter in the next two chapters. For this reason, we now want todescribe some aspects of symbolic dynamics viewed as an independent subject.

We let S = {1, 2, 3, ..., N}, N > 2 be our collection of symbols. We willbuild our sequences from elements of S. Note that for the purpose of constructing

sequences the elements of S could be anything, e.g., letters of the alphabet, Chinese

characters, etc. We will use positive integers since they are familiar, easy to writedown, and we have as many of them as we desire. At this time we will assume that

S is finite (i.e., N is some fixed positive integer > 2), since that is adequate for many

purposes and will enable us to avoid certain technical questions in our exposition

which at the moment we believe might be unnecessarily distracting. However, after

discussing symbolic dynamics for a finite number of symbols, we will return tothe case where N can be arbitrarily large and describe the technical modifications

necessary to make our results go through in this case also.

2.2a. The Structure of the Space of Symbol Sequences

We now want to construct the space of all symbol sequences, which we will refer

to as EN, from elements of S and derive some properties of EN. It will be conve-nient to construct EN as a cartesian product of infinitely many copies of S. Thisconstruction will allow us to make some conclusions concerning the properties of

EN based only on our knowledge of S and the structure which we give to S. Also,

this approach will make the generalization to an infinite number of symbols quitesimple and straightforward later on.

Now we want to give some structure to S; specifically, we want to make S into

a metric space. Since S is a finite set of points consisting of the first N positiveintegers, it is very natural to define the distance between two elements of S to bethe absolute value of the difference of the two elements. We denote this as follows:

d(a,b)-Ja-bi , Va,bES. (2.2.1)


Thus S is a discrete space (i.e., the open sets in S defined by the metric consistof the individual points which make up S so that all subsets of S are open) andhence it is totally disconnected. We summarize the properties of S in the following

proposition.

Proposition 2.2.1. The set S equipped with the metric (2.2.1) is a compact,totally disconnected metric space.

We remark that compact metric spaces are automatically complete metricspaces (see Dugundji [1966]).

Now we will construct EN as a bi-infinite cartesian product of copies of Seo

EN- fl Si whereS'=S Vi. (2.2.2)i=-oo

So a point in EN is represented as a "bi-infinity-tuple" of elements of S

s E EN s = {...,9-n, 9-1, so,91,...,sn,...I

or, more succinctly, we will write s as

where si E S V i(2.2.3)

wheresiES Vi. (2.2.4)

A word should be said about the "decimal point" that appears in each symbol

sequence and has the effect of separating the symbol sequence into two parts withboth parts being infinite (hence the reason for the phrase "bi-infinite sequence").

At present it does not play a major role in our discussion and could easily be leftout with all of our results describing the structure of EN going through just thesame. In some sense, it serves as a starting point for constructing the sequences by

giving us a natural way of subscripting each element of a sequence. This notation

will prove convenient shortly when we define a metric on EN. However, the realsignificance of the decimal point will become apparent when we define and discuss

the shift map acting on EN and its orbit structure.In order to discuss limit processes in EN it will be convenient to define a

metric on EN. Since S is a metric space it is also possible to define a metric onEN. There are many possible choices for a metric on EN; however, we will utilize

the following:

fors={...s-n...s-1.9091...sn...} S={...5-n...s-1.9091...Sn...}EEN(2.2.5)


the distance between s ands is defined as

001 lsi - s{d(s,9) = E 21'1 1+lsi-sili=-oo

(2.2.6)

(Note: the reader should check that satisfies the four properties which, by

definition, a metric must possess.) Intuitively, this choice of metric implies that two

symbol sequences are "close" if they agree on a long central block. The following

lemma makes this precise.

Lemma 2.2.2. For s, s E EN, 1) Suppose d(s, s) < 1/(2M+1); then si = si forall l it < M. 2) Suppose si = si for lil < M; then d(s,s) < 1/(2M-1)

PROOF: The proof of 1) is by contradiction. Suppose the hypothesis of 1) holdsand there exists some j with l j j < M such that s j # 9j. Then there exists a termin the sum defining d(s, s) of the form

1sj - 9jl

2121 1 +

But

1+

Si - sjl

sj - sjl>1

2sj - sjl

and each term in the sum defining d(s, s) is positive, so we have

1 s2 sjl 1 1d(s, s) >

2121 1 +S j - sj l

- 2I2I+1 2M+1

but this contradicts the hypothesis of 1).

We now prove 2). If si = si for l it < M we have

d(s's)s

sitl

+21i 1 +si 21'1

1 +sisa siliI-co i=M+1

i=-(M+1)

(2.2.7)

(2.2.8)

but si - sil / (1 + lsi - sil) < 1, so we get


d(s, s) < 21 _ 1

i=M+12i - 2M-1 (2.2.9)

Armed with our metric, we can define neighborhoods of points in EN anddescribe limit processes. Suppose we are given a point

giES Vi,

and a positive real number e > 0, and we wish to describe the "E-neighborhoodof 9", i.e., the set of s E EN such that d(s,g) < c. Then, by Lemma 2.2.2, givenc > 0 we can find a positive integer M = M(E) such that d(s, s) < c implies

Si = si V jil < M. Thus, our notation for an c-neighborhood of an arbitraryg E EN will be as follows:

JJM(e) (g) _ { s E EN I si = gi , v lil < M , si,; E S V i } . (2.2.10)

Before stating our theorem concerning the structure of EN we need the follow-

ing definition.

Definition 2.2.1. A set is called perfect if it is closed and every point in the set isa limit point of the set.

The following theorem of Cantor gives us information concerning the cardinality

of perfect sets.

Theorem 2.2.3. Every perfect set in a complete space has at least the cardinalityof the continuum.

PROOF: See Hausdorff [1957].

We are now ready to state our main theorem concerning the structure of EN.

Proposition 2.2.4. The space EN equipped with the metric (2.2.6) is1) compact,

2) totally disconnected,

and3) perfect.

PROOF: 1) Since S is compact, EN is compact by Tychonov's theorem (Dugundji[1966]). 2) By Proposition 2.2.1, S is totally disconnected, and therefore EN is


totally disconnected, since the product of totally disconnected spaces is likewisetotally disconnected (Dugundji [1966]). 3) EN is closed, since it is a compactmetric space. Let g E EN be an arbitrary point in EN; then, to show thats is a limit point of EN, we need only show that every neighborhood of s con-tains a point s 94 s with s E EN. Let be a neighborhood of s and let

s = sM(E)+1+1 if sM(E)+1 N, and s = sM(e)+1-1 if N. Then thesequence {. s-M(e)-2 s -M(e) s-1 sOsl ... sM(E) s" SM(e)+2 ... } is con-

tained in RM(E) (s) and is not equal to s; thus EN is perfect.

We remark that the three properties of EN stated in Proposition 2.2.4 areoften taken as the defining properties of a Cantor set of which the classical Cantor

"middle-thirds" set is a prime example.

Next we want to make a remark which will be of interest later when we use EN

as a "model space" for the dynamics of maps defined on more "normal" domains

than EN (i.e., by "normal" domain we mean the type of domain which might arise

as the phase space of a specific physical system). Recall that a map, h : X -' Y, of

two topological spaces X and Y is called a homeomorphism if h is continuous, one-

to-one, amd onto and h-1 is also continuous. Now there are certain properties oftopological spaces which are invariant under homeomorphisms. Such properties are

called topological invariants; compactness, connectedness, and perfectness are three

examples of topological invariants (see Dugundji [1966) for a proof). We summarize

this in the following proposition.

Proposition 2.2.5. Let Y be a topological space and suppose that EN and Y arehomeomorphic, i.e., there exists a homeomorphism h taking EN to Y. Then Y iscompact, totally disconnected, and perfect.

2.2b. The Shift Map

Now that we have established the structure of EN, we want to define a map of EN

into itself, denoted by a, as follows:

For C EN, we define

Q(s) - {... s-n ... sn ...} (2.2.11)

or, [a(s)]i = si+1.


The map, a, is referred to as the shift map, and when the domain of or is taken

to be all of EN, it is often referred to as a full shift on N symbols. We have thefollowing proposition concerning some properties of a.

Proposition 2.2.6. 1) a(EN) = EN. 2) a is continuous.

PROOF: 1) is obvious. To prove 2) we must show that, given c > 0, there

exists a 6(e) such that d(s,s) < 6 implies d(a(s),a(s)) < c for s, s E EN.Suppose e > 0 is given; then choose M such that 1/(2M-2) < E. If we then

let 6 = 1/2M+1, we see by Lemma 2.2.2 that d(s,s) < 6 implies si = s{ for

lil < M; hence [a(s)]t = [a(s)]g, Jil < M - 1; then, also by Lemma 2.2.2, we have

d(a(s),a(s)) < 1/2M-2 < E.

We now want to consider the orbit structure of a acting on EN. We have thefollowing proposition.

Proposition 2.2.7. The shift map a has1) a countable infinity of periodic orbits consisting of orbits of all periods;

2) an uncountable infinity of nonperiodic orbits; and

3) a dense orbit.

PROOF: 1) This is proved exactly the same way as the analogous result obtained

in our discussion of the symbolic dynamics for the Smale horseshoe map in Section

2.1c. In particular, the orbits of the periodic symbol sequences are periodic, andthere is a countable infinity of such sequences. 2) By Theorem 2.2.3 EN isuncountable; thus, removing the countable infinity of periodic symbol sequences

leaves an uncountable number of nonperiodic symbol sequences. Since the orbits of

the nonperiodic sequences never repeat, this proves 2). 3) This is proved exactly the

same way as the analogous result obtained in our discussion of the Smale horseshoe

map in Section 2.1c; namely, we form a symbol sequence by stringing together allpossible symbol sequences of any finite length. The orbit of this sequence is dense in

EN since, by construction, some iterate of this symbol sequence will be arbitrarily

close to any given symbol sequence in EN.

2.2c. The Subshift of Finite Type

In some applications which will arise in Chapter 3 it will be natural to restrict thedomain of a in such a way that it does not include all possible symbol sequences.


This will be accomplished by throwing out symbol sequences in which certain sym-

bols appear as adjacent entries in the sequence. In order to describe this restriction

of the domain of a the following definition will be useful.

Definition 2.2.2. Let A be an N x N matrix of 0's and 1's constructed by thefollowing rule: (A)ij = 1 if the ordered pair of symbols ij may appear as adjacent

entries in the symbol sequence, and (A)ij = 0 if the ordered pair of symbols ij may

not appear as adjacent entries in the symbol sequence. The matrix A is called the

transition matrix. (Note: by the phrase "ordered pair of symbols if we mean thatwe are referring to the symbols i and j appearing in the pair with i immediately to

the left of j.)

The collection of symbol sequences defined by a given transition matrix isdenoted EN and may be concisely written as

EN = { 8={... S-n ... Sn ...} E EN I (A)3s.8s.tl = 1 v i }

(2.2.12)

We remark that it should be clear that EN C EN and that the metric (2.2.6)serves to define a topology on EN with Lemma 2.2.2 also holding for E.EXAMPLE 2.2.1. Let /0 1\

A f\0 1/1 (2.2.13)

then EA consists of the collection of bi-infinite sequences of 1's and 2's where the

combination of symbols 11 and 21 does not appear in any sequence.

We would like to characterize the structure of EN similarly to the way in which

Proposition 2.2.4 characterizes the structure of EN. However, two different transi-

tion matrices may define two EN's which have very different topological structures.

EXAMPLE 2.2.2. Let

A= (2.2.14)

then EA consists of all possible bi-infinite sequences of 1's and 2's and, therefore,

has the topological structure described in Proposition 2.2.4.

Let

A' _ (2.2.15)

then EA, consists of precisely two points, namely the sequences { . 1111.1111 . . }

and { . 2222.2222 . . .}. (Notation: if all of the entries of A are one, then EN = N,and we disregard the A in the notation for the space of symbol sequences.)


It is possible to impose restrictions on the transition matrix A in such a way

that the properties of EN described in Proposition 2.2.4 also hold for EN. We now

want to describe the nature of the necessary requirements on A for this to hold.

Definition 2.2.3. The transition matrix A is called irreducible if there is an integer

k > 0 such that (Ak),lj 0 for all 1 < i,j < N.

Recall from our previous discussion that many of our results concerning thestructure of EN and the orbit structure of a involved constructions using symbolsequences of finite length. In this regard the following definition will be useful.

Definition 2.2.4. Let A be a transition matrix and let al sk, si E S i =1, ... , k be a finite string of symbols of length k. Then si sk will be called an

admissible string of length k if (A)sisi+l = 1, i = 1,...,k - 1.

Let K > 0 be the smallest integer such that (AK)ij # 0 for all 1 < i,j < N. Then we have the following lemma.

Lemma 2.2.8. Suppose A is an irreducible transition matrix, and let K be asdescribed above; then, given any i, j E S, there exists an admissible string oflength k< K-1, sl sk, such that isl . ski is an admissible string of length

k+2.

PROOF: The ij element of A is given by

N(A)i31 (A)3132 ... (A)SK-2SK_1(A)sK-1j

sl ,...,3K _ 1(2.2.16)

where each element of the sum is either 0 or 1. Then, since (AK)ij 54 0, theremust exist at least one sequence si 8K_1 such that

(A)151(A)1152 ... (A)sK-2sK-1(A)sK_lj = 1.

Then each element of this product must also be 1; therefore, isl . 9K-1i is anadmissible string.

We make the following remarks regarding Lemma 2.2.8.

1) It follows from the proof of Lemma 2.2.8 that for any i, j E S there exists amaximal integer given by K - 1 and an admissible string of length K - 1,

81 sK_1, such that isl . s%_1 j is an admissible string of length K+ 1.


However, it is certainly possible that for any particular i, j E S j appearsearlier in the sequence than the last place. This is the reason for stating thelemma in terms of an admissible string of length k < K - 1. However, forsome constructions the use of this maximal integer will play an important role

(e.g., see Proposition 2.2.9).

2) For i, j E S regarded as fixed, we will often use the phrase "admissible stringof length k connecting i and j," or, when the length of the string does notmatter, "admissible string connecting i and j."

We are now in a position to prove the following result concerning the structure

of E.

Proposition 2.2.9. Suppose A is irreducible; then EN equipped with the metric(2.2.6) is

1) compact,

2) totally disconnected,

and

3) perfect.

PROOF: 1) In order to show that EN is compact, it suffices to show that EN isclosed since closed subsets of compact sets are compact (see Dugundji [19661).

Let {st} be a sequence of elements of EN, i.e., a sequence of sequences, such

that {s4} converges to s. EN is closed if s E EN. The proof that s E EN is bycontradiction. Suppose s 0 EN; then there must exist some integer M such that(A)sM9M+1 = 0. Now {st} converges to s, so there exists some integer M such that

for i > M d(si, s) < 1/2M+2. So by Lemma 2.2.2 for i > M we have s' = sj forall 1jj < M + 1 and, since si E EN, we have (A) s,

8= (A)SMSM+l = 1.

M M+1This is a contradiction which arises from the fact that we have assumed E.2) This is obvious since the largest connected component of EN is a point and the

same must hold for any subset of EN.

3) In 1) we showed that EN is closed; hence, in order to show that EN is perfect,

it remains to show that each point of EN is a limit point.Let s E ENA; then to show that s is a limit point of EN we need to show that

every neighborhood of s contains a point s s with s E EN. Let VM(()(9) bea neighborhood of 9 = { s_M . 9-1.9091 9M ...}. Now from the remark


following the proof of Lemma 2.2.8, there exists an integer K such that for anyi, j E S there exists an admissible string of length K-1 given by s1 sK_1 suchthat is1 sK-13 is an admissible string of length K + 1. Now consider theM + K entry ins given by sM+K. Let sM+K = 9M+K + 1 if 9M+K N or

sM+K = sM+K - 1 if 9M+K = N. Then consider the sequence

s = {... s-M ... s-1.9091... 9M31 ... SK-1sM+Ks1 ... 9K-18M+K+1 ...}

where s1 .. sK_1 is the admissible string connecting 9M and sM+K+ sl "' SK-1is the admissible string connecting sM+K and sM+K+l, and the " " befores_M and after 9M+K+1 indicates that the infinite tails of s are the same as s.Now it should be clear by construction that s E EA, s E .W M(E) (9), and s 54 s.

Now we want to consider the orbit structure of the shift map a restricted toE. In this case a is called a subshift of finite type (note: the phrase "finite type"comes from the fact that we are considering only a finite number of symbols). Wehave the following proposition.

Proposition 2.2.10. 1) a (EA) = E. 2) a is continuous.

PROOF: 1) is obvious and the proof of 2) follows from the same argument as that

given for the proof of continuity of the full shift given in Proposition 2.2.6.

Now we give the main result concerning the orbit structure of a restricted toENA

Proposition 2.2.11. Suppose A is irreducible; then the shift map or with domainEN has:

1) a countable infinity of periodic orbits,

2) an uncountable infinity of nonperiodic orbits, and

3) a dense orbit.

PROOF: 1) Recall the construction of the countable infinity of periodic orbits for

the full shift. In that case, we merely wrote down the periodic symbol sequencesof length 1, 2, 3, ... and the result immediately followed. Now for the case ofsubshifts of finite type it should be clear that this construction does not go through.

However, a similar procedure will work.

Let i, j E S; then by Lemma 2.2.8 there exists an admissible string Si . . . sk

such that is1 Ski is also an admissible string. For a more compact notation


we will denote the admissible string connecting i and j by 51 sk - sij (note:from the proof of Lemma 2.2.8 it should be clear that for a given i, j E S theremay be more than one admissible string connecting i and j; henceforth, for eachi, j E S we will choose one admissible string sij and consider it fixed). The

construction of the countable infinity of periodic sequences proceeds as follows.

a) Write down all sequences of elements of S of length 2, 3, 4, 5.... which beginand end with the same element. It should be clear that there is a countableinfinity of such sequences.

b) Choose a particular sequence constructed in a). Now between each pair ofentries ij in the sequence place the admissible string sjj which connects these

two elements of S. Repeating this procedure for each element constructed in

a) yields a countable infinity of admissible strings. Since each admissible string

begins and ends with the same element of S we can place copies of each admis-

sible string back-to-back in order to create a bi-infinite periodic sequence. In

this manner we construct a countable infinity of admissible periodic sequences.

Repeating this procedure for all possible sij for each i, j should yield all ad-missible periodic sequences.

2) Since EN is perfect, by Theorem 2.2.3 it has the cardinality of at least thecontinuum. Thus, subtracting the countable infinity of periodic sequences awayfrom EN leaves an uncountable infinity of nonperiodic sequences.

3) The construction of the sequence whose orbit under o is dense in ENT is verysimilar to that in the case of the full shift. Write down all admissible stringsof length 1, 2, 3, ...; then connect them all together in one sequence by usingconnecting admissible strings provided by Lemma 2.2.8. The proof that the orbit of

this sequence is dense in ENT is exactly the same as that for the analogous situation

with the full shift, see Section 2.Ic.

2.2d. The Case of N = oo

We now want to consider the case of an infinite number of symbols. In particular,we want to determine how much of the previous results concerning the structure of

the space of symbol sequences as well as the orbit structure of the shift map arestill valid. We will only consider the case of the full shift but we will make somebrief comments concerning the subshift at the end of our discussion.


We begin by discussing the structure of the space of symbol sequences, in this

case denoted E°O. Let S = {1, 2, 3, ... , N, ...} be our collection of symbols.

Then E°O is constructed as a Cartesian product of infinitely many copies of S.Therefore, it should be clear that the first casualty incurred by allowing an infinite

number of symbols is the loss of compactness for E°O since S is now unbounded.

In fact this is the only problem and is quite easily handled. We can compactify Sby the usual one point compactification technique (see Dugundji [1966]) of adding

a point at infinity. Thus, we have S = {1, 2, ..., N, ..., oo} where oo is thepoint such that all other integers are less than infinity. We now construct E°°as an infinite Cartesian product of S as in Section 2.2a. Hence, by Tychonov's

theorem (Dugundji [1966]), E°° is compact and clearly E°O C E°°. E°° is called

a compactification of E°O. The metric (2.2.6) still works for E°° if we define

IN - cc]

1 + IN - cc]=1, NES (2.2.17)

and100-001

= 0 . (2.2.18)1+100-00[Thus, neighborhoods of points in E°O are defined exactly as in (2.2.10). Also,

Lemma 2.2.2 and Proposition 2.2.4 apply for E°O (in particular, E°O is a Cantorset). Regarding the orbit structure of the shift map acting on E°O, Propositions2.2.6 and 2.2.7 still hold. Now, although the consideration of an infinite number of

symbols does not greatly change our results concerning the structure of the space of

symbol sequences or the orbit structure of the shift map, we will see in Section 2.3

and in Chapter 3 that the symbol oo does have a special meaning when we utilizesymbolic dynamics to model the dynamics of maps.

Now let us make a few comments regarding subshifts on a space of symbolsequences having an infinite number of symbols. Recall from our discussion of sub-

shifts of finite type that the transition matrix as well as powers of the transitionmatrix (specifically, the concept of irreducibility) played a central role in determin-

ing both the structure of EN and the orbit structure of the shift acting on EN.Thus, the immediate problems faced when dealing with subshifts of infinite type in-

volve multiplication of infinite matrices as well as defining a concept of irreducibility

for an infinite transition matrix. Since we will have no cause to utilize subshifts of

infinite type in this book, we will leave the necessary generalizations as an exercise

for the interested reader.


2.3. Criteria for Chaos: The Hyperbolic CaseIn this section we will give verifiable conditions for a map to possess an invariant

set on which it is topologically conjugate to the shift map acting on the space of bi-

infinite sequences constructed from a countable set of symbols. Our plan follows that

of Conley and Moser (see Moser [1973]); however, our criteria will be more general

in the sense that they will apply to N dimensional invertible maps (N > 2) andthey will allow for subshifts as well as full shifts. The term "hyperbolic case" needs

explanation. Roughly speaking, this term arises from the fact that at each pointof the domain of the map there is a splitting of the domain into a part whichis strongly contracting (the "horizontal" directions) and a part which is strongly

expanding (the "vertical" directions). This results in the invariant set of the mapbeing a set of discrete points; a more precise definition will be given later. Theoutline of this section is as follows:

1) We begin by developing introductory concepts, specifically the generalization of

the horizontal and vertical rectangles described in our discussion of the Smale

horseshoe in Section 2.1 and concepts for describing their behavior under maps.

2) We state and prove our main theorem.

3) We introduce and discuss the idea of a sector bundle, which provides an alter-

native and more easily verifiable hypothesis for our main theorem.

4) We give a second set of alternate conditions for verifying the hypotheses of our

main theorem that are more convenient for applications near orbits homoclinic

to fixed points.

5) We define and discuss the idea of a hyperbolic invariant set and show how itrelates to our previous work.

2.3a. The Geometry of Chaos

In order to understand what the essential elements of such criteria might be, let usrecall our discussion of the Smale horseshoe map given in Section 2.1. We saw that

the horseshoe map contained an invariant Cantor set on which the dynamics waschaotic. Furthermore, we were able to obtain very detailed information concerning

the orbit structure of the Smale horseshoe using symbolic dynamics. These results

followed from two properties of the map:

1) The map possessed independent expanding and contracting directions.


2) We were able to locate two disjoint "horizontal" rectangles with horizontal and

vertical sides parallel to the contracting and expanding directions, respectively,

which were mapped onto two "vertical" rectangles with each vertical rectangle

intersecting both horizontal rectangles and with the horizontal (resp. verti-cal) boundaries of the horizontal rectangles mapping onto the horizontal (resp.

vertical) boundaries of the vertical rectangles.

These two properties were responsible for the existence of the invariant Cantor set

on which the map was topologically conjugate to a full shift on two symbols. (We

remark that the reason the map was topologically conjugate to a full shift was that

the image of each horizontal rectangle intersected both horizontal rectangles.) So

the fact that a map possesses a chaotic invariant set is largely due to geometriccriteria and is not a consequence of the specific analytical form of the map (i.e.,there is not a "horseshoe function" analogous to a trigonometric function or elliptic

function, etc.). Now our goal will be to weaken the above two properties as much as

possible and extend them to higher dimensional maps in order to establish criteria

for a map to possess an invariant set on which it is topologically conjugate to asubshift.

We consider a map

f:D-,]Rnx1Ryn (2.3.1)

where D is a closed and bounded n + m dimensional set contained in R' x R.We will discuss continuity and differentiability requirements on f when they areneeded. We note that the domain of f is not required to be connected and, inseveral examples in Chapter 3, we will see that it is necessary to consider mapswhose domains consist of several connected components. However, we will not be

interested in the behavior of f on its entire domain, but rather in how it acts ona set of disjoint, specially defined "horizontal slabs". (Note: this situation is com-

pletely analogous to the Smale horseshoe discussed in Section 2.1. In that example

the map was defined on the unit square, but all of the complicated dynamical conse-

quences were derived from a knowledge of how the map acted on the two horizontal

rectangles HO and H1.)

We will now begin the development of the definition of horizontal and vertical

slabs which will be the analogs of the Hi, Vi, i = 0, 1, in our discussion of theSmale horseshoe in Section 2.1. Following this, we will define the width of the slabs


and various intersection properties of horizontal and vertical slabs. These will be of

use in proving our main theorem, which will provide sufficient conditions in order

for f to possess an invariant set on which it is topologically conjugate to a subshift

of finite type.

We begin with some preliminary definitions. The following two sets will be

useful for defining the domains of various maps:

Dx = {x E R' for which there exists a y E Rm with (x, y) E D}(2.3.2)

Dy = {y E 1Rm for which there exists an x E 1R' with (x, y) E D} .

So Dx and Dy represent the projection of D onto R.' and 1Rm, respectively, seeFigure 2.3.1.

Figure 2.3.1. D, Dx, and Dy in 1R7z+m; n = 2, in = I.


Let Ix be a closed, simply connected n dimensional set contained in Dx, and

let Iy be a closed, simply connected m dimensional set contained in Dy. We will

need the following definitions.Definition 2.3.1. A uh-horizontal slice, H, is defined to be the graph of a function

h: Ix IR.'n where h satisfies the following two conditions:

1) The set TI = { (x, h(x)) E R' x R' I x c Ix } is contained in D.

2) For every x1, x2 E Ix, we have

l h(xl) - h(x2)1 <- lih Ixl - x21 (2.3.3)

for some 0 < µh < 00-

Similarly, a µv-vertical slice, V, is defined to be the graph of a function v: Iy ->

Rn where v satisfies the following two conditions:

1) The set V = { (v(y), y) E ]Rn x 1R' I y E Iy } is contained in D.

2) For every yl, Y2 C Iy, we have

Iv(y1) - v(y2)I <- /A- IYl - 1121

for some 0 < µv < oo.

(2.3.4)

See Figure 2.3.2 for an illustration of Definition 2.3.1 (Note: hereafter in givingfigures to describe the definitions of µh-horizontal slices, ph-horizontal slabs, widths

of slabs, etc., we will not show the domain D of f in the figures so that they do not

become too cluttered.)

Next we want to "fatten up" these horizontal and vertical slices into n + mdimensional horizontal and vertical "slabs." We begin with the definition of Fth-horizontal slabs.

Definition 2.3.2. Fix some µh, 0 < µh < oo. Let H be a µh-horizontal slice,and let J'n C D be an m-dimensional topological disk intersecting H at any, butonly one, point of H. Let Ha, a E I, be the set of all µh-horizontal slices thatintersect the boundary of J'n and have the same domain as II where I is someindex set (note: it may be necessary to adjust the domain Ix of ft, or equivalently,

adjust J'n in order for this situation to be obtained). Consider the following set in1Rn x ]Rm.


Y1

Figure 2.3.2. µh-Horizontal and µv-Vertical Slices in 1R.u x R"`; n = 2, m = 1.

SH = { (x, y) E 1Rn X 1R'7 I x E Ix and y has the property that, for each x E Ix,given any line L through (x, y) with L parallel to the x = 0 plane, thenL intersects the points (x,ha(x)), (x,hp(x)) for some a, ft E I with(x, y) between these two points along L. }

Then a µh-horizontal slab H is defined to be the closure of SH.

When we discuss the behavior of µh-horizontal slabs under maps it will beuseful to have the notion of horizontal and vertical boundaries.

Definition 2.3.3. The vertical boundary of a µh-horizontal slab H is denoted 8vHand is defined as

8vH-{(x,y)EHI xEBIx}. (2.3.5)

The horizontal boundary of a uh-horizontal slab H is denoted 8hH and is defined

as

8hH - 8H - 8,,H . (2.3.6)


We remark that it follows from this as well as Definition 2.3.2 that 8vH isparallel to the x = 0 plane. See Figure 2.3.3 for an illustration of Definitions 2.3.2

and 2.3.3.

Figure 2.3.3. Horizontal Slab in R' x IRm; n = 2, m = 1.

Before proceeding further, let us give some motivation for Definition 2.3.2. We

will see later on that the main properties we need for H are that it is an n + mdimensional compact set such that any point on 8hH lies on a µh-horizontal slice.

In Definition 2.3.2 these properties are manifested as follows.

By construction 8hH is made up of Irh-horizontal slices. Therefore, any point

lying on 8hH lies on a µh-horizontal slice. We remark that it should be clearthat I is an uncountable set.The line L is used to fill out the space "between" the µh-horizontal slices that

make up 8hH. By moving L through 8Iy the vertical boundary of H is created.

In this way one obtains an n + m dimensional compact set.

We will be interested in the behavior of ph-horizontal slabs under maps. Inparticular, we will be interested in the situation where the image of a µh-horizontal

slab intersects its preimage (note: the "preimage of the image" is just the slabitself). For describing this situation the following definition will be useful.


Definition 2.3.4. Let H and H be µh-horizontal slabs. H is said to intersect Hfully if H C H and 8vH c avH.

See Figure 2.3.4 for an illustration of Definition 2.3.4.

(a)

(b)

Figure 2.3.4. a) H Does Not Intersect H Fully. b) ft Intersects H Fully.

Next we will define µv-vertical slabs.

Definition 2.3.5. Fix some Ecv, 0 < µv < oo. Let H be a µh-horizontal slab, andlet V be a FLv-vertical slice contained in H such that aV C ahH. Let Jn c H bean n dimensional topological disk intersecting V at any, but only one, point of V,

and let Va, a e I, be the set of all µv-vertical slices that intersect the boundaryof J' with 0 7' C ahH where I is some index set. We denote the domain of thefunction va(y) whose graph is V a by I. Consider the following set in Rn x R'.

Sv = { (x, y) E Rn X R" ` I (x, y) is contained in the interior of the set bounded

by Va, a E I, and ahH I.Then, a µv-vertical slab V is defined to be the closure of SV.


We remark that the main properties we need for µv-vertical slabs are that they

are n + m dimensional compact sets such that any point on the vertical boundary

lies on a µv-vertical slice (cf. the discussion following Definition 2.3.3).

The horizontal and vertical boundaries of uv-vertical slabs are defined as fol-lows.

Definition 2.3.6. Let V be a uv-vertical slab. The horizontal boundary of V,denoted ahV, is defined to be V n ahH. The vertical boundary of V, denoted avV,

is defined to be aV - ahV .

See Figure 2.3.5 for an illustration of Definitions 2.3.5 and 2.3.6.

vl

a,.v jn

H

x

Figure 2.3.5. Vertical Slab in R' X Rm; n = 2, in = 1.

Notice in Figures 2.3.4 and 2.3.5 that we depict the nh-horizontal and uv-vertical slabs as slightly warped cubes or tubes. Definitions 2.3.2 and 2.3.5 certainly

allow much more pathological behavior of the boundaries; however, for convenience

we will continue to draw the slabs as in Figures 2.3.4 and 2.3.5.

Next we define the widths of µh-horizontal and µv-vertical slabs.


Definition 2.3.7. The width of a kh-horizontal slab H is denoted by d(H) and is

defined as follows:

d(H) = sup I ha(x) - hf(z) . (2.3.7)x EIa

a,pE1

Similarly, the width of a µv-vertical slab V is denoted by d(V) and is defined asfollows:

d(V)= sup Iva(y)-v/i(y)IyEII

(2.3.8)

where Iy = Iy n Iy

The following lemmas will be very useful later on.

Lemma 2.3.1. a) If H1 D H2 : H3 1) ... is an infinite sequence of µh-horizontal

slabs with Hk+1 intersecting Hk fully, k = 1, 2, ..., and d(Hk) ---f 0 as k --> no,

then oo Hk = H' is a µh-horizontal slice with aH°° C avHl. b) Similarly,k=1

if V1 D V2 D V3 D . . is an infinite sequence of µv-vertical slabs contained in00

a µh-horizontal slab H with d(Vk) -+ 0 as k --+ no, then n V k = V °O is ak=1

µv-vertical slice with 19V °O C ahH.

PROOF: We will only prove a) since the proof of b) is quite similar.Let J'n be an m-dimensional topological disk contained in Hl as described in

Definition 2.3.2. Then the collection of functions y = h(x), x E Ix, satisfying theLipschitz condition (2.3.3) whose graphs form µh-horizontal slices that intersect J"i

form a complete metric space with the metric obtained from the supremum norm.

Let {hak(x)}, ak E Ik be the set of functions whose graphs form the horizontalboundary of Hk. Consider the infinite sequences of functions

ha0k 1' t hak3 k 1' Ihak' hakIk 1(2.3.9)

where ak, ak E Ik are regarded as fixed for each k. Now the condition

Hk Hk+l, Hk+1 intersecting Hk fully for k = 1,2....

with d(Hk) - 0, k -+ no (2.3.10)

implies that the three sequences in (2.3.9) are Cauchy sequences. Since the ele-ments of these Cauchy sequences lie in a complete metric space, it follows that they


each converge to a limit h°O(x). The limit functions for the three sequences in(2.3.9) must be identical since the first two sequences are subsequences of the third

sequence. Moreover, the limit function must satisfy the condition

Jh"(xl) - hoo(x2)I < ih Ix1 - x21 , X1, X2 E Ix, 0<µh<oo. (2.3.11)

Thus, we have shown that all the µh-horizontal slices that make up the bound-

ary of Hk converge to a unique function h°O(x) as k --+ oo with h°°(x) satis-fying (2.3.11). So the graph of h°O(x) is a µh-horizontal slice denoted H°° withaH°O C avH1, since the domain of h°°(x) is Ix.

Lemma 2.3.2. Let H be a µh-horizontal slab. Let H be a ph-horizontal slicewith aH C avH, and let V be a µv-vertical slice with aV C ahH such that0 < µvµh < 1. Then H and V intersect in precisely one point.

PROOF: The lemma will be proved if we show that there is a unique point x E Ixsuch that x = v(h(x)) where Ix = closure {x E Ixlh(x) is in the domain of

Now, since Ix is a closed subset of R', it is a complete metric space and, since

V C H v o h maps Ix into ix, hence v o h maps the complete metric space i.into itself. If we show that v o h is a contraction map, then it follows from thecontraction mapping theorem (see Chow and Hale [1982]) that v o h has a unique

fixed point in Ix and, therefore, the lemma is proved. To show that v o h is acontraction map, choose x1, x2 E Ix, in which case we have

Iv(h(x1)) - v(h(x2))l :5 /iv Ih(xl) - h(x2)1

<_Avµh1x1-x21

(2.3.12)

So, since 0 < µvµh < 1, v o h is a contraction map.

At this point we would like to comment on the motivation behind our somewhat

involved definitions of µh-horizontal and µv-vertical slabs given in Definitions 2.3.2

and 2.3.5.

The main motivating factor is to define objects which display the propertiesdescribed in Lemma 2.3.1 and Lemma 2.3.2. So, roughly speaking, horizontal slabs

are n + m dimensional objects whose "horizontal sides" are foliated by horizontal

slices resulting in the intersection of a countable infinity of nested horizontal slabs

being a horizontal slice. Similarly, a vertical slab is an n + m dimensional object


whose vertical sides are foliated by vertical slices resulting in the intersection of a

countable infinity of nested vertical slabs being a vertical slice. We will see thatthis property, along with the fact that horizontal and vertical slices intersect in aunique point, is of crucial importance in explicitly constructing the invariant set of

the map f.

2.3b. The Main Theorem

We are now at the point where we can give conditions sufficient for the map f tohave an invariant set on which it is topologically conjugate to a subshift of finitetype.

Let S = {1, 2, ..., N}, N > 2, and let Hi, i = 1, ... , N, be a set of disjointN

ph-horizontal slabs with DH U Hi. We assume that f is one-to-one on DHi=1

and we define

f(Hi)nHVji di,jESand

Hi n f-1(Hj) = f-I(Vji) = Hij , Vi,jES. (2.3.13)

Notice the subscripts on the sets Vji and Hi j. The first subscript indicates whichparticular uh-horizontal slab the set is in, and the second subscript indicates forthe Vji into which uh-horizontal slab the set is mapped by f-1 and for the Hi,into which µh-horizontal slab the set is mapped by f.

Let A be an N x N matrix whose entries are either 0 or 1, i.e., A is a transition

matrix (see Section 2.2) which will eventually be used to define symbolic dynamics

for f. We have the following two "structural" assumptions for f.

Al. For all i, j E S such that (A) ij = 1, Vji is a p,-vertical slab containedin Hj with avVji C a f (Hi) and 0 < µvµh < 1. Moreover, f maps Hijhomeomorphically onto Vji with f -1(avVji) C 8vHi.

A2. Let H be a Ah-horizontal slab which intersects Hi fully. Then f (H) n H1

H, is a ph-horizontal slab intersecting H1 fully for all i E S such that(A)ij = 1. Moreover,

d(Hi) < vh d(H) for some 0 < vh < 1 . (2.3.14)


Similarly, let V be a µv-vertical slab contained in Hj such that also V C Vji for

some i, j E S with (A)ij = 1. Then f (V) n Hk Vk is a Ecv-vertical slab

contained in Hk for all k E S such that (A) jk = 1. Moreover,

d(Vk) < vv d(V) for some 0 < vv < 1 . (2.3.15)

See Figures 2.3.6 and 2.3.7 for a geometric interpretation of Al and A2.

011?igure 2.3.6. An Example of Horizontal Slabs and Their Images under f; A = 0 0 p

Let us now make the following remarks concerning Al and A2.

1. Al is the global hypothesis dealing with the nonlinear nature of f. It assuresthat the appropriate boundaries of the images and preimages of the Hi underf are aligned along the appropriate contracting and expanding directions. A2


Figure 2.3.7. Hij and Vji for 1 < i, j < 3, (A){2 = 1.

gives specific rates of contraction and expansion of the Hi under f along thehorizontal and vertical directions, respectively.

2. Regarding Al, it is important to have C 49f (Hi), since otherwise

f -1(a vji) would not be contained in

3. In A2 let H = Hj; then it follows that the Hij are µh-horizontal slabs inter-secting Hi fully. Moreover, the horizontal (resp. vertical) boundaries of theHij map to the horizontal (resp. vertical) boundaries of the Vji under f. Thecorrespondence of appropriate boundaries of the Hij and Vji under f and f

is very important.

4. It is important to realize that Al and A2 are hypotheses which concern onlyone forward and backward iterate of f. We will see that Al and A2 implyresults on all iterates of f.

Theorem 2.3.3. Suppose f satisfies Al and A2; then f possesses an invariant setof points A C DH on which it is topologically conjugate to a subshift of finite type

with transition matrix A, i.e., there exists a homeomorphism (k: A -4 EN such


that the following diagram commutes

A

ENA

Moreover, if A is irreducible then A is a Cantor set.

(2.3.16)

PROOF: The proof is broken down into several steps.

1) Geometrically construct the invariant set, A, of f and verify that it isnonempty.

2) Based on the geometrical construction of A, define a map 4': A -. EN

3) Prove that 0 is a homeomorphism.

4) Prove that the diagram (2.3.16) commutes, i.e., 4' o f = a o 0.

We begin with the first step.

1) Construction of the Invariant set A.

The invariant set A consists of the points in DH that remain in DH underall forward and backward iterations by f. If we denote the set of points thatremain in DH under all backward iterations by A_00, and the set of points thatremain in DH under all forward iterations by A+00, then the invariant set A isthe set of points that is common to both A_m and A+oo or, in other words,A = A-,, n A+00. In constructing A we will construct and determine the nature of

A-oo and A+oo separately, and then take their intersection in order to obtain A.

1a) Construction of A_,,,.

We want to construct and determine the nature of the set of points whichremain in DH under all backwards iterations by f, i.e.,

{pEDH I f-Z(p)EH8_iI s-iES, i = 0, 1,2,...,n,...}=A-co

This will be accomplished via an inductive construction where we construct sequen-

tially the set of points remaining in DH under 1, 2, ... , n, ... backwards iterationsby f, utilize Al and A2 to determine the nature of the set constructed at each step,and then consider the set obtained in the limit as n -* oo.


We begin by writing down expressions for the set of points which remain in

DH under 1, 2, ..., n, ... backwards iterations by f.

A-i ° U (f (H3_1) n H30) VSp8_1

aiES aiESi=0, -1 i=0,-1

={PEDH I pEHso, f(p)EH8-1; s0,s-lES}

A-2 = U (f (V3_13-2) n H30)aiES

i=0,-1,-2

(2.3.17)

= U (f2 (H3-2) n f (H3-1) n H30) = U v803-13-2aiES aiES

i=0,-1,-2 i=0,-1,-2

_ {p E DH I p E Hso,f-1(P) E Hs_l ,f-2(p) E Hs-2 ; so,s-l,s-2 E S } .

(2.3.18)

A-rz = U (f (v3_1...3.. ) n H3o)aiES

i=0,-1,. ,-n

= U (f n l(H3_nJ ) n f n-1 (H3-n+l) n ... n H30) = U v30...3_n

aiES aiESi=01-1,...,-n i-0,-1,...,-n

_ {PEDH I f-'(p) EH3-i; s-iES, i =0,1,...,n} .

(2.3.19)

Now, by Al, A_1 consists of a collection of disjoint µv-vertical slabs Vs08_1

contained in H30 for all sp,9_1 E S such that (A)8_130 = 1.Proceeding to A_2, we use the information obtained concerning A_1 at the

previous step and appeal to A2 to conclude:

i) A-2 consists of a collection of disjoint µv-vertical slabs V303-13-2 con-

tained in Hso for all sp, s-1, s-2 E S such that (A)3_23-1(A)3-130 =1.

ii) d (V30 8_13_2) < vv d (V303_1). (2.3.20)

iii) By definition of A_2 and A-1 it follows that

V303_13_2 C V303-1 . (2.3.21)


To determine the nature of A_3, we use the information obtained concerning

A_2 at the previous step and appeal to A2 to conclude:

i) A_3 consists of a collection of disjoint {Lv-vertical slabs

contained in Hso for all sp,s_l,s_2,s_3 E S such that

(A)3-33-2(A)3_23-1(A)3_13p = 1.

ii) d (v303-13-25-3) vv d (v3O3-13-2) < Vy d (VsOS-1)iii) By definition of A_3, A_2 and A_1 it follows that

vso$_1s_2S_3 C VSOS-1s-2 C VSOS_1.

Vsps_1S-23-3

(2.3.22)

(2.3.23)

Continuing to argue in this manner, we determine the nature of A_n, by using

the information obtained concerning A_n+l at the (n - 1)th step and appeal toA2 to conclude:

i) A-n consists of a collection of disjoint pv-vertical slabs V8p3_1.__5_n con-

tained in Hso for all so, s_1, , s_, E S such that (A)3_ns_n+1

...(A)3_13a = 1.ii) (L(VeOa_1...8_n) Gv d(VcOa_1...e-n+1)<...<yv-1 d(V(2.3.24)

iii) By definition of A-k, k = 1, 2, ..., n, we have

v3p8_1...3_n C V3p3_1...3_n+1 C ... C Vso$_1 . (2.3.25)

Before proceeding to discuss the limit as n -> oo we make the importantremark that, at each stage of the construction process, each iv-vertical slab canbe labelled uniquely by an admissible string of elements of S determined by thetransition matrix A and having a length of one plus the number of the step. Fur-thermore, all possible admissible strings of the appropriate length are realized at

each step due to the assumption A2.Now in the limit as n --+ oo we obtain the set

A-oo={pEDHI f -i(P)EH3_i; s_iES, i = 0, 1, 2,...,n,...}1(2.3.26)

and we can immediately make the following conclusions concerning the nature of

i) Each element of A_,,,, Vso3_1...8_, ..., s_i E S, i = 0, 1, ...,, can be labelledby a unique infinite sequence of elements of S allowed by the transition matrix

A. Furthermore, all possible such sequences are realized.


00ii) Since V303_1...3_n... = n Vso...s_n where V30.._3_, are µv-vertical slabs

n=1contained in Hip for all so, , s-n E S such that ... (A)S-150 =

1 and V30 ...s_n C Vso...s_n+l with d(Vso...3_n) -+ 0 as n -- oo, by

Lemma 2.3.1 we can conclude that A-, consists of a set of µv-verticalslices, Vso...s_, ..., with 3V30...3_n"' C ahHso. The cardinality of this setis determined by the transition matrix A. In particular, if A is irreducible,A_00 consists of an uncountable infinity of uv-vertical slices.

lb) Construction of A+,,.The construction of A+oo is virtually identical to the construction of A_00,

with the obvious modifications.

We begin by writing down expressions for the set of points that remain in Hunder 1, 2, ..., n, ... forward iterations by f.

Al = U (f 1(H31) n H30) = U H3031

aiES aiESi=0,1 i=0,1

={PEDH I PEHsO, f(P) EH3l; $o, s1 E S1 .

A2 _ I '(f-1 (H3132) n H30)aiES

i=0,1,2

U (f -'(H32) n f -1 (H31) n H30) U H303132aiES aiES

i =0,1,2 i=0,1,2

={ P C DH I P E H30, f (p) E H31, f 2(P) E H32 ;

An - U (f -1 (H31...sn) n Hso)

s0'sl's2 C S }.

aiESi=0,1,../,n

= U (f -n (H3,) n f-n+1 (Hsn-1) n ... n H30) = U H30...3,

aiES aiESi=0.1,...,n i=0.1,...,n

= {PE DH fa(P)EE Hs siES, i=0,1,...,n}.

(2.3.27)

(2.3.28)

(2.3.29)

Now, by A2, Al consists of a collection of disjoint µh-horizontal slabs H3031 in-

tersecting H30 fully for all so, sl E S such that (A)3031 = 1.


For A2, we use the information obtained concerning Al at the previous stepand appeal to A2 to conclude:

i) A2 consists of a collection of disjoint µh-horizontal slabs H303132 inter-

secting H3o fully for all so, 81, s2 E S such that (A)3p31(A)3132 = 1.

ii) d (H-10,102) < Vh d (H3031). (2.3.30)

iii) By definition of A2 and Al we have

H,08132 C H30 31 . (2.3.31)

Continuing to argue in this manner, we determine the nature of An by using

the information obtained concerning An-1 at the (n - 1)th step and appeal to A2to conclude:

i) An consists of a collection of disjoint !Eh-horizontal slabs H30...3n in-

tersecting Hip fully for all so, , sn E S such that (A)3p31(A)3132

ii) d (H3pS1...3rz) < vh d (H8pS1...3, n-1 d(H3031). (2.3.32)

iii) By definition of Ak, k = 1, 2, ..., n, we have

H30...3n C H30...3n_1 C ... C H3p31 . (2.3.33)

As in the construction of A-o., before proceeding to discuss the limit asn -+ oo we make the important remark that at each stage of the constructionprocess each uh-horizontal slab can be labelled uniquely by an admissible string of

elements of S determined by the transition matrix A and having a length of oneplus the number of the step. Furthermore, all possible admissible strings of theappropriate length are realized at each step due to the assumption A2.

Now in the limit as n --+ oo we obtain the set

A+oo = { p E DH I ft(p) E H3ti ; si E S , i = 0, 1, 2, ..., n, ... } , (2.3.34)

and we can immediately make the following conclusions concerning the nature of

A+,,.i) Each element of A+oc, H30...3n..., si E S, i = 0, 1, ...,, can be labelled by

a unique infinite sequence of elements of S allowed by the transition matrix A.

Furthermore, all possible such sequences are realized.


00ii) Since Hso...5.... = n Hso...s,,. where Hso...In are ph-horizontal slabs inter-

n=)secting Hso fully for all so, , sn E S such that (A)8031 ... (A)sn-1sn = 1

and Hso...sn+1 C Hso...In with d (Hso...sn) -+ 0 as n -+ oo, by Lemma2.3.1 we can conclude that A+,o consists of a set of ph-horizontal slices,Hso...sn... with 8H30...sn... C avHso. The cardinality of this set is deter-mined by the transition matrix A. In particular, if A is irreducible, A+oo

consists of an uncountable infinity of ph-horizontal slices.

lc) Construction of the Invariant Set A = A-oo fl A+00.From la) and lb) we have seen that A+oo consists of a collection of pv-

vertical slices Vs....8_n... with "so.-s..... C ahHso, and A_0, consists of acollection of ph-horizontal slices H50...3, ... with aHso...In... C ahHso wherethe subscripts on the slices are infinite sequences of elements of S admitted by the

transition matrix A. Thus, by Lemma 2.3.2, we see that A - A_co flA+eo consists

of a set of points corresponding to the intersection of the Vso...s_n... and theHI0...sn.... The cardinality of A (as well as other properties) depends on the nature

of the transition matrix A. In particular, if A is irreducible, then A consists of an

uncountable infinity of points.

2) Definition of the map 0: A --i E.For any point p E A we have

p = Vgp...g_n... fl Hso...gn... (2.3.35)

where Vso...3_n... is a pv-vertical slice with 3Vso...8_, ... C ahHso defined by

Vso...s_,...={pEDHI f-a(p)EHs_j ; siES, i=0,1,2,...}, (2.3.36)

and Hso...In... is a ph-horizontal slice with 8Hso...3, ... C ahHso defined by

Hso...In...={pEDHI f2(p)EH3 ; siES, i=0,1,2,...}, (2.3.37)

and the infinite sequences subscripting V30...3_, ... and Hso...In... satisfy

(A)sas;+1 = 1 , for all i. (2.3.38)

Thus, we define a map from A into EN as follows:

q:A -' EN

P H s = (2.3.39)


where the infinite sequences so s,, - and s_1 ... s_, are obtained from the

respective /.ch-horizontal and µv-vertical slices whose intersection is p; thus s E E.

This map is well defined since the Hi are disjoint.

From the definition of V30...3_, .. and Hso...sn... we see that the bi-infinitesequence associated to p contains a considerable amount of information concerning

the dynamics of p under iteration by f. In particular, if the ith entry of s is si thenwe know that fa(p) E H3a.

3) Prove that 0 is a homeomorphism.

The proof of this is virtually identical to the proof of the similar assertion given

in our discussion of the Smale horseshoe map in Section 2.1.

4) Prove that Qi o f = or o 0.

This is an immediate consequence of the definition of ¢. Let p E A with0(p) = { s_n s_1.sps1 sn . .}. By the construction of A, p is the uniquepoint in H such that f 5(p) E Hsi, i = 0, ±1, ±2, ... and therefore 0 o f (p) ={ s_n s_iso.sl . . . sn . . .}. But, by the definition of the shift map or, we haveao4)(p) = { s_n s_lsp.sl ... sn ...}. Thus, we have established that 4.o f (p)a o 4)(p) for any PEA.

Let us make the following remarks concerning Theorem 2.3.3.

1) Recall the consequences of f IA being topologically conjugate to aIEN (seeA

Section 2.2). In particular, if A is irreducible, A is a Cantor set of points andf IA exhibits the same rich dynamics as al EN described in Section 2.2.

A2) It is important to note that, although Theorem 2.3.3 describes a very rich

dynamical structure for f, it by no means tells the complete story of the dy-namics of f. Many important global questions remain involving the dynamics

in a neighborhood of A (see the comments at the end of Section 2.1).

3) An obvious question is what does one look for in the phase space of a map in

order to show that Al and A2 hold for that map? One might presume that anontrivial amount of knowledge concerning the dynamics of the map is needed.

In Chapter 3 we will see that special types of orbits, called homoclinic andheteroclinic, often give rise to conditions for which Al and A2 hold. Regarding

the condition A2, for theoretical purposes (e.g., as in proving Theorem 2.3.3)

our form of the statement of the condition is often the easiest to utilize. How-

ever, for computational purposes, A2 can be difficult to implement, therefore


we next address the question of devising an alternative condition to A2 which ismore computationally oriented.

2.3c. Sector Bundles

The condition A2 gives uniform estimates on the contraction of the widths of Ah-

horizontal slabs under f -1 and the contraction of the widths of AV-vertical slabsunder f. Typically, when one thinks of expansion or contraction properties onethinks of the behavior of the jacobian of the map, its eigenvalues and eigenvectors,

and how they vary over the region in question.

Recall the geometrical point of view described in Section 1.3 of the derivative

of the map at a point acting on tangent vectors emanating from that point. We will

quantify the expansion and contraction properties of f by describing how it acts on

tangent vectors which are aligned in certain directions or sectors, and we now begin

the development of these ideas. First, however, we need an additional hypothesis

on f.Recall that in Al or A2 no differentiability properties were stated for f, and

they are obviously needed if we are to consider the derivative of f.

Hypothesis: Let )( = U Hij and V = U V.i, then f is C1 on N and f isijES i,jES

(A)ij=1 (A)ij=1C1 on V.Note that f (4) = V.

Now choose a point z0 = (x0, y0) E V U N. The stable sector at z0, denotedSzp, is defined as follows:

Szo = {(ez0, 0) E IR' X X R' 1117zol Ah IezoI} . (2.3.40)

The unstable sector at z0i denoted Szp is defined as follows:

So = {(e,,, 77,0) E 1Rn x im I iezo1 <A, InzzI}z . (2.3.41)

Geometrically, we think of (ezpfrizo) as a vector emanating from the point z0.Thus, SZ0 and Szo appear geometrically as cones with vertex at z0, see Fig-ure 2.3.8.

For z0 E N we have f (zo) E V. Now recall from Section 1.3 that D f (zo)maps tangent vectors at zo to tangent vectors at f (zo). Applying this to all vectors

2.3. Criteria for Chaos: The Hyperbolic C:.se 129

Figure 2.3.8. Stable and Unstable Sectors at z0.

in Szo, we see that D f (zo) maps the cone S ,'O to some cone with vertex at f (z0).

Similarly, D f -1(zo) maps the cone SZ0 to some cone with vertex at f -1(zo). Wewill be interested in the behavior of D f and D f -1 on the cones So and Szo,respectively, as zo varies over the regions X and V, respectively. We define thefollowing sets:

Ss - SsN zozoEN

SSV - U Sszo

zoEN

Su= U SoN

zoENSU U Szo

zoEN

(2.3.42)

These sets are called sector bundles or cone fields. We have the following hypothesis:

A3. Df(Su) C Sv and Df(SV) C S. Moreover, if (Czpfr)zo) C Szp and

Df(zo)(ezo,,7zo)=(ef(zo)'77f(zo))CS'zp),then we have lrlf(zo)I > Ir/z0I

Similarly, if (Ezo,''7zo) E SZO and Df-1(z0)(Ezo,r7zo) -- (ef-1(zo)'1f-1(zo)) E


S f_i(z ), then we have I f_1(ZO) > I µJ ezpl where 0 < It < 1 - /iv/h, seeFigure 2.3.9.

We will shortly show that A3 can be substituted for A2, however first wewill derive a result which will be useful for estimating the widths of images ofµh-horizontal and /sv-vertical slabs under f.

Figure 2.3.9. Images of Sectors under D f and D f -1.

Let H be a µh-horizontal slab. Let Hl and H2 be disjoint µh-horizontal slices

contained in H with 8H1 and 8H2 contained in 8vH. We denote the domain ofthe functions h1 (x) and h2 (x) of which Hl and H2 are the graphs by I. Let V1and V2 be disjoint µv-vertical slices contained in H with 0V1 and 8V2 containedin BhH. We denote the domains of the functions v1(y) and v2(y) of which V1and V2 are the graphs by Iy and Iy, respectively. Let

Ilhl - h2II = sup ihl(x) - h2(x)I ,xEly

(2.3.43)

IIv1 -veil = sup Ivl(y) -v2(y)I .yEIy nIy

(2.3.44)


By Lemma 2.3.2 , Hl and V1 intersect in a unique point which we call zl =(x1, y1), and H2 and V2 intersect in a unique point which we call z2 - (x2,y2)See Figure 2.3.10 for an illustration of the geometry.

z2=(x2,y2)

Figure 2.3.10. Intersecting Horizontal and Vertical Slices.

We have the following lemma.

Lemma 2.3.4.

2) Iy1-y2I-

11_uvuh[11hl-h211+Ah1111-v2II1

PROOF: We have

1x1 - x21 = Ivl(yl) - V2(y2)1 < v1(yl) - V1 (Y2)1 + Ivl(y2) - v2(y2)I

Using (2.3.4) and (2.3.44), (2.3.45) becomes

Ix1 --2I <- AvIy1-Y21+11111 -11211

We also have

(2.3.45)

(2.3.46)

Iyl - Y21 = Ihl(xl) - h2(x2)I < Ihi(xi) - hl(x2)I + hl(x2) - h2(x2)I. (2.3.47)


Using (2.3.3) and (2.3.43), (2.3.47) becomes

Y1 - Y21 : µh 1-1 - x21 + 11hl - h211 (2.3.48)

Substituting (2.3.48) into (2.3.46) gives 1), and substituting (2.3.46) into (2.3.48)

gives 2).

Theorem 2.3.5. If Al and A3 hold with 0 <A <

Vh=Vv =1-!yµ .1 - µhµv then A2 holds with

PROOF: We will prove that the part of A2 dealing with horizontal slabs holds,since the part dealing with vertical slabs is proven similarly. The proof proceeds in

several steps:

1) Let H be a ph-horizontal slice contained in Hj with aft C avHj. Then weshow that f -1 (H) n Hi = Ht is a µh-horizontal slice contained in Hi withoHi c avHi for all i E S such that (A)ij = 1.

2) Let H be a µh-horizontal slab which intersects Hj fully. Then we use 1) toshow that f-1 (H) n Hi - Ht is a µh-horizontal slab intersecting Hi fully forall i E S such that (A)ij = 1.

3) Show that d (II{) < A d(H).

We begin with step 1). Let H be a jah-horizontal slice contained in Hj withaft C avHj. We denote the region in the y = 0 plane over which Hj is definedby Iy, i.e., Iy is the domain of the function h(x) whose graph is H.

Since aH C avHj, by Lemma 2.3.2 we know that H intersects Vji witha (H n Vji) C avVji for all i E S such that (A)ij = 1. Now Al holds so thatf-1 (avVji) c avHi; therefore, f -1 (a (H n Vji)) C avHi. So f -1 (H n Vjj)

consists of a collection of n dimensional sets with a (f -1 (H n Vji)) C avHi, seeFigure 2.3.11.

We now show that f -1 (H n Vji) are uh-horizontal slices. By A3, sinceD f -1 maps Sv into S, it follows from the mean value theorem that for any twopoints (x1, y1), (x2, y2) E f-1 (H n Vji) we have

Iy1 - Y21:5 Ah I X1 - x21. (2.3.49)

This shows that f -1 (H n Vji) is the graph of some function h(x) defined over

Iz and satisfying

h(x1) - h(x2) I Ah Ixl - x21 . (2.3.50)


Figure 2.3.11. Image of H under f -1.

Step 2). Let H be a µh-horizontal slab intersecting Hj fully. Therefore,

av (H n Vji) C BvVji

for all i E S such that (A) ij = 1. Now applying the result of step 1) to the

horizontal boundary of H n Vji, we see that the f-1 (H n Vji) Hi are disjointµh-horizontal slabs intersecting Hi fully for each i E S such that (A)ij = 1.

Step 3). We now show that d (Hi) < 1 d(H). Let H be a µh-horizontal slab

intersecting Hj fully, and let f (H n Vji) - E. Fix i and let PO = (x0, Yo)

and pl = (xl, yl) be two points on the horizontal boundary of Hi which have thesame x coordinate, i.e., x0 = x1, such that

d (Hi) = Ip0 - p1I = Iyo - yl I (2.3.51)

See Figure 2.3.12.

Consider the line

p(t) = (1 - t) p0 + tpl , 0 < t < 1 (2.3.52)


and the image of p(t) under f, which is the curve z(t) = f (p(t)). Writing z(t) _(x(t), y(t)), since A3 holds, we have

Iy(t)IAt)I>0, 0<t<1, 0<14 <1-µv/Lh, (2.3.53)

from which we conclude that

Ipo - pil ! o ly(0) - y(1)I (2.3.54)

By Al, z(O) = (x(O),y(O)) and z(l) = (x(1),y(1)) are points contained inthe horizontal boundary of H, see Figure 2.3.12. Hence z(0) and z(1) lie onhorizontal slices which are represented as graphs of the functions ho(x) and h1(x),

respectively. Since p(t) is parallel to the x = 0 plane the tangent vector to p(t),p(t), is contained in Sjt for 0 < t < 1. Then, by A3, the tangent vector to z(t),z(t) = D f (p(t)) p(t), is contained in Sv for 0 < t < 1. Therefore, z(O) and z(1) lieon some Fiv-vertical slice V C Vii. So we can apply Lemma 2.3.4 to (2.3.54) and

obtainlp0 - pl I 1 _ IIh0 - h111

(2.3.55)

< µ d(H) .1 - AvAh

Since Ipo - pl l = d(H), and this argument holds for all i E S such that (A)i? = 1,this completes the proof.

2.3d. Alternate Conditions for Verifying Al and A2

We now want to give a set of alternate conditions which imply Al and A2 ofTheorem 2.3.3. We will see in Chapter 3 that these conditions are often more easily

applied near orbits homoclinic to fixed points of ordinary differential equations than

Al, A2, and A3. These conditions are actually higher dimensional generalizations of

those given by Afraimovich, Bykov and Silnikov [1984], which they obtained during

their study of the Lorenz equations.

First we will need to include a differentiability hypothesis on f .

Hypothesis: f is a C' diffeomorphism of DH onto f(DH).

As notation, for (x, y) E D we let

f(x,y) = (fl(x,y),f2(x,y)) = (x,9) (2.3.56)

2.3. Criteria for Chaos: The Hyperbolic Case

Figure 2.3.12. Width of H under f -1

and our first condition for f is as follows.

Al.

135

IIDzf1II < 1 (2.3.57)

II(Dyf2)-lII < 1 (2.3.58)

1- II(Dyf2)-1IIIIDxf1il > 2 IlDyfiIIIIDzf21III(Dyf2)-1112 (2.3.59)

1-(IIDzf1II+I1(Dyf2)-'II)+IIDxf1IIII(Dyf2)-'II > IDzf21IIIDyfiI1 (Dyf2)-lI1(2.3.60)

where11 ' II = sup and is some matrix norm. Al is the condition dealing

(x,y)EDHwith stretching and contraction rates and, in the applications which we address in

Chapter 3, they will be readily verifiable.

We next need to introduce some condition governing the behavior of horizontal

and vertical slabs and their respective boundaries under f. However, first we willelaborate on some implications of Al which will be needed later on.


Alternate Representation for f. By A1, since II(Dyf2)-'II < 1, we can utilize theimplicit function theorem and rewrite y = f2(x, y) in an alternate form by solving

for y as a function of x and y, i.e., we have

y = f2(x,y) (2.3.61)

which we can substitute into (2.3.56) and obtain

fl (x, 9) = (1(x,12 (x, y))

y = f2(x,f2(X,Y))

(2.3.62)

(2.3.63)

This will be useful when we begin to examine the behavior of vertical and horizontal

slices under f and f -1. The following estimates are trivial applications of theimplicit function theorem.

IIDzf1II <- IIDxfiIl + IIDzf2IIIIDYf1IIII(Dyf2)-'II

IIDyf1II < IIDyf1IIII(Dyf2)-'II

IIDzf2II < IIDzf2IIII(Dyf2)_1II

IDyf2II < II(DYf2)-1II

(2.3.64)

(2.3.65)

(2.3.66)

(2.3.67)

We will now give a series of lemmas whose motivation at present is not obvious

but will become more so later on.

Lemma 2.3.6. Suppose Al holds. Then the inequalities

IlDyfl ll II(Dyf2)-111µh - (1- I1Dzfl II II(Dyf2)-1II)/2h + IIDzf2II II(Dyf2)_1II < 0(2.3.68)

IIDzf2IIII(Dyf2)-1II/4 - (1 - IIDzfiIIII(Dyf2)-11)µ-+ IIDyfiIIII(Dyf2)-1II < 0(2.3.69)

have positive solutions. Moreover, these solutions lie in the intervals 0 < µh in << /Lv < /tmaxmin/Lh < /rm ax and 0 < /Lv

a = IIDzf1II1I(DYf2)-111,

respectively, where, setting

max 1 - a ± (1- a)2 - 4IIDyf1 II IIDzf2II II (Dyf2)-' 112

/ min(2.3.70)

h 2IIDyf1 1111 (Dyf2) -III


ax - 1-a±(Dyf2)-111

(2.3.71)

PROOF: This is a trivial calculation noting that by Al we have

1 - 1Dxf1II II(Dyf2)-111 > 0

(1 - IID.f11111(Dyf2)-111)2 -4IIDyfillllDxf2IIII(Dyf2)-1112 > 0.

Lemma 2.3.7. Suppose Al holds, and let ph > 0 satisfy (2.3.68) and /tv >0 satisfy (2.3.69). Then it follows that

ph < 1/(IIDyf1IIII(Dyf2)-lII)

A. < 1/(IIDaf2IIII(Dyf2)-111)

(2.3.72)

(2.3.73)

PROOF: Equation (2.3.72) follows by examining it a" and noting that by Al thenumerator is smaller than 2. A similar argument applied to pmax gives (2.3.73). El

Lemma 2.3.8. Suppose Al holds. Then there exists ph satisfying (2.3.68) and pvsatisfying (2.3.69) such that

1 - II(Dyf2)-1IIph

< IIDyf11111(Dyf2)-1i1

(2.3.74)

1 - IIDxf1II (2.3.75)AVC IIDxf211

PROOF: We give the proof for (2.3.74) with the proof for (2.3.75) following a similar

line of reasoning.

By Lemma 2.3.6, for (2.3.74) to hold, it is sufficient that (once again setting

a =IID2f1II11(Dyf2)-111 )

1 - a - 1 - a) 2 - 4IIDyf1II llDzf2IIII(Dyf2)-1112 1 (I(Dyf2)-1II2IIDyf11111(Dyf2)-111 < IIDyfl1111(Dyf2)-111

(2.3.76)

or

(1 - a)2 - 411Dyf11111D2f21111(Dyf2)-1112 > 2I1(Dyf2)-111 - a- 1 . (2.3.77)


If the right hand side of (2.3.77) is negative, then we are finished. If it is positive,

then squaring both sides and subtracting away similar terms leads to (2.3.60).

Lemma 2.3.0. Suppose Al holds with µh satisfying (2.3.68). Then there existsAV satisfying (2.3.69) such that

0<PvPh<1.

PROOF: From (2.3.70) and (2.3.71) it follows that

AmaxAh in = AminAm ax = 1

and from (2.3.79) the lemma follows.

(2.3.78)

(2.3.79)

11

We can now state the condition describing how f behaves on horizontal andvertical slabs.

A2. Hi, i = 1,.. . , N are µh-horizontal slabs with Ph satisfying (2.3.68) and

(2.3.74). For all i, j E S such that (A)ij = 1, V?i is a AV-vertical slab withA v satisfying (2.3.69), (2.3.75), and (2.3.78). Moreover, we require 9 Vji cof (IIi) and f-1(avVji) C avHi.

Our goal will now be to show that Al and A2 imply Al and A2 and henceTheorem 2.3.3. However, first we will need two preliminary lemmas.

Lemma 2.3.10. Suppose Al and A2 hold, and let H be a µh-horizontal slicecontained in Hj with 8H C 8 Hj with µh satisfying (2.3.68). Then f -1(H) nHi is also a µh-horizontal slice contained in Hi with 8(f -1(H) nHi) C BvHi forall i E S such that (A)ij = 1 and with Ph satisfying (2.3.68).

PROOF: The proof is accomplished in several steps:

1) Describe f -1(H) n H.2) Show that for each i E S such that (A)ij = 1 f -1(H) n Hi is the graph of

a function of x.

3) Show that f -1(H) n Hi is µh-horizontal and that ph satisfies (2.3.68).

We begin with Step 1). Note that f -1(H) n Hi = f -1(H n Vji). Now, since8H C 8vHj, we know that H intersects Vji with 8(H n Vji) C BvVji for alli E S such that (A) ij = 1. Now A2 holds so that f-1(avVji) C 9vHi; therefore,


f -1(a(HnVji)) c a,Hi. So f -1(HnVj2) consists of a collection of n dimensionalsets with a(f -1(H n Vji)) c see Figure 2.3.11.

Step 2). Fix any i E S such that (A)ij = 1 and consider H n Vji. Now, sinceH is a µh-horizontal slice, any two points (sl, yl) , (-t2, 92) E H n Vji satisfy

Iyl - 921 <- /h Izl - X21 (2.3.80)

Therefore, H n Vji is the graph of a Lipschitz function y = H(x) with Lipschitzconstant µh.

Now consider f -1(H n Vji). We want to show that this set is the graph of afunction. Now any point (x, y) E f -1(H n Vji) must satisfy

9 = f2(x,y) = H(f1(x,y)) = H(a) . (2.3.81)

If we denote X = {x E 1Rn 13 y E Rm with (X, Y) E f-1(H n Vji)}, then(2.3.81) has at least one solution for each x E X. We now show that this solutionis unique. Recall that since Al holds, then by the implicit function theorem analternate expression for the map f = (fi, f2) is given by

-* = f1(x,9) (2.3.82)

y=f2(x,y) (2.3.83)

Now, also by the implicit function theorem, (2.3.82) defines y as a function of x

under the condition

/hIIDyflll < 1

or, using (2.3.65), (2.3.84) becomes

uh < 1/IIDyf1IIII(Dyf2)-1II

which follows from Lemma 2.3.7. Thus, when (2.3.85) holds we have

y = y(x)

and substituting (2.3.86) into (2.3.83) gives

y = f2(x,9(x)) = h(x).

(2.3.84)

(2.3.85)

(2.3.86)

(2.3.87)


Hence, we have shown that f -1(H n Vji) is the graph of a function y = h(x).

Step 3). We now show that y = h(x) is Lipschitz with Lipschitz constant µh.

Let (x1,91), (x2,92) E f-1(H n Vja) with their respective images under f

denoted by (x1, 91), (x2+92) E H n Vji. Then the following relations hold.

91 = H(fl(x1,91))

92 = H(fl(x2,92))

Y1 = f2(xi,91)

Y2 = 12(x2,92)

(2.3.88)

(2.3.89)

(2.3.90)

(2.3.91)

Using (2.3.88)-(2.3.91) along with (2.3.64)-(2.3.67) we obtain the following estimate

yl - 92I<(IIDxf1Il + IDxf2IIIIDyf1IIII(Dyf2)-'II) ph

Ix1 - x2I (2.3.92)1- Lhll Dyflllll(Dyf2)-1II

which we use to obtain

Iyl - y2I = h(x1) - h(x2)I

AhII Dxflil l1(Dyf2)-1Il +IIDxf2IIII(Dyf2)-1II (2.3.93)

1- IhlIDyf1II II(Dyf2)-1IIIxl - x2I .

(Note: positivity of 1 - µh II Dyflll II (Dy f2) -1 II follows from Lemma 2.3.7.) Thus,

from (2.3.93) we see that h(x) is Lipschitz with Lipschitz constant µh provided

BhllDxf1IIII(Dyf2)-1II + IIDxf21III(Dyf2)-1II < Ah ,

1 - IhlLDyf1IIII(Dyf2)-1II(2.3.94)

that is, µh must satisfy

IIDyf1IIII(Dyf2)-1ll/h -(1- IIDxfl ll II(Dyf2)-1ll),h + IlDxf2II ll (Dy f2)1II < 0

which it does by hypothesis.

Lemma 2.3.11. Suppose Al and A2 hold, and let V be a p -vertical slice contained

in Hj with aV C BhH2 and with liv satisfying (2.3.69). Then f (V) n Hk is alsoa µv-vertical slice contained in Hk with a(f(V) n Hk) C ahHk for all k E Ssuch that (A) jk = 1 and with uv satisfying (2.3.69).

PROOF: The proof is very similar to the proof of Lemma 2.3.10 and proceeds inthree steps.


1) Describe f (V) n H.

2) Show that for each k E S such that (A) jk = 1 f (V) n Hk is the graph ofa function of y.

3) Show that f (V) n Hk is /.tv-vertical and that µv satisfies (2.3.69).

We begin with Step 1). Note that f(V) n Hk = f(V n Hjk). Now, sinceBV C BhHj, we know that V intersects Hjk with B(V n Hjk) C ahHjk for allk E S such that (A) jk = 1. Since A2 holds, we know that f (BhHjk) C BhHk;therefore, f (B(V n Hjk)) c BhHk. So f (V n Hjk) consists of a collection of in.dimensional sets with B(f (V n Hjk)) C BhHk.

Step 2). Fix k E S such that (A)jk = 1, and consider V n Hjk. Since V is a

µv-vertical slice any two points (x1, yi), (x2, y2) E V n Hjk satisfy

Ix1 -x21:µvIyi-Y2I (2.3.95)

Therefore, V n Hjk is the graph of a Lipschitz function x = v(y) with Lipschitzconstant µv.

We now want to show that f (V n Hjk) is the graph of a function. For

(v (y), y) E V n Hjk we have

fl (V (y), Y)

f2 (v(y), y) .

By the implicit function theorem, when

II(Dyf2)-liIIIDxf2llµv < 1

(2.3.96)

(2.3.97)

(2.3.98)

we can solve equation (2.3.97) for y as a function of y, i.e., y = y(y) and (2.3.98)holds by Lemma 2.3.7. Thus, substituting y = y(y) into (2.3.96) gives

x = fi (v(y(y)),y(y)) = V (Y)

Therefore, f (V n Hjk) is the graph of x = V(9).

(2.3.99)

Step 3). We now show that x = V (p) is Lipschitz with Lipschitz constant µv.


Let (x1,y1), (x2,y2) E V n Hjk and denote their respective images under f

by (±1, yl), (22,92) E f (V n Hjk). Then the following relations hold.

xl = fl(v(y1),y1) (2.3.100)

02 = f1(v(y2),y2) (2.3.101)

Y1 = f2(00,91) (2.3.102)

Y2 = f2 (v M), 92) (2.3.103)

Using (2.3.100)-(2.3.103) along with (2.3.64)-(2.3.67) we obtain the following esti-

mate

y1 - y2III 1c

)Il-1 11µv1 - Dzf21111(Dyf )

which we use to obtain

191 -921 (2.3.104)

1±1 - x21 = IV (y1) - V(92)1

(2.3.105)< IIDyf1III1(Dyf2)-'II +µv1IDzf1IIII(Dyf2)-III Iyl -y2I1- IIDzf2II II (Dyf2)-I 11µt?

So x = V (q) is Lipschitz with Lipschitz constant µv provided

IDyf1IIII(Dyf2)-III+,VIIDxf1IIII(Dyf2)-III <µv 2.31061- IIDzf2IIII(Dyf2)-'11µv

(. )

or

IDzf2IIII(Dyf2)-111µv - (1- IIDxf1III(Dyf2)-111)µv + IIDyf1IIII(Dyf2)-'II < 0

and (22.3.107) holds by hypothesis.

We are now ready to prove our main theorem.

Theorem 2.3.12. Al and A2 imply Al and A2.

(2.3.107)

0

PROOF: That A2 implies Al is obvious; thus, we only need to show that Al andA2 imply A2.

We begin with the part of A2 dealing with horizontal slabs. The proof consists

of two steps.

1) Let H be a µh-horizontal slab which intersects Hj fully. Then we show thatf-1(H) n Hi - H i is a µh-horizontal slab intersecting Hi fully for all i E S

such that (A)ij = 1.


2) Show that d(Hi) < vhd(H) with 0 < vh < 1.

We begin with Step 1). This follows immediately by applying Lemma 2.3.10

to the horizontal boundary of H.

Step 2). For fixed i let pl = (xl, y1), P2 = (x2, y2) be two points on the horizontalboundary of Hi with x1 = x2 such that

d(Hti) = IPl - P21 = I Y1 - y2I . (2.3.108)

We denote their respective images under f by (al, yi) and (-t2,92). Now, by A2,

(x1, y1), P2,92) E BhH; hence 91 = Hl(.t1) and 92 = H2(x2) where the graphof the functions y = H1(x) and y = H2(x) are µh-horizontal slices.

Now we have the relations

Y1 = f2(x1,91) (2.3.109)

Y2 = f2(x2,y2) (2.3.110)

yl = Hl(z1) = H,(fl(xl,91)) (2.3.111)

92 = H2 P2) = H2(fl(x2,92) )- (2.3.112)

Subtracting (2.3.109) from (2.3.110) and using (2.3.67) along with the fact thatx1 = x2 yields the estimate

Iy1 - y2I < II(Dyf2)-1II I91 - 92I . (2.3.113)

Subtracting (2.3.111) from (2.3.112) and using (2.3.65) along with the fact that H1

and H2 are µh-horizontal and x1 = x2 yields the estimate

1191-92I 1-FihilDyf1IiII(Dyf2)-1IIIIH1-H2II

(2.3.114)

Combining (2.3.114), (2.3.113), and (2.3.108) yields

II(Dyf2)-1II I(12)d (Hi)(H2) <1 - uhIlDyf1IIII(Dyf2)

1 IIII Hl - H2II 1 _ hIIDIIII( Yf2) 1 II

d(H)

(2.3.115)

and we haveII(Dyf2)-'Ii

(2.3.116)1- µhlIDyfiIIII(Dyf2)-flj

<1,


since Ah satisfies (2.3.74).

Now the part of the proof dealing with vertical slabs proceeds along the same

lines in two steps.

1) Let V be a Fiv-vertical slab contained in Hi such that also V c V32 for somei, j E S with (A) {j = 1. Then we show that f (V) n Hk - Vk is a µv-verticalslab contained in Hk for all k E S such that (A) jk = 1-

2) Show that d(Vk) < vvd(V) with 0 < vv < 1.

We begin with Step 1). This follows immediately by applying Lemma 2.3.11

to the vertical boundary of V.

Step 2). For fixed k let PI = (21, 91), P2 = (22,92) be two points on the verticalboundary of Vk with yl = 92 such that

d(Vk) = IP1 - P2I = Ixl - x2I (2.3.117)

We denote their respective images under f -1 by (xl, yl) and (x2i y2). Now, by A2,

(x1,91), (x2,92) E avV; hence x1 =V1(yl) and X2 =V2(y2) where the graphsof the functions x = VI (y) and x = V2(y) are jAv-vertical slices. The followingrelations hold:

x1 = fl(VV(yl),yl)

x2 = f1(V2(y2),y2)

Yi = f2(VI(yl),91)

Y2 = f2(V2(y2), 2)

(2.3.118)

(2.3.119)

(2.3.120)

(2.3.121)

Subtracting (2.3.120) from (2.3.121) and using (2.3.66) along with the fact that

91 = 92 yields the estimate

Iyl -y2I <_IIDxf2II II(D f2)-'II

IIVI - V2II1- IIDxf2IIII(yf2) Iluv(2.3.122)

Subtracting (2.3.118) from (2.3.119) and using (2.3.64) along with the fact that

91 = 92 yields the estimate

Ixl - x2I <_ (IIDxf1II + IIDxf2IIIIDyf1IIII(Dyf2)-'II)(iv Iyl - y2I + IIVI - V2 11)(2.3.123)


Combining (2.3.123) and (2.3.122) gives

d(Vk) _ Ixi - 221 (2.3.124)

(II Dyfl II II (Dyf2)-' II + IlDxfl II I

(Dyf2)_'11y.

IIDxf2II + IlDxfi Il) IIVi - V2111 - I1Dxf2II II(Dyf2)-1l1/.v

(1 vll Dxf2II + IDxf1Il)! Vi - V2I1 (2.3.125)

< (iiv IDxf2II + I Dxf1 )d(V) (2.3.126)

and lzvt Dxf2II + IDxf1II < 1 by (2.3.75).

2.3e. Hyperbolic Sets

In Chapter 1 we introduced the idea of a hyperbolic fixed point of a map or flowand, more generally, the idea of a normally hyperbolic invariant manifold. We now

want to show that the invariant set A constructed in Theorem 2.3.3 shares someproperties similar to these invariant sets. We begin by giving the definition of ahyperbolic invariant set of a map.

Definition 2.3.8. Let f : ]R'i --> lR' be a C' (r > 1) diffeomorphism, and let Abe a closed set which is invariant under f. We say that A is a hyperbolic act if for

each p E A there exists a splitting lR" = Ep (D Ep such that

1)

Df (p) - Ep = Ef(P) (2.3.127)Df (P) - Ep = Ef (P)

2) There exist positive real numbers C and A with 0 < A < 1 such that

if p E Ep, I Df'i(p)epl < CA' IEPI

if '7p E Ep , Df -n(p)?Ipl < CA'd InPI(2.3.128)

3) Ep and EP vary continuously with p.

Let us make the following remarks concerning Definition 2.3.8.

1) It should be clear from Definition 2.3.8 that the orbits of points in A have awell-defined asymptotic behavior. Specifically, orbits whose initial points areslightly displaced from p along directions in Ep converge exponentially to the


orbit of p as n -+ oo, and orbits whose initial points are slightly displaced from

p along directions in EP converge exponentially to the orbit of p as n --> -oo.

2) It is important to note that the constant C does not depend on p; in this case,the set is sometimes said to be uniformly hyperbolic. Much of dynamical sys-

tems theory has been built around uniform hyperbolicity assumptions; however,

recently Pesin [1976], [1977] has developed a theory of nonuniform hyperbolicity

which is still waiting to be exploited for the purposes of applications.

Finally, we remark that in much of the literature of dynamical systems theory

the constant C is taken to be one. This can be done by utilizing a specialmetric called an adapted metric. This trick is due to Mather, and a discussion

of it can be found in Hirsch and Pugh [1970]. Although it is of tremendoustheoretical use, because the main purpose of this book is to develop techniques

which are applicable to specific dynamical systems arising in applications, we

will not state the theorems in terms of an adapted metric, since computations

with such a metric may be somewhat unwieldy.

3) Continuity of the splitting 1Rn = EP 0) EP can be stated in several equivalent

ways. One statement, sufficient for our purposes, is as follows: let p be fixed,

and choose a set of basis vectors for EP and EP; then the splitting is said tobe continuous if the basis vectors vary continuously with p. More discussion of

this point can be found in Nitecki [1971] or Hirsch, Pugh, and Shub [1977].

Note that the idea of a hyperbolic invariant set of a map is developed in terms of

the structure of the linearized map. We will see that this structure has implications

for the nonlinear map. First we begin with some definitions. For any point p E A,

e > 0, the stable and unstable sets of p of size a are defined as follows:

WE(p)={p'EE AI Ifn(p)-fn(p')I <e forn>0}(2.3.129)

WE (P) pI C A I f -n(p) - f -n (P) < e forn > 0 }

Now from Theorem 1.3.7, we have seen that if p is a hyperbolic fixed point thefollowing hold.

1) For a sufficiently small, WE (p) is a Cr manifold tangent to EP at p and havings

the same dimension as p. WE (p) is called the local stable manifold of p.

2) The stable manifold of p is defined as follows:00

Ws (P) = U f -n (W'sn=0

(2.3.130)


Similar statements hold for WE (p).

The invariant manifold theorem for hyperbolic invariant sets (see Hirsch, Pugh,

and Shub [1977]) tells us that a similar structure holds for each point in A.

Theorem 2.3.13. Let A be a hyperbolic invariant set of a Cr (r > 1) diffeomor-phism f. Then for e > 0 sufficiently small and for each point p E A the followinghold.

1) WE (p) and WE (p) are Cr manifolds tangent to EP and EP, respectively, at

p and having the same dimension as EP and EP, respectively.

2) There are constants C > 0, 0 < A < 1, such that, if p' E WE (p), thenfn(p) - fn(p') I < Can Ip - p'I for n > 0 and, if p' E WE (p), then

If-n(p)-f-n(p,)I <Canlp-p'I for n>0.3)

f (WE (p)) C We (f (p))(2.3.131)

f-1 (WE (p)) C W, (f-1(p))

4) WE (p) and WE (p) vary continuously with p.

PROOF: See Hirsch, Pugh, and Shub [1977].


1) The constants C > 0 and 0 < A < 1 do not necessarily need to be the sameas those appearing in Definition 2.3.8.

2) What it means for WE (p) and WE (p) to vary continuously with p is bestexplained within the context of function space topologies, see Hirsch [1976] or

Hirsch, Pugh, and Shub [1977].

3) For any point p E A the stable and unstable manifolds of p are respectivelydefined as follows:

00

n=000

Ws(p) = U f-n (WE (fn(p)))

Wu(p) = U fn (WE (f-n(p)))

n=0

(2.3.132)

Now, in practice, verifying the conditions of Definition 2.3.8 for an invariantset of a map is quite difficult. Fortunately, there is an equivalent formulation ofhyperbolicity due to Newhouse and Palis [1973], which we now describe.


As above, let f: R" --' Rn be a C' (r > 1) diffeomorphism, and let A be aclosed set which is invariant under f. Let R' = EP ® EP be a splitting of 1R'i for

p E A, and let µ(p) be a positive real valued function defined on A. We define theµ(p) sector, denoted S,,(p), as follows:

Sµ(p) = { (ep,rlp) E EP ® EP I C p(p) I77P1 } (2.3.133)

and we define the complementary sector, S',,(P), as follows:

S, (P)1Rn - SJ(P)

Then we have the following theorem.

(2.3.134)

Theorem 2.3.14. Let f : Rn -4 1Rn he a Cr (r > 1) diffeomorphism, and letA C 1R't be a closed set which is invariant under f. Then A is a hyperbolic invariant

set if and only if there exists a splitting Rn = EP (D EP for each p E A, an integer

n > 0, constants C > 0, 0 < a < 1 with Can < 1, and a real valued functionEt: A -+ R+ such that the following conditions are satisfied:

1) sup max({t(p),µ(p)-1)} < no. (2.3.135){pEA

2) For each p E A, we have

a) Dfn(P) . S.U(P)

C S1.(fn(P))and

b) if ep E Sµ(P) I Can I PI

c) if ep E Sµ(p) , Df-n(P)epl <_CAnIEpl(2.3.136)

The proof of this theorem can be found in Newhouse and Palis [1973]. Theorem

2.3.14 tells us that in order to establish hyperbolicity for A we need only find bundles

of sectors S = U Sµ(p), S' = U S1(p), such that D f maps S into S whilepEA pEA

expanding each vector in S, and Df maps S' into S' while contracting each vectorin S'. See Figure 2.3.13 for an illustration of the geometry of Theorem 2.3.14.

Now, regarding our map, the reader should notice that, if Al and A3 hold,then the invariant set A is a hyperbolic invariant set, since the conditions of A3are weakened versions of the necessary and sufficient conditions for a set to behyperbolic given in Theorem 2.3.14.


Figure 2.3.13. Geometry of Theorem 2.3.14.

The reason for discussing the concept of hyperbolic invariant sets is that they

have played a central role in the development of modern dynamical systems theory.

For example, the ideas of Markov partitions, pseudo orbits, shadowing, etc. all

utilize crucially the notion of a hyperbolic invariant set. Indeed, the existence of a

hyperbolic invariant set is often assumed a priori. This has caused great difficulty to

the applied scientist since, in order to utilize many of the techniques or theorems of

dynamical systems, theory he must first show that his particular system possessesa hyperbolic invariant set. Theorem 2.3.5 allows one to establish this fact. For

more information on the consequences and utilization of hyperbolic invariant sets

see Smale [1967], Nitecki [1971], Bowen [1970], [1978], Conley [1978], Shub [1987],

and Franks [1982].

2.3f. The Case of an Infinite Number of Horizontal Slabs

In some applications it may arise that our map contains an invariant Cantor set onwhich it is topologically conjugate to a full shift on an infinite number of symbols.

We now discuss the necessary modifications of Al, A2, A3, Al, and A2 in order to

provide conditions whereby this occurs in a specific map.

Let Hi, i = 1, 2, ..., N, ..., be a collection of µh-horizontal slabs containedin the domain, D, of f. Everything will go through if we choose the Hi such thatthe following two conditions hold.

1) Jim d(H1) = 0.ti-+oo


2) For each i, f (Hi) intersects Hj for j = 1, ..., N, ... .

Note that condition 2) implies that we will be dealing with full shifts and will have

no need for a transition matrix. Now, if we let S = {1, 2, ..., N, ...}, then Al,A2, A3, Al, and A2 are stated exactly the same except that the phrases "such that

(A)ij = 1" are deleted. Theorems 2.3.3 and 2.3.5 and their proofs go through inthe same manner except the conclusion of Theorem 2.3.3 is that f has an invariant

Cantor set on which it is topologically conjugate to a full shift on an infinite number

of symbols.

Although there is little difficulty in extending the results of the previous sections

to the case of an infinite number of symbols, the dynamical consequences require

careful interpretation. In practice, dynamical situations in which conditions 1) and

2) hold usually require that the Hi converge to the boundary of D as i -> oo. Thisis important since, in showing that Al and A3 hold for the Poincare map associated

with some ordinary differential equation, often the boundary of D corresponds to

the stable and unstable manifolds of some invariant set. In this case, orbits starting

on aD could not return to D under iteration by the Poincare map. We will seethis in some examples in Chapter 3, and there we will be careful to explain thedynamical consequences of the symbol "oo."

2.4. Criteria for Chaos: The Nonhyperbolic Case

In Section 2.3 we gave sufficient conditions for a map to possess a chaotic invariant

Cantor set of points. The conditions involved the decomposition of a subset of the

domain of the map into contracting (horizontal) directions and expanding (vertical)

directions along with more global conditions pertaining to the compatibility of the

map with these directions (i.e., horizontal slabs mapping to vertical slabs). In this

situation we will derive analogous conditions for the case when not all directions

are strongly contracting or expanding, which we will call the "nonhyperbolic case."

These conditions can be viewed as a generalization of those given in Wiggins [1986a].

The main difference in our results for the nonhyperbolic case as opposed to thehyperbolic case is that the conditions we give analogous to Al, A2, and A3 inSection 2.3 will lead to our map having a chaotic invariant Cantor set of surfaces as

opposed to points. An invariant set of surfaces arises due to the fact that, since not


all directions are expanding or contracting, the intersection of the set of forward and

backward iterates of the map (i.e., the map's invariant set) need not be points. We

now begin our development of the criteria, which will parallel as closely as possible

the development of the criteria for the hyperbolic case given in Section 2.3.

2.4a. The Geometry of Chaos

We consider a map

f:DxSl -* ]R'bxR.mx]RP (2.4.1)

where D is a closed and bounded n+m dimensional set contained in ]Rfz x ]Rm, and

Sl is a closed, connected, bounded set contained in HP. The extra p dimensions will

correspond to dimensions experiencing neutral growth behavior. We will discuss

continuity and differentiability properties of f as they are needed.

We will now begin the development of the analogs of the horizontal and vertical

slabs described in Section 2.3 and then proceed to define widths of the slabs andto discuss various intersection properties of the slabs. Our development of theseideas will closely parallel that of Section 2.3 with the addition of the necessarymodifications in order to accommodate the neutral growth directions. We beginwith some preliminary definitions. The following sets will be useful for defining the

domains of various maps.

Dx = {x E ]R't for which there exists a y E IRm with (x, y) E D}

Dy = {y E R' for which there exists a x E R' with (x, y) E D}(2.4.2)

These sets represent the projection of D onto ]R' and 1Rm, respectively. See Figure

2.4.1 for an illustration of the domain D x Sl of f.We remark that, hereafter, in giving illustrations to describe the analogs of

slices, slabs, etc. from Section 2.3 we will not explicitly show D x Sl in order toreduce the clutter in the diagram.

Let Ix be a closed simply connected n dimensional set contained in Dx, and

let Iy be a closed simply connected m dimensional set contained in Dy. We willneed the following definitions.

Definition 2.4.1. A µh-horizontal slice, H, is defined to be the graph of a Lipschitz

function h: Ix x Sl -> Rm where h satisfies the following two conditions:

1) The set II = { (x, h(x, z), z) E ]Rn X ]Rm x RP I X E Ix, Z E Sl} is contained inDxSl.


Dxf

DXx11

Dyx12

(b)

Figure 2.4.1. a) DxfinR'xR'xRP, n=m=p=1.b)Dzx1Z and Dyx11 inRnxR'nxRP, n=m=p=1.

2) For every x1, x2 E Ix, we have

I h(xl, z) - h(x2, -)I < µh x1 - x21 (2.4.3)

for some 0 < kh < oo and for all z E U.Similarly, a µv-vertical slice, V, is defined to be the graph of a Lipschitz function

v: Iy X U -. Rn where v satisfies the following two conditions:

1) The set V = { (v (y, z), y, z) E Rn x Rm x RP I y E Iy, z E n) is contained in

D.

2) For every y1, y2 E Iy, we have

Iv(y1, z) - v(y2, z) I -< All y1 - Y21 (2.4.4)


for some 0 < µ.,, < oo and for all z E 12.

So Definition 2.4.1 can be viewed as a parametrized version of Definition 2.3.1

with the conditions (2.4.3) and (2.4.4) holding uniformly in the parameters, seeFigure 2.4.2.

Figure 2.4.2. Horizontal and Vertical Slices in 1i.n xR.m x ]Rp, n = m = p = 1.

Next we "fatten up" these horizontal and vertical slices into n + m + p dimen-

sional horizontal and vertical "slabs." We begin with the definition of µh-horizontal

slabs.

Definition 2.4.2. Fix some µh, 0 < µh < oo. Let H be a ph-horizontal slice andlet Jm':Ii x n --> 1R' be a continuous map having the property that the graph ofJ' intersects H in a unique, continuous p dimensional surface. Let Ha, a E I, bethe set of all µh-horizontal slices that intersect the boundary of the graph of J'nin continuous p dimensional surfaces and have the same domain as H, where I issome index set (note: as in Definition 2.3.2, it may be necessary to adjust Ix orequivalently J'n in order for the Ha to be contained in D x f2).

Consider the following set in lRn x lR' x R.p.

SH = { (x, y, z) E lRn x lRm x lRp I X E Ix, z E 12 and y has the property thatfor each x E Ix, z E 12, given any line L through (x, y, z) with L parallel

to the x = 0, z = 0 plane then L intersects the points (x, ha(x, z), z),

(x) hP (x, z), z) for some a,,8 E I with (x, y, z) between these two pointsalong L }.


Then a µh-horizontal slab H is defined to be the closure of SH.

Boundaries of Ah-horizontal slabs are defined as follows.

Definition 2.4.3. The vertical boundary of a uh-horizontal slab H_ is denotedavH and is defined as

avH - {(x, y, z) E Hi x c aIx} . (2.4.5)

The horizontal boundary of a µh-horizontal slab H is denoted ahH and isdefined as

ahH - aH - avH . (2.4.6)

We remark that it follows from Definitions 2.4.2 and 2.4.3 that avH is parallelto the x = 0 plane, see Figure 2.4.3 for an illustration of the geometry of Definitions

2.4.2 and 2.4.3.

ahH

Figure 2.4.3. Horizontal Slab in Rn x R' x RP, n = m = p = 1.

For describing the behavior of µh-horizontal slabs under maps the followingdefinition will be useful.

Definition 2.4.4. Let H and H be Ech-horizontal slabs. H is said to intersect Hfully if H C H and avH C avH.

See Figure 2.4.4 for an illustration of Definition 2.4.4.

Now we will define µv-vertical slabs.


Figure 2.4.4. a) H Does Not Intersect H Fully.b) H Does Intersect H Fully.

Definition 2.4.5. Fix some kv, 0 < µv < oo. Let H be a µh-horizontal slab andlet V be a µv-vertical slice contained in H with aV C ahH. Let J't: Ix x Il --p lRm

be a continuous map having the property that the graph of J92 intersects V in aunique, continuous p dimensional surface. Let V a, a E I, be the set of all N-v-vertical slices that intersect the boundary of the graph of J' with aVa C ahHwhere I is some index set. We denote the domain of the function va(y, z) whosegraph is V' by I' x D.

Consider the following set in lRn x IRm x RP.

SV = { (x, y, z) E IR' X lRm X RP I (x, y, z) is contained in the interior of the set

bounded by V', a E I, and ahH }.


Then a µv-vertical slab V is defined to be the closure of SV.

The horizontal and vertical boundaries of µh-horizontal slabs are defined as

follows.

Definition 2.4.6. Let V be a ,iv-vertical slab. The horizontal boundary of V isdenoted 8hV and is defined to be V n BhH. The vertical boundary of V is denoted

8vV and is defined to be 8V - ahV.

See Figure 2.4.5 for an illustration of Definitions 2.4.5 and 2.4.6.

Figure 2.4.5. Vertical Slab in ]Rl X ]R.m X RP, n = m = p = 1.

Let us make a general remark concerning the boundaries of horizontal andvertical slabs. In each case the part of the boundary corresponding to the "ends" of

the slabs (i.e., to be more precise, for a horizontal slab { (x, y, z) E aH l z E ahh }and for a vertical slab { (x, y, z) E aV I z c Oil }) is defined to be in the horizontal

boundary. This may appear strange, but it causes no problems, since we will require

that this part of the boundary will always behave the same under f and f -1, i.e.,by definition it will always map to the part of the boundary corresponding to theends of the slabs. We remark that, in the important special case in which i is atorus, the "ends" of the slabs do not exist, since a torus has no boundary.

Next we define the widths of µh-horizontal and µv-vertical slabs.

Definition 2.4.7. The width of a µh-horizontal slab H is denoted d(H) and isdefined as follows:

d(H) = sup ha(x, z) - hp(x, z) (2.4.7)(z,z)EII x12

aOEI


Similarly, the width of a pv-vertical slab V is denoted by d(V) and is defined asfollows:

d(V) = sup Iva(y,z) - v1(y,z)(Y,z)Ely x 12

a,flEI

(2.4.8)

where Iy = Iy fl Iy .

We now give two key lemmas.

Lemma 2.4.1. a) If H1 ID H2 I ID Hn D is an infinite sequence of ph-

horizontal slabs with Hk+1 intersecting Hk fully, k = 1, 2,..., and d(Hk) . 000

as k --> oo, then (" Hk H' is a ph-horizontal slice with all' c 8H1.A:=1

b) Similarly, if V 1 ID V, I ID V n ID . . . is an infinite sequence of pv-vertical

slabs contained in a ph-horizontal slab H with d(Vk) --+ 0 as k -> oo, then00n V k = V oo is a pv-vertical slice with OV °O C 8hH.

k=1PROOF: The proof of this lemma is very similar to the proof of Lemma 2.3.2. Weleave the details to the reader. pLemma 2.4.2. Let H be a ph-horizontal slab. Let H be a ph-horizontal slicewith eH C 8H, and let V be a pr-vertical slice with OV c 8hH such that0 < pvph < 1. Then H and V intersect in a unique p dimensional Lipschitzsurface.

PROOF: The proof of this lemma is similar to the proof of Lemma 2.3.2. We let

H = graph h(x, z) for x E Ix , z E 11(2.4.9)

V = graph v (y, z) for y E Iy , z E SZ .

Let ix = closure { x E Ix I h(x, z) is in the domain of v ( , ) for each z E fZ }. Nowif H intersects V, there must be a point x such that x = v(y, z) with y = h(x, z)for some z- E 1. So the lemma will be proved if we can show that the equationx = v (h(x, z), z) has a solution for each z E ft and the solution is Lipschitz in z.Now, for each z E Il, v (h( , z), z) maps ix into is, since V C H with 8V C 8hH.Since ix is a closed subset of the complete metric space R' it is likewise a complete

metric space. Recall that for each x1, x2 E Ix, z E 0, we have

Iv(h(xl,z),z) - v(h(x2,z),z) I < phlh(xl,z) - h(x2,z)I

<_pvphxl-x2I(2.4.10)


Thus, for each z E ft, v (h(-, z), z) is a contraction map of the complete metric

space ix into itself since 0 < uv/Lh < 1. So, by the contraction mapping theorem,

the equation x = v (h(x, z), z) has a unique solution for each z E [I. Now, since

v and h are Lipschitz, it follows from the uniform contraction principle (Chow and

Hale [1982]) that this solution depends on z in a Lipschitz manner.

We are now at the point where we can give conditions sufficient for our map to

possess a chaotic Cantor set. However, first let us recall the role that the analogs

of Lemmas 2.4.1 and 2.4.2 in Section 2.3 played in obtaining the result for the hy-

perbolic case. In the hyperbolic case all directions were either strongly contracting

or expanding, and this fact, coupled with Lemmas 2.3.1 and 2.3.2 and the struc-

tural assumptions Al and A2 on f, led to the existence of a chaotic Cantor set ofpoints. In the present situation the reader might guess that analogous structuralassumptions on f along with Lemmas 2.4.1 and 2.4.2 might instead lead to a chaotic

Cantor set of p dimensional surfaces for the map. We will shortly show that this is

indeed the case.

Let S = 11, 2, ... , N}, N > 2, and let Hi, i = 1, ... , N, be a set of disjointN

1 h-horizontal slabs with DH = U Hi. We assume that f is one-to-one on DH,i=1

and we define

f(Hi)nH-=Vji, Vi,jESand (2.4.11)

Hi n f-1(Hj) = f 1(Vji) = Hij Vi,jES.

Notice the subscripts on the sets Vji and Hij. The first subscript indicates whichparticular µh-horizontal slab the set is in, and the second subscript indicates forthe Vji into which µh-horizontal slab the set is mapped by f -1 and for the Hi jinto which µh-horizontal slab the set is mapped by f.

Let A be an N x N matrix whose entries are either 0 or 1, i.e., A is a transition

matrix (see Section 2.2) which will eventually be used to define symbolic dynamics

for f. We have the following two "structural" assumptions for f.

Al. For all i, j E S such that (A)ij = 1 Vii is a pv-vertical slab containedin Hj with avVji C a f (Hi) and 0 < µvµh < 1. Moreover, f maps Hijhomeomorphically onto Vji with f -1(avVji) C avHi.


A2. Let H be a µh-horizontal slab which intersects Hi fully. Then f (H) n Hi

Hi is a µh-horizontal slab intersecting Hi fully for all i E S such that(A)ij = 1. Moreover,

d(Hi) < vhd(H) for some 0 < vh < 1 . (2.4.12)

Similarly, let V be a,iv-vertical slab contained in Hj such that also V C Vji for

some i, j E S with (A)ij = 1. Then f (V) fl Hk - Vk is a µv-vertical slab

contained in Hk for all k E S such that (A) jk = 1. Moreover,

d(Vk) < vvd(V) for some 0 < vv < 1 . (2.4.13)

See Figures 2.4.6 and 2.4.7 for an illustration of the geometry of Al and A2.

The reader may be struck by the fact that Al and A2 read just as in the hyper-

bolic case discussed in Section 2.3; however, note that the definition of horizontaland vertical slabs along with the domain of f have been modified to account forneutral growth directions.

2.4b. The Main Theorem

We now state our main theorem which gives sufficient conditions in order for our

map to possess a chaotic invariant set.

Theorem 2.4.3. Suppose f satisfies Al and A2; then f possesses an invariantset of p dimensional Lipschitz surfaces. Moreover, denoting this set of surfaces

by A, there exists a horneomorphism 0: A -r EN such that the following diagramcommutes.

A f, A

01EA EN

Let us make the following remarks regarding Theorem 2.4.3.

(2.4.14)

1) If A is irreducible then A is a Cantor set of surfaces, see Section 2.2.

2) Let us discuss in more detail the expression A f A. The phrase "A is aninvariant set of p dimensional Lipschitz surfaces" means that given any pointr E A f (r) is also an element of A, and hence a p dimensional Lipschitz

surface. Thus, the expression A I * A implies that points in the restricted


I

1

F.li I /1

---------------Il

(a)Enlarged cross-section

(b)

f (H2)

1 1 0

Figure 2.4.6. a) Horizontal Slabs and Their Images under f, A = 1 0 1

0 1 1

b) Enlarged Cross-sectional View for z Fixed.


domain of f are to be taken as points in A, i.e., as p dimensional Lipschitzsurfaces (note: in the sense that the domain of a map is part of the definition

of the map, it might make more sense to rename f when it is viewed as being

restricted to A; however, we do not take this approach).

3) In order for 0 to be a homeomorphism it is necessary to equip A with a topology.

There are two ways of doing this.

The first way uses the fact that elements of A can be written as the graphs of

Lipschitz functions. Let the graphs of the Lipschitz functions ul(z) and u2 M,

z E Sl, represent two elements of A. Then the distance between the graphs of

ul and u2 is defined to be

d(u1,u2) = sup lul(z) - u2(z)IzEfl

This metric suffices to define a topology on A.

A second way of equipping A with a topology would be simply to "mod out"

the z direction and use the quotient topology.4) An important special case is when fl is a torus. In this case A becomes a set

of tori.

5) Suppose A is irreducible; then we can make the following conclusions:

a) There exists a countable infinity of periodic surfaces in A.

b) There exists an uncountable infinity of nonperiodic surfaces in A.

c) There exists a surface in A that at some point along its orbit is arbitrarily

close to every other surface in A.

Thus, one might think of the dynamics in directions normal to the surfaces as being

chaotic in the same sense as in the hyperbolic case described in Section 2.3 while

the dynamics in directions tangent to the surfaces is unknown.PROOF: (of Theorem 2.4.3). The proof of this theorem proceeds precisely the same

as the proof of Theorem 2.3.3. See Wiggins [1986a] for details of the case when 1

is a torus.

2.4c. Sector Bundles

We now want to give a more computable criterion for verifying the stretching and

contraction estimates which appear in A2. This criterion will be analogous tothe condition A3 given in Section 2.3 for the hyperbolic case. The condition will


H23H 2

H21

V23 V21

N

H33

V32 (b)

V11 V12

Figure 2.4.7. a) Hij and Vjj for 1 < i, j < 3, (A)ij = 1.b) Enlarged Cross-sectional View for z Fixed.

H11

H12H1

R`n

LRn


likewise be phrased in terms of the action of the derivative of f acting on tangentvectors. The idea will be to give essentially the same conditions on the stretching

and contracting directions but make them uniform in z. We will need the following

additional requirements on f :

R1. Let 31 U Hij and 1 = U Vji; then f is C1 on K and f -1 isi,j,ES i,j,ES

(A)ij=1 (A)j,=1Cl on V.

R2. Consider the definition of ph-horizontal slices and Etv-vertical slices given in

Definition 2.4.1. We strengthen the Lipschitz requirements (2.4.3) and (2.4.4)

as follows:

a) For every xl, x2 E Ix, z1, z2 E fZ we have

(2.4.15)

for some 0<µh<oo, 0<µh! Ithb) For every y1, y2 Ely, z1, z2 E ft we have

Iv(yl, zl) - v(y2, z2) I ! A- I Y1 - Y2I + Av Izl - z2I

for some 0 < µv < oc, 0 < µv < µv.

(2.4.16)

We next define stable and unstable sectors at a point.

Choose a point po - (x0, y0, z0) E V U M. The stable sector at po, denotedSPo, is defined as follows

SsPo = { (APO,'tpo,XPo) E Rnx]R.'nx]RP I InPoi phI£PoI , InPo I -< ih IXPo I}

(2.4.17)

and the unstable sector at po, denoted Spo, is defined as follows

SPo (CPo,I?po,XPo) E ]R"X]R XRP I I£PoI IA. lnPoI' I4ol: i.vIXPoI}(2.4.18)



Figure 2.4.8. Stable and Unstable Sectors (Note: We Representthe Horizontal Slabs as Cut Open for Clarity).

We define sector bundles or cone fields as follows:

Ss =X U

SsPO

s8 -

PoE)1s8

V PO

PoEV(2.4.19)Su = Su

)1 PO

Su -POE)/

SuV - Po

poEV

We have the following hypothesis:

A3. D f (Su) C Su and D f -1(SV) C S. Moreover, if (40 , 17po, Xpo) E Spo and


Df(po) (ef(po)+17f(Po),Xf(Po)) E Su , then we havef(PO)

1. 0<Au <1-7Lv h-phPv-Th17f(po)

2. 1 <117 f(po)

Xf(PO)

for all p0 E X, E SP0.

Similarly, if (epo577po5XPo) E Spo and

D f-1(p0) (CPo511Po,XPo) =

then we have

E c'3 -1(Po),

1. of- 1(Po) >_ s I ePo I 0 < µs < 1 - 7ivJih - l VAh - FAv

2. 1 <f-1(PO)

I Xf-1(Po)

for all PO E V, (epo,17Po, XPo) E Spo.

We make the following remarks concerning A3.

1) The conditions Df (S') C SV and Df -1(SV) C S, imply the preservationof the horizontal and vertical directions under f and f -1, respectively, as well

as the fact that the images of vertical slices under f and horizontal slices under

f-1 may not "roll-up" in the z direction.2) The condition 71f(PO) > (Au) 117PoI for any POE I, (4o,f7Po,XPo) E Spo

implies that the vertical directions are uniformly expanded under f. Similarly,the condition

I

ef-1(po) > Iepol for any PO E V, (Cpo,r7Po,XPo) E SPo

implies that the horizontal directions are uniformly expanded under f -1.3) The condition 1 < 171f(PO) / Xf(P0)I implies that the "shear" in the z di-

rection experienced by a µv-vertical slice under mapping by f is bounded.Similarly, the condition 1 < e f_1(PO) / IXf_1(po)I implies that the "shear"in the z direction experienced by a ph-horizontal slice under mapping by f -1

is bounded. We will see that these conditions are important for estimating the

widths of images of slabs.


Figure 2.4.9. The Geometry of Hl, H2, Vl, and V2 in H.

Now the idea will be to show that A3 can be substituted for A2; however, first we

will derive a preliminary estimate which will be useful for estimating the widths of

images of slabs under f .

Let H be a lzh-horizontal slab. Let H1 and H2 be disjoint Juh-horizontal slices

contained in H with 811 and aH2 contained in BvH. We denote the domain of the

functions hl (x, z) and h2 (x, z) of which Hl and H2 are the graphs by I2 x IL LetVl and V2 be disjoint µv-vertical slices contained in H with aV1 and alt contained

in ahH. We denote the domains of the functions vl (y, z) and v2 (y, z) of whichVl and V2 are the graphs by Iy x 1Z and Iy x n , respectively. Let

Ilhl - h2II = sup Jhl(x,z) -h2(x,z)J(z,z)EII X l

1lv1 - v2II = sup Ivl(y,z) -v2(y,z)I(y,z)E(IInIt)xtl

(2.4.20)

By Lemma 2.4.2, Hl and Vl intersect in a unique p dimensional continuous Lips-chitz surface which we call r1, and H2 and V2 intersect in a unique p dimensional


continuous Lipschitz surface which we call r2. Let (xl, yl, zj) and (-2, Y2, z2) bearbitrary points on r1 and r2, respectively. See Figure 2.4.9 for an illustration ofthe geometry.


Lemma 2.4.4.

Ixi - x21 < 1 - 1vNh [(AV + l1vl-ih) I zl - z21 + j,v Ilhl - h211 + 11111 - v2111

Iy1 - Y21 C 1- FivAh

[(Ah + JVILh) Iz1 - z21 + 11h Ilvi - 0211 + IIhi - h2 III .1

PROOF: We have

xl - x21 = Iv1(y1,z1) - v2(y2,z2)I <- Ivl(yi,zl) - vl(y2,z2)I + Ivi(y2,z2) - 02(y2,z2)I

<_ AV yi - 1121 + µv 1--l - z21 + lv1 - 0211 (2.4.21)

and

y1 - Y21 = Ihl(xl,zl) - h2(x2,z2)I Ihl(xl,zl) - hl(x2,z2)I +I hl(x2,z2) - h2(x2,z2)l(2.4.22)

µhlxl-x2l +fzhlzl-z21 +IIhi-h211

Substituting (2.4.22) into (2.4.21) gives the first inequality, and substituting (2.4.21)

into (2.4.22) gives the second inequality.

Theorem 2.4.5. If Al and A3 hold then A2 holds with vh = 1 _ AU{6vILh-Ahµv-FLh

and vv = 1 _ AskvIh - LvPh - Av

PROOF: The proof proceeds in much the same way as the analogous Theorem 2.3.5

for the hyperbolic case. However, we will include the details, since the geometryassociated with the nonstretching directions is somewhat different. We will provethat the part of A2 dealing with vertical slabs holds, since the part dealing withhorizontal slabs is proven similarly. The proof proceeds in several steps:

1) Let V be a µv-vertical slice contained in Hj such that also V C Vjj with8V C ehHj for some i, j E S with (A)ij = 1. Then we show thatf(V) n Hk is a µv-vertical slice with 3(f(V)) C ahHk for all k E S suchthat (A) jk = 1.

2) Let V be a ltv-vertical slab contained in Hj such that also V C Vji for somei, j E S with (A) ij = 1. Then we use 1) to show that f (V) n Hk - Vk is aµv-vertical slab contained in Hk for all k E S such that (A) jk = 1.


Figure 2.4.10. The Geometry of V and f (V).

3) Show that d(Vk) < As d(V).1 - AvAh - µvAh - µv

We begin with Step 1). Let V be a /iv-vertical slice contained in Hj suchthat also V C Vji with 8V C ehHj for some i,j E S with (A)ij = 1.Then, by Lemma 2.4.2, V intersects Hjk with 8 (V n Hjk) C BhHjk for all

k E S such that (A) jk = 1. Now Al holds so that f (BhHjk) C BhVkj; therefore,

f (a (V n Hjk)) C BhVkj for each k E S such that (A) jk = 1. So f (V n Hjk)

consists of a collection of m + p dimensional sets with a (f (V n Hjk)) C BhVkj,see Figure 2.4.10.

We now argue that f (V n Hjk) are µv-vertical slices. By A3, Df maps S'

into Sv for all pp E M. Thus, for any (xl, yl, zl), (x2, y2, z2) E f (V n Hjk) wehave

Ix1-X2I <_ILvIYl-Y2I ,

x1 - x21 <- Pv z1 - z21(2.4.23)

So (2.4.23) allows us to conclude that, for each k E S such that (A)jk = 1,

2.4. Criteria for Chaos: The Nonhyperbolic Case

WI lot--- I

169

Figure 2.4.11. V and V2 (Note: V3 Has Been Left Out of the Figure for Clarity).

f (V n Hjk) can be expressed as the graph over the (y, z) variables of a Lipschitzfunction V (y, z) such that

V ( Y 2 , 4 - 2 ) I <_ All Iyl - y2I + iiv Iz1 - z2I . (2.4.24)

Step 2) Let V be a µv-vertical slab contained in Hj such that also V C Vji for

some i,j E S with (A)2j = 1. Then 8h (V n Hjk) C 8hHjk for all k E S suchthat (A) jk = 1. Applying the result of Step 1) to the vertical boundaries of each

V n Hjk, we see that f (V n Hjk) = Vk is a µv-vertical slab contained in Hk foreach k E S such that (A) jk = 1. Moreover, the Vk are disjoint.

Step 3) We now show that d (Vk) < 1 - uvµhA

/whh - uvd(V). Fix k and let

PO = (x0, y0, z0) and P1 = (x1, y1, z1) be two points on the vertical boundary ofVk having the same y and z coordinates, i.e., yo = yl and z0 = z1, such that

d (Vk) = IP0 - p1I = Ix0 - xl I , (2.4.25)

see Figure 2.4.11.


Consider the line

p(t) = (1 - t) p0 + tP1 , 0 < t < 1 (2.4.26)

and the image of p(t) under f -1 which is the curve w(t) = f (p(t)). By Al,w(0) and w(1) are points contained in the vertical boundary of V, see Figure 2.4.11.

Therefore, w(0) is contained in the graph of v0(y,z), and w(1) is contained in the

graph of v1 (y, z) where v0 and v1 are IL,-vertical slices. Since p(t) is parallel to the

y = z = 0 plane the tangent vector to p(t), p(t), is contained in Sy for 0 < t < 1.Therefore, w(t) lies on some Ith-horizontal slice H with H intersecting the vertical

boundary of V.

Also by A3, the tangent vector to w(t) =(x(t), y(t), z(t)), w(t)=D f -'(p(t)) ji(t),

is contained in S' for 0 < t < 1 with

I±(t)I ?its W01

and

1 < Ix(t)I

for 0 < t < 1.From (2.4.27) and (2.4.28) we conclude that

and

Ipo - P11 <- /ts Ix(o) - x(1)I

(2.4.27)

(2.4.28)

(2.4.29)

Iz(0) - z(1)I < x(0) - x(1)1 . (2.4.30)

Using Lemma 2.4.4 we obtain

Ix(O) - x(1)I < 1 -ft1

vlth[(A- + pvi h) 140) - z(1)I + IlvO - v1II] . (2.4.31)

Substituting (2.4.30) into (2.4.31) gives

Ix(0) - x(1)I <- 1 IIvO - viii . (2.4.32)1-Itvlth-JAvith-Ftv

So from (2.4.25), (2.4.29), and (2.4.32) we obtain

d(Vk) <As

d(V) (2.4.33)1 - ItvFth - ltvJh - /2v

which is true for each k E S such that (A)jk = 1.

CHAPTER 3Homoclinic and Heteroclinic Motions

In this chapter we will study some of the consequences of homoclinic and hetero-

clinic orbits in dynamical systems. Part of the motivation for the study of thesespecial orbits comes from the fact that, in recent years, it has become apparent that

homoclinic and heteroclinic orbits are often the mechanism for the chaos and tran-

sient chaos numerically observed in physical systems. We will comment on specific

examples as we go along.

3.1. Examples and Definitions

The purpose of this first section is to introduce the idea of homoclinic and hetero-

clinic motions. We will do this by first giving some examples of specific physical

systems which exhibit homoclinic and heteroclinic motions so that the reader may

develop some intuition. After this we will give specific mathematical definitions for

homoclinic and heteroclinic orbits.

EXAMPLE 3.1.1. The Simple Pendulum

We consider a mass m, suspended via a weightless rigid bar of length L froma support and moving under the influence of gravity as shown in Figure 3.1.1 (we

also neglect dissipative effects such as wind resistance).

The equation of motion of the pendulum in rescaled, dimensionless variables

can be written as

B+sinO=0 (3.1.1)

or, as a systemB=V

(B, v) E T 1 x R. (3.1.2)v = -sin O

172 3. Homoclinic and Heteroclinic Motions

Figure 3.1.1. The Simple Pendulum.

V

9

e

Identify7r

Figure 3.1.2. Phase Space of the Simple Pendulum.

The phase space of the simple pendulum is the cylinder, T1 x It, and has thestructure as shown in Figure 3.1.2 with 0 = 7r and 0 = -ir identified.

From Figure 3.1.2 we see that the pendulum has two equilibrium positions,one stable at (0, v) = (0, 0) corresponding to the mass hanging straight down,and one unstable at (B, v) = (7r, O) - (-?r, o) corresponding to the mass standing

upright vertically. Also, we see that there are two orbits which connect the unstable


equilibrium to itself. These correspond to trajectories which approach the unstable

equilibrium position asymptotically in time (Note: no trajectory may reach theunstable equilibrium position in finite time, since the equilibrium position itself is

a solution to (3.1.2), and we have uniqueness of solutions). There are two suchtrajectories since the pendulum may rotate either clockwise or counterclockwise.

These two special orbits are said to be homoclinic to the unstable fixed point at(B, v) = (0, 0).

It should be apparent that the homoclinic orbits consist of the (nontransverse)

intersection of the stable and unstable manifolds of (B, v) = (0, 0). We will see

that this characterization of homoclinic orbits is quite useful. In this example, thehomoclinic orbits do not signal any complicated motions but merely separate two

qualitatively distinct motions, namely, the librational motions inside the homoclinic

orbits and the rotational motions outside the homoclinic orbits. Recall that in thecontext of planar ordinary differential equations the name separatrix is often given

to what we have called the homoclinic orbits. This is because the one dimensional

orbits separate the two dimensional phase plane into two disjoint parts.

EXAMPLE 3.1.2. The Buckled Beam

The system consisting of a long slender cantilevered beam buckled in the field

of two permanent magnets has been extensively studied both experimentally andtheoretically by Moon and Holmes (see Moon [1980], Holmes [1979]). The experi-

mental apparatus is shown in Figure 3.1.3.

It has been shown that, in certain parameter ranges, the first mode of oscillation

of the beam is adequately described by the following normalized version of Duffing's

equation

s-x+x3=0 (3.1.3)

or, as a systemx=y

(x, y) E R1 X R1 .y=x-x3The phase space of this system appears as in Figure 3.1.4.

(3.1.4)

The system has an unstable equilibrium point at (x, y) = (0, 0) corresponding

to the beam being at a position midway between the two magnets. This unstableequilibrium point is connected to itself by two homoclinic orbits corresponding to

motions which approach the unstable equilibrium point asymptotically in both time


Rigid Frame

Beam

Magnets

Figure 3.1.3. Elastic Beam in Magnetic Field.

Figure 3.1.4. The Phase Space of the Beam.

directions. Thus, the homoclinic orbits are characterized by the (nontransverse)intersection of the stable and unstable manifolds of (x, y) = (0,0) (note: thereare two homoclinic orbits in this system as a result of the reflectional invarianceof the system). As in the case of the simple pendulum, the homoclinic orbits are

not indicative of any complicated motions but merely form a boundary (separatrix)

between two qualitatively distinct motions.

Suppose, however, that we force this system horizontally with a small amplitude


periodic force given by ry cos wt, ry small. In this case the equation of motion is given

by

x=y3

(z, y) E R1 x R1 (3.1.5)z-z +'ycos wt

or, as the suspended system

y

z-z3+rycosO z,y)O) E R1xR1xT1 . 3.1.6)

8=w

It can be shown (see Guckenheimer and Holmes [1983] or Chapter 4) that theunstable equilibrium point in the unforced system now becomes an unstable periodic

orbit of period 27r/w for -y sufficiently small. Furthermore, for certain values of the

parameters ry and w, the two dimensional stable and unstable manifolds of thisunstable periodic orbit may intersect transversely in the phase space R1 x R1 x T1

to yield a picture like that shown in Figure 3.1.5.

Figure 3.1.5. The Phase Space of the Forced Beam.

In this case the points in the intersections of the stable and unstable manifolds

of the unstable periodic orbit lie on orbits which approach the unstable periodic


orbit asymptotically in both directions of time and are said to be homoclinic to the

unstable periodic orbit.

The resulting complicated geometrical phenomena associated with these homo-

clinic orbits is made a bit clearer by instead considering an associated two dimen-

sional Poincare map (see Section 1.6). We construct a two dimensional cross-section

E to the three dimensional phase space of (3.1.6) as follows

E_{(x,y,0)1B=0E(0,21r]) . (3.1.7)

Then the Poincare map of E into itself is defined by

P: E --+ El l .(x(0), y(0)) '-' (x(w), y( \2w ))

(3.1.8)

In terms of the Poincare map, the unstable periodic orbit is manifested as an un-stable fixed point whose stable and unstable manifolds intersect as in Figure 3.1.6.

Figure 3.1.6. Homoclinic Orbits of the Poincare Map.

Figure 3.1.6 depicts the familiar homoclinic tangle first discovered by Poincare

[1899] during his studies of the three body problem. We will see that this phenomena

implies the presence of Smale horseshoes and their attendant chaotic dynamics.

For more details concerning the dynamics of this particular example see Guck-

enheimer and Holmes [1983], Chapter 2.

EXAMPLE 3.1.3. Rigid Body Dynamics

Euler's equations of motion for a free rigid body are given by

ml =12 - 13

m2m3

rn2 =

12-313 -11

1113mim3 (m1,m2,m3) E R1 xR1 x]R1 (3.1.9)

ms = 11 - 12m1rn21112


where Il > 12 > 13 are the moments of inertia about the principal, body fixedaxes and mi = Iiwi, i = 1,2,3, where wi is the angular velocity about the ithprincipal axis (see Goldstein [1980]).

These equations have two constants of motion given by

1_2 2 2m2 m3

H 2 I+ 12 13)1 2

12 =mi+m2+m3.(3.1.10)

Thus, the orbits of (3.1.9) are given by the intersection of the ellipsoids H = constant

with the spheres 12 = constant. The flow on the sphere has saddle points at (0, +1, 0)

and centers at (0, 0, ±1). The saddles are connected by four orbits, as shown inFigure 3.1.7. These four orbits have the property that trajectories through points

on the orbits approach one of the saddles as t - +oo and the other saddle ast -> -oo. These orbits are said to be heteroclinic to the fixed points (0, ±1, 0).

Figure 3.1.7. The Phase Space of (3.1.9), 1 Fixed.


When the free rigid body is perturbed by adding attachments, chaotic motions

may be created similar to those in Example 3.1.2. We refer the reader to Holmes and

Marsden [1983], Koiller [1984], and Krishnaprasad and Marsden [1987] for details

and examples.

EXAMPLE 3.1.4. Point Vortices in a Time Varying Strain Field

The fluid flow induced by a pair of translating point vortices separated by a

distance 2d and with circulation +r is sketched in Figure 3.1.8. The motion isviewed in a frame moving with the velocity of the vortices, v = r/47rd ez.

The stream function for this flow can be found in Lamb [1945] and is given by

00r log(x-xv)2+(y-yv)2 _ ry

47r [(x - xv)2 + (y + yv)2 47rd

where (xv, yv) is the position of the vortex in the upper half plane; also note that tlij

is symmetric about the x-axis. For the velocity field defined by (3.1.11) (xv, yv) _(0, d), and the equations for fluid particle motions are given by

dx 300dt = aydy _ -3 b0dt 3x

(3.1.12)

Equations (3.1.12) have stagnation points p± = (± , 0) which are connected toeach other by three streamlines ;¢o+, 000, 00- defined by

00(x,y)=0, IxI </d. (3.1.13)

Fluid particle paths starting on 000, 00-, and 00+ are said to be heteroclinic top+ and p_. Specifically, fluid particles starting on ?/i0+ and 00_ approach p_ ast -+ +oo and p+ as t -oo, and fluid particles starting on 000 approach p_ ast -r -oo andp+as t-++oo.

Next we consider the effect of adding a time-periodic potential flow, i.e.,

%b = 00 + OE (3.1.14)

with

0e = exyw sinwt + vey (3.1.15)


Figure 3.1.8. Flow Induced by a Pair of Translating Vortices.

and where the constant translation speed vE is included in anticipation of a co-ordinate change and is determined by requiring that the vortices have zero driftvelocity. Such a flow satisfies the Euler equations and is produced, for example, by

the motion of a vortex pair in a wavy-walled channel. The resulting motion of the

vortices is relatively simple. Introducing the dimensionless parameter,

a = r (3.1.16)27rwd2

and the dimensionless variables (x/d, y/d) -> (x, y), wt --> t and veldw ---> v,, we

compute the motion of the vortices with (xv(0), y.(0)) _ (0,1) to obtain

t

xv(t) = exp(-ecost) exp(ecost') {2 [exp(-e(cost' - 1)) - 1] + vE} dt'f0

yv(t) = exp(c(cost - 1))

wherea exp(E}

vE = 2 1 -IO(E)

(3.1.17a)

(3.1.17b)

(3.1.18)

and where Io(E) is the modified Bessel function of order zero.


The equations for fluid particle motion are given by

i= ay (x, y; xv(t), Y. (0) +eay (x, y,t)

axo(x,y;xv(t),yv(t)) - a c(x,y,t)(3.1.19)

where the expressions for (xv(t),yv(t)) in (3.1.19) are given in (3.1.17a,6). Equa-

tions (3.1.19) have the form of a time periodic planar vector field of period 27r and

is most conveniently analyzed by studying the associated two dimensional Poincare

map (cf. Section 1.6) given by

P: (x(to), y(to)) -* (x(t0 + 2xr), y(to + 27r)) (3.1.20)

where to is the section time for the map. For e = 0 the streamlines of the flowshown in Figure 3.1.9 are the invariant curves of the map. In particular, this maphas two hyperbolic saddle points at

p± = (±V3,0) (3.1.21)

and the unstable manifold of p+ coincides with the stable manifold of p_. Thesemanifolds are also heteroclinic orbits and are defined by the streamlines 00± defined

above in dimensional form.

Now for e # 0 and small, the fixed points p± persist, denoted p±,,. Thestreamline 000 persists unbroken; however, the remaining branches of the stableand unstable manifolds of p±,, intersect in a discrete set of points leading to acomplicated geometric structure as shown in Figure 3.1.9.

The heteroclinic points of the Poincare map (i.e., the points asymptotic top+,e in positive time and p- ,E in negative time) are responsible for chaotic parti-

cle trajectories as well as mixing and transport properties of this flow. For moreinformation on this problem see Rom-Kedar, Leonard, and Wiggins [1988].

EXAMPLE 3.1.5. Traveling Wave Solutions of Partial Differential Equations

Consider a. partial differential equation in one space and one time variabledenoted x and t, respectively. Then transforming to the variable z = x+ct gives rise

to an ordinary differential equation whose solutions represent traveling waves with

propagation speed c. Homoclinic or heteroclinic orbits in the ordinary differentialequation represent solitary waves in the partial differential equation. For more

3.1. Examples and Definitions

Figure 3.1.9. The Poincare Map for (3.1.19), c # 0, Small.

181

information see Conley [1975], Feroe [1982], Glendinning [1987], Hastings [1982],Kopell [1977], and Smoller [1983].

EXAMPLE 3.1.6. Phase Transitions

In continuum mechanics homoclinic and heteroclinic orbits often arise as struc-

tures separating two distinct phases of the continua. More specifically, they may

arise in the phase space of the Euler-Lagrange Equation associated with minimizing

some type of energy functional of a system. For more information see Carr [1983],

Coullet and Elphick [1987], Slemrod [1983], and Slemrod and Marsden [1985].

We now want to give a general definition for homoclinic and heteroclinic orbits.

Definition 3.1.1. Let V be an invariant set of a dynamical system (map or flow).Let p be a point in the phase space of the dynamical system, and suppose thatthe orbit of p approaches V asymptotically as t -> -oo and V asymptotically ast --4 +oo; then the orbit of p is said to be homoclinic to V.

Let V1 and V2 be disjoint invariant sets of a dynamical system, and suppose

that the orbit of p approaches V1 asymptotically as t --* -oo and V2 asymptotically

as t -r +oo; then the orbit of p is said to be heteroclinic to Vl and V2.

Definition 3.1.1 is too general for us to get an analytical handle on the orbitstructure near homoclinic and heteroclinic orbits. However, if V, V1, and V2 are


such that they possess stable and unstable manifolds (e.g., they are hyperbolic ornormally hyperbolic), then Definition 3.1.1 can be alternately stated as follows.

Definition 3.1.2. Let V, V1, V2, and p be as above. The orbit of p is said to behomoclinic to V if p lies in both the stable and unstable manifolds of V.

The orbit of p is heteroclinic to V1 and V2 if p lies in the unstable manifold of

V1 and the stable manifold of V2.

The point p is referred to as a homoclinic (resp. heteroclinic) point and, if the

stable and unstable manifolds of V (resp. V1 and V2) intersect transversely at p,

then p is called a transverse homoclinic (resp. heteroclinic) point.

In this book we will be interested in invariant sets which are either fixed points,

periodic orbits, or invariant tori. In all cases we will assume that the invariant sethas some type of hyperbolic structure.

3.2. Orbits Homoclinic to Hyperbolic Fixed Points ofOrdinary Differential Equations

We will now begin our study of the orbit structure near orbits homoclinic to hyper-

bolic fixed points of ordinary differential equations. We will attempt to answer the

following three questions:

1) Does there exist chaotic behavior near the homoclinic orbit?

2) Does the behavior persist for nearby systems, e.g., if system parameters arevaried?

3) What are the effects of symmetries in the system?

Only in the simplest cases will we be able to give complete answers to all threequestions.

Before beginning our analysis of specific systems, we will describe the general

technique of analysis in Section 3.2a. In 3.2b we will derive a classical bifurcation

result for planar systems, which can be found in Andronov et al. [1971]. In 3.2c we

study third order systems and show how Smale horseshoes may arise near homoclinic

orbits. In 3.2d we study two examples of homoclinic orbits in fourth order systems.

In 3.2e we study the orbit structure near orbits homoclinic to hyperbolic fixed points

3.2. Orbits Homoclinic to Hyperbolic Fixed Points of O.D.E.s 183

in fourth order Hamiltonian systems, and in 3.2f we discuss some known results in

dimensions > 4.

3.2a. The Technique of Analysis

Before proceeding to specific systems we want to describe the basic idea behindour method of analysis as well as some general results which will simplify our later

work.

We will be considering ordinary differential equations of the form

i = F(z) , z E IRs+u (3.2.1)

where F: U 1Rs+u is Cr (r > 2 is adequate) on some open set U C lR.s+u.

We have the following assumptions on (3.2.1).

Al. Equation (3.2.1) has a hyperbolic fixed point at z = zo. In particular, weassume that the matrix DF(zo) has s eigenvalues having negative real partsand u eigenvalues having positive real parts.

A2. Equation (3.2.1) has a homoclinic orbit connecting zo to itself, i.e., there exists

a solution ¢(t) of (3.2.1) such that lim q5(t) = lim 4(t) = zo.t--->+oo t-.-oo

Our goal will be to study the orbit structure near the homoclinic orbit. This

will be accomplished by constructing a Poincare map near the homoclinic orbit.The Poincare map will consist of the composition of two maps; one given by the

(essentially) linear flow near the fixed point and the other given by an (essentially)

rigid motion along the homoclinic orbit outside a neighborhood of the fixed point.

Assumptions Al and A2 alone are not always sufficient to allow for the construction

of such a Poincare map. The homoclinic orbit must be a nonwandering set (cf. Sec-

tion 1.1k). This is always the case in dimensions two and three but not necessarily

in dimensions four and higher. In our present construction we will introduce this as

an assumption, and it will be necessary for us to verify this fact for specific systems.

We now want to describe the construction of the Poincare map in a neighbor-

hood of the homoclinic orbit. This will be accomplished in a series of steps.

Steps 1-3. In this series of steps we transform the fixed point to the origin and show

how the local stable and unstable manifolds of the origin can be used as

local coordinates.

184 S. Homoclinic and Heteroclinic Motions

Step 4. We study the geometry near the origin and set up cross-sections to the

vector field.

Step 5. We construct the Poincare map near the homoclinic orbit.

Step 6. We construct an approximate Poincare map which we can more readilycompute.

Step 7. We give results which show how the dynamics of the approximate Poincare

map are related to the dynamics of the exact Poincare map.

We now begin the construction.

Step 1. Transform the Fixed Point to the Origin.

This is a trivial step which we include for completeness. Under the affine (i.e.,

linear plus translation) transformation, w = z - z0, (3.2.1) becomes

w = F(w + z0) - G(w) , (3.2.2)

and it is clear that (3.2.2) has a fixed point at w = 0.

Step 2. Utilize the Linear Stable and Unstable Eigenspaces as Coordinates.

By assumption A2, the (s+u) x (s + u) matrix DG(0) has s eigenvalues having

negative real parts and u eigenvalues having positive real parts. Thus, from linear

algebra we can find a linear transformation such that DG(0) has the following form

ADG(O) =

Osu(3.2.3)

(O's B

where A is an s x a Jordan block such that all the diagonal entries have negativereal parts, B is a u x u Jordan block such that all the diagonal entries have positive

real parts, and Osu (resp. Ous ) represents an s x u (resp. u x s ) matrix whoseentries are all zero. Utilizing this same linear transformation the nonlinear system

(3.2.2) can be put in the form

(l;,rl) E Rs X Ru (3.2.4)Brl+F2(C,+])

where F1 and F2 are Cr-1 and satisfy

F1(0, 0) = F2 (0, 0) = DF1(0, 0) = DF2 (0, 0) = 0. (3.2.5)

3.2. Orbits Homoclinic to Hyperbolic Fixec Points of O.D.E.s 185

(Note: we want to make a remark concerning (3.2.5). F1 is an s vector, F2 is a uvector, DF1 is an s x (s + u) matrix, and DF2 is a u x (s + u) matrix, so strictlyspeaking (3.2.5) is incorrect, since the equality sign has no meaning. However,(3.2.5) has a symbolic meaning in the sense that F1, F2, DF1, and DF2 are allequal to the zero element in the appropriate space. Another way of writing (3.2.5)

would be F1, F2 = 0 (I I2 + In I2) = 0 (2). )

Step 3. Utilize Stable and Unstable Manifolds as Coordinates.

Consider the linearized system

(3.2.6)

From Section 1.3 we know that there exists an s dimensional linear subspace Esgiven by rl = 0 and a u dimensional linear subspace Eu given by e = 0 suchthat solutions of (3.2.6) starting in Es decay exponentially to the origin as t -,+oo, and solutions starting in Eu decay exponentially to the origin as t --. -oo.For the nonlinear problem (3.2.4) the stable and unstable manifold theorem (seeTheorem 1.3.7) tells us that there exists Cr manifolds WS and W" intersecting atthe origin and tangent to Es and E", respectively, at the origin which have theproperties that solutions of (3.2.4) starting in W' decay exponentially to the originas t -, +oo and solutions of (3.2.4) starting in Wu decay exponentially to theorigin as t -, -oo. Since W' and W' are tangent to Es and Eu, respectively,locally they can be represented as graphs, i.e.,

W8loc = graph4s(e)Wu

= graphOu(,7)

where c5s(e) and fu(rl)

(3.2.7)

are C' maps of .A18CR8-,Ru and .VuCRu-,R8,respectively, which are defined in sufficiently small neighborhoods, Jds and JJu, of

the origin. Eventually we will be interested in comparing the linear flow generated

by (3.2.6) with the nonlinear flow generated by (3.2.4) near the origin. For thispurpose it is useful to use Wloc and Wloc as coordinates rather than Es and E.This is accomplished by the following transformation

(x, y) = (e - -0u('7)"7 - -0s(£)) . (3.2.8)


Under (3.2.8), the nonlinear equation (3.2.4) becomes

i=Ax+ fl(x,y)

y = By + f2 (x, y)

where fl and f2 are 0(2) and also

(x)y) E R' X Ru (3.2.9)

f1(0, y) = f2(x,0) = 0 . (3.2.10)

Equation (3.2.10) reflects the fact that y = 0 is the local stable manifold of the

origin and x = 0 is the local unstable manifold of the origin. We emphasize the

fact that (3.2.10) is only valid locally in some neighborhood V' x Nu C R' x Ru,

since the transformation (3.2.8) is only a local transformation defined on N' x .A/u.

In Sections 3.2b-3.2f we will assume that the equations under consideration havebeen transformed to the form of (3.2.9). See Figure 3.2.1 for an illustration of the

geometry of the transformation (3.2.8).

Wuloc

(x(i;,'1),Y (,)1))

t

Figure 3.2.1. Geometry of the Transformation (3.2.8).

x

Step 4. The Geometry of the Vector Field Near the Origin.

We will denote the homoclinic trajectory obtained after 4(t) has undergonethe transformAtions in Steps 1, 2, and 3 above by 0(t); so we have lim 0(t) _

lim Vi(t) = 0. We remark that the vector field in the local coordinate systemt- +-00 1,

(3.2.8) may be joined smoothly to the vector field outside of .W3 x Nu by an

Y


appropriate choice of "bump functions," see Spivak [1979]. Consider the following

s + u - 1 dimensional sets

CE ={(x,y) EIRS x R' IxI=E

CE = { (x, y) E 1R8 x Ru I IxI < c,(3.2.11)

and their associated closures

IX IyI <(3.2.12)

CE ={(x,y)EIRS xR."IIxl<E, IyI=E}.

We assume that a is chosen sufficiently small such that CE and CE are contained in

,VS x )J'. We define the following neighborhood of the origin

.N = { (x, y) E R,S x R." I IxI < c , Iyi < E } ; (3.2.13)

then it is easy to see that CE and CE form the boundary of N.

Let ns(x,y) denote the unit vector normal to CE at the point (x,y) E C.Similarly let nu(x, y) denote the unit vector normal to CE at the point (x, y) E CE C.

Recall from Section 1.6 that CE and Cu are cross-sections of the vector field (3.2.9)

provided

n5(x,y)- (Ax+f1(x,y),By+f2(x,y)) 0, V (x, Y) ECc

nu(x,y) . (Ax+fl(x,y),By+f2(x,y)) 54 0, d(x,y) E CE .

We have the following proposition.

(3.2.14)

Proposition 3.2.1. Fore sufficiently small, CE and CE are cross-sections to thevector field (3.2.9). Moreover, (3.2.9) points strictly into the interior of .N on CEand (3.2.9) points strictly to the exterior of .N on CE .

PROOF: This is an easy computation using the fact that A has eigenvalues withnegative real parts, and B has eigenvalues with positive real parts.

In computing the Poincare map it will be useful to have expressions for theintersection of the stable manifold with CE and the intersection of the unstablemanifold with CE . These are given as follows:

SE = { (x, y) E IRS X 1Ru I IxI = E, IyI = 0}(3.2.15)

SE = { (x, y) E IRS X Ru I IxI = 0, IyI = E}.


Figure 3.2.2. Geometry of the Vector Field Near the Origin, s = 2, u =

See Figure 3.2.2 for an illustration of the geometry of the vector field near the origin.

Step 5. Construction of the Poincari Map.

We now will describe the construction of the Poincar6 map defined in a neigh-

borhood of the homoclinic orbit. As mentioned earlier, the Poincare map will consist

of the composition of two maps; one defined in a neighborhood of the origin and the

other defined outside of a neighborhood of the origin along zi(t). We will discuss

each map separately.

a) The Map Near the Origin.

Consider the sets CE -SE and Cu-S'. By Proposition 3.2.1, for e sufficientlysmall all points in CE - SE reach Cu - SE under the flow generated by (3.2.9).


Let us denote the flow generated by (3.2.9) by

0(t, x0, y0) = (x(t, x0, y0), y(t, x0, y0)) (3.2.16)

Suppose (x0, y0) E CE -SE ; then (x0, y0) reaches C,-S,' in a time T = T (x0, y0)

which is a solution of the equation

Iy(T,x0,y0)I=E.

(Note: T (x0, yo) -i +oo logarithmically as yo -, 0.)We define the map

PO:CE - SE - CE - SE

(x0, y0) '-' (x(T (x0, y0), x0, y0), y(T (x0, y0), x0, y0))

(3.2.17)

(3.2.18)

where T(x0, y0) is the solution of (3.2.17) with (x0, y0) regarded as fixed.

b) The Map Along 0(t) Away from the Origin.

Let a and p denote points of intersection of the homoclinic orbit with CEand CE, respectively. Let Ua be a neighborhood of a in Cu, and let Up be aneighborhood of /3 in C. Now, since a and p lie on the homoclinic orbit sli(t),there exists a finite time r such that cb(r,a) _ /3. Since the flow is C' (r > 2) thenwe can choose Ua sufficiently small such that 4 (r(u),u) C Up, u E Ua where r (u)

is the time necessary for a point u E Ua to reach Up. Thus, we define the map

P1:Ua -+ Up

u 0(r(u),u) .

(3.2.19)

c) The Poincare Map.

Now Up C CE and Ua E CE . Suppose it is possible to choose an open setVp C Up such that

Po(Vp) C Ua . (3.2.20)

If this can be done we define the Poincare map to be

P-P1oPp:Vp-->Up . (3.2.21)

See Figure 3.2.3.


Figure 3.2.3. The Poincare Map, P = P1 o P0.

The condition (3.2.20) cannot always be satisfied with only the assumptionsAl and A2 given in the beginning of this section (however, it can always be satisfied

in dimensions 2 and 3), and in the following sections we will treat its applicability

on a case by case basis.

Step 6. The Approximate Poincare Map.

Our method for studying the orbit structure near the homoclinic orbit willconsist of constructing a Poincare map similar to that just described and thenstudying its dynamics. However, from the definition of the map, it is clear that to

construct it we must first solve for the flow generated by (3.2.9). This cannot bedone in general. Instead, we will construct an approximate Poincare map which will

reproduce the dynamics of the exact Poincare map we are interested in studying.


The approximate Poincare map consists of the composition of two maps.

a) The Approximate Map Near the OriginThe flow generated by the vector field (3.2.9) linearized about the origin is

given by

x0, y0) = (eAtx0, eBty0) ; (3.2.22)

and we define the map

P0:CE-SE --SCE -SEAT BT

(3.2.23)

where T solves

yo)(x0, y0) (e xo)e

I eBT yo) = e . (3.2.24)

Intuitively, it should be reasonable that PD is "close" to P0 for c sufficiently small,

since nearer the origin the vector field looks more and more linear. We will make

this precise in Step 7.

b) The Approximate Map Along sli(t) Away from the Origin

Consider the map P1 defined in (3.2.19). Taylor expanding P1 about a gives

P1 (a + u') = P1 (a) + DP1(a)u' + 0 (I u' 12)

=/3+DP1(a)u'+O(Iu'I2).

We define the map

(3.2.25)

PL : Ua - Up(3.2.26)

a+u'+--> /3+DP1(a)u'.

c) The Approximate Poincare Map

As in Step 5, we suppose that it is possible to choose Vp C Up such thatPO(Vp) C U. Then we define the approximate Poincare map, PL, as

PL=P1 (3.2.27)

Next we demonstrate how well PL approximates P.

Step 7. The Relation Between the Exact and Approximate Poincard Maps.

First we want to show that the map constructed near the origin is approximated

to within an error of 0 (c2) if the flow generated by the linearized vector field isused for its construction.


We begin by rescaling the coordinates as follows

x=Ex

y=Ey.

0<E«1.In this case (3.2.9) becomes

x = Ax+ EJ1(Ex,Ey) = Ax+fl(x,9;E)

y = By+ Ef2(Ex,Ey) = B9+f2(x,9;E)

wherelim 11(x,y;E)=0E-,0

lim 12 =0.

(3.2.28)

(3.2.29).

(3.2.30)

We denote the flow generated by (3.2.29) by

(t, x0, y0, E) = (x(t, xo, yo, E), y(t, x0, y0, E)) , (3.2.31)

and it should be clear that

(t, x0, y0, 0) = (eAtxo, eBtyo) , (3.2.32)

i.e., at c = 0 (3.2.9) reduces to the linearized equations. Thus, the rescalingby e has the effect of "magnifying" a neighborhood of the origin. In the rescaledcoordinates the cross-sections to the vector field take the form

Ci = { (x, y) E R8 x 1Ru 121 = 1 , 191 < 1 }(3.2.33)

Ci={(x,y)EIR8xRU Ixj <1, 19I=1}

and the intersection of the stable manifold with Ci and the unstable manifold with

Cu are given by

Sl = { (x, y) E 1R8 x Ru (xj = 1, jyj = 0}

Si ={(x,y)EIR8xRu121=0, 191=1}.

Then, in the scaled coordinates, the map near the origin becomes

PO:Cj - Sl -Sl(x0, 90) `-' (T (x0, y0, C),20, y0, E)

(3.2.34)

(3.2.35)

3.2. Orbits Homoclinic to Hyperbolic Fixed Points of O.D.E.s

where T (z0, y0, E) is the solution of

193

I y(T,.xo, y0, E)' = 1 . (3.2.36)

For c = 0 we denote the map by

PL:C3-SsC"-Su

0 1 1 1 1

(xo,go)'-* (eATLxo,eBTLgo)

where TL is the solution of

We now want to show thatsome preliminary lemmas.

(3.2.37)

eBTLgO = 1 . (3.2.38)

PO - Pf = 0 W. However, we will first need

Lemma 3.2.2. The solution of (3.2.36), T (x0, y0, c), is a Cr function of (t0, go, e)

for (xo, 90) E Ci - Si and for c sufficiently close to zero.

PROOF: The equation for the time of flight of a point (a0, y0) E Ci - Sl toCi - Si is given by

h(T, x0, y0, E) = Iy(T, .0, y0, E) I - 1(3.2.39)

_(yl(T,X0,y0,E)1J 2

+ ... + (yu(T,X0,y0,E)\12- 1 = 0

where (T, x0, yo, e) E ]R1 x Ci - S1 x IRl. We will use the implicit function theorem

to show that T is a Cr function of (x0i y0, c).

Now equation (3.2.39) has a solution at E = 0 for each (x0, 90) E Ci - Si ,namely,

h (TL, .to, yo, 0) =eBTL

y0 - 1 = 0. (3.2.40)

A simple calculation using (3.2.39) gives that

Dth(TL, 2o, y0, 0) =Y (TL, -to, yo, 0) I

By(TL,.x0, y0, 0) . (3.2.41)

So, since B has eigenvalues with nonzero real parts, (3.2.41) is nonzero for each(x0, 90) E Ci - S8, or, more geometrically, (3.2.41) is non-zero since Ci is a cross-

section of the vector field. Hence, by the implicit function theorem, for c sufficiently

close to zero and for each (.to, 90) E Ci - Si T = T (x0, y0, c) is Cr in (xo, y0i c).


Lemma 3.2.3. DET (20, 9o, 0) and DET (20, y0, 0) are bounded in Ci - Sl .

PROOF: We have shown that T is Cr in Ci - Sl. However, a problem mayarise as Si is approached (i.e., as 1901 -. 0), since in this case the time of flightapproaches oo logarithmically. Thus, for proving the lemma it suffices to show that

lim DET (20, 90, 0) and lim DET (20, 90, 0) are bounded.IOI 0

,01o

We can compute these derivatives directly using (3.2.39) and the implicit func-

tion theorem. We begin by computing DET (20, y0, 0).

Using Lemma 3.2.2 and (3.2.39), we obtain

DET(2o,yo,0) = -[Dth(T(2o,yo,0),xo,yo,0)]-1DEh(T(2o,yo,o),2o,y0,0)(3.2.42)

and, from (3.2.41), we obtain

Dth(T(x0,y0,0),2o,y0,0) =9(TL,20, o, 0)I

By(TL,20,Yo,0), go,

eBTLy0 . Be13TLy0eBTL y0)

and

(3.2.43)

BTL -DEh(T(20,y0,0),20,y0,0) = IeBTLyoI .DEy(TL,x0,go,0) (3.2.44)

Thus, using (3.2.43) and (3.2.44) we have

DET (20, y0, 0) _-IeBTLy0

. BeBTLyO]-1IeBTLyo . DEy(TL, 2o, 90+0)]. (3.2.45)

Now, in order to show that (3.2.45) is bounded as I90I -> 0, it suffices to show two

things:

lim Dth(TL, x0, y0, 0) is bounded; (3.2.46a)Iyol -0

lim DE9(TL, x0, 90,0) is bounded. (3.2.46b)IYol 0

(3.2.46a) follows from the geometrical fact that, for each (20,90) E Ci - Si, wehave

I eBTLgo I = 1.

We remark that it is necessary to consider the limit superior rather than the limitfor (3.2.46a), since the limit may change as IYoI -. 0 along different eigendirections

of B.


The fact that (3.2.46b) holds relies heavily on the following:

1) 12(-t,0,E)=0,

2) eAtzO < Ke-at Ix0I

eBtyol < Keat IYol

t>0,t<0.

For some constants K, a > 0.

Now 2) follows from the fact that the origin is a hyperbolic fixed point (see Hale

[1980]), and 1) follows from the choice of the local stable and unstable manifoldsas coordinates. Along with the fact that f2 is Cr-1, r > 1, the latter also impliesthe existence of a constant K > 0 such that

I f2(2+y)I !5 K(IXI IyI + IyI2)-

For (x,y)E.W3x.W".Using (3.2.29) we have

I f2 (x, y, E) < KE[[xI 191 + I9I21

From (3.2.30) we obtain

and hence

I12(x,y,E) -12(2+9,0)I < K[IxII9I+Iy12]

IDEf2(2,y,0)I :5K[IxIIyI+I9I2] (3.2.47)

Now, using (3.2.47), we can solve for DE7(TL, xo, y0, 0) directly using the variation

of constants formula (see Arnold [1973] or Hale [1980]) to obtain

TL

DEy(TL,x0,Yo,0) = eBTL 1 e-BsDEJ2(eAsxo,eBsgo,0)ds. (3.2.48)

0

Using 2), (3.2.47), and (3.2.48) it can easily be shown that (3.2.46b) is bounded as

Iol - 0.We next compute DET(.to,y0,0). Using Lemma 3.2.2 and (3.2.39) we obtain

DET(xo, 90,0) = -[Dth]-1 [Dt h(DET)2 + 2(DEDth)DET + DE h] (3.2.49)


where all derivatives in (3.2.49) are evaluated at (T(-to, go, 0),.to, go, 0). Now we

have already shown that

lim Dth(T (z0, y0, 0), x0, y0, 0) is bounded; (3.2.50)Igol -i 0

and

, go

lil

m>0

DET (2oi y0, 0) is bounded. (3.2.51)-Simple calculations using (3.2.39) give

BTL

2 BeBTLyoeBTLgo(

eBTLyoBeBTLyo)

Dt hl e=o BT BeBTL yo (3.2.52)I e L 901 I e BTL 901 2

DED h = DE y (T (y0, y0, E), -to, y0, E) . By(T(x0, 90) E), x0, y0, (3.2.53)(t( 19 (T (x0, y0, e), xo, yo, ) I

DEh = DE y(T(xo, yo, E), -t0' yo, E)Dey(T (xo, yo, E),.t0, yo, E) . (3.2.54)

y (T (x0, y0, E), x0, y0, E) I

Now, using arguments similar to those given above (i.e., 12(±,O,c) = 0, hyperbolic-

ity of the fixed point, and the variation of constants formula), we can conclude that

the limit superiors as 1901 - 0 (3.2.52), (3.2.53), and (3.2.54) are bounded; hence,

lim DET (.t0, y0, E) is bounded. (3.2.55)Iyo l - 0

We leave the details to the reader.

We now use Lemmas 3.2.2 and 3.2.3 to prove the following proposition.

Proposition 3.2.4. PO - Po = 0(E).

PROOF: By Lemma 3.2.2, for each (t0, yo) E Ci - Sl we can Taylor expand Tas follows

T (±o, yo, c) = T (xo, yo, 0) + ET1(.to, yo, 0) + 0 (E2) (3.2.56)

where T1(to,go,0) = DET(to,y0,0) andT(xo,go,0) = TL.Now, using the expression (3.2.56) in Po(xo, yo) and Taylor expanding about

E = 0, we obtain

P0(xo,yo,E) _ (.t(TL,.tO,yo,o),y(TL,.tO,y0,0))

+

0 (E2) .(3.2.57)


From (3.2.32) we have (z(TL, x0, yo, 0), y(TL, x0, yo, 4)) = (eATL2o, eBTLyo), sothat (3.2.57) can be written as

Po(xo, g0, E) = 3'0 (z0, y0)

+ e(DEZ(TL,.to,yo,0)+TDtt(TL,.to,Yo,O),DEy(TL,xo,yoo)+TDt+J(TL,2o,+Jo,O))

+ 0(f2).(3.2.58)

Now Lemma 3.2.3 and the mean value theorem applied to (3.2.58) proves the propo-

sition.

Transforming back to the unscaled, original coordinates we obtain the following

result.

Proposition 3.2.5. Po - PO = 0(E2).

PROOF: This is an obvious consequence of Proposition 3.2.4 and the rescaling.

We also have the following important result.

Proposition 3.2.6. I DPo - DPO 0 (e), I DPo -DPO = 0 (e2).

PROOF: DPO - DP0L = 0(,) follows a proof similar to that given in Proposi-

tion 3.2.4. The main step is to show that DT and D2T are bounded as Iyol -+ 0

analogous to Lemma 3.2.3. We leave the details to the reader. The relationDPo - DPD = 0(e2) follows by transforming back to the original unscaled coor-

dinates.

Now the relationship between the approximate and exact maps along 0(t) isrelatively trivial. Recall that in Step 5 we defined a map P1 along 0(t) from aneighborhood Ua C CE - SE into Up C CE - SE . Taylor expanding P1 about

the point u = a gave

P1(a+u') = P1(a) +DP1(a)u'+ 0(u12)(3.2.59)

_ 0 + DP1(a)u' + 0(u'2).

We defined an approximation to P1 by

P1 : Un -,Up(3.2.60)

a +ul -+(3+DP1(a)u',

and we have the following result.


Proposition 3.2.7. I P1 - Pl = 0(e2).

PROOF: This is a trivial consequence of the fact that the diameter of the sets CEand CE is 0 (e).

We are now in a position to show the relationship between the exact andapproximate Poincare maps. We assume that it is possible to choose Vp C Up such

thatPoPo(Vp) C Ua and (Vp) C U., (3.2.61)

and we have defined the Poincare maps

P=_ PI oP0:Vp--+Up (3.2.62)

PL=P1 Pa P:Vp-+Up. (3.2.63)

We have the following result.

Proposition 3.2.8. p - PL = 0 (e2), J DP - DPLI = 0(e2)

PROOF: This is a simple consequence of Propositions 3.2.5, 3.2.6, and 3.2.7.

The results that we need, which will relate the dynamics of PL to P, are asfollows.

Proposition 3.2.9. Suppose PL has a hyperbolic fixed point of (xo, yo). Then,for a sufficiently small, P has a hyperbolic fixed point of the same stability type at

(x0, yo) + 0 (E2).

PROOF: This follows from an application of the implicit function theorem to the

map in the scaled coordinates (2,9). El

Proposition 3.2.10. Suppose pL satisfies Al and A2, Al and A3, or Al and A2of Section 2.3. Then fore sufficiently small, P also satisfies Al and A2, Al and A3,

or Al and A2.

PROOF: This follows from the definition of Al, A2, A3, Al, and A2 and the factthat the maps as well as their first derivatives are close as described in Proposi-tion 3.2.8.

Finally, we remark that this entire analysis goes through in the case where(3.2.1) depends in a Cr (r > 2) manner on the parameters.


µ<o

µ=o

µ>o

Figure 3.2.4. Behavior of the Homoclinic Orbit as µ is Varied.

3.2b. Planar Systems


i=ax+f1(x,y;A)(x, y, µ) E 1R1 x R1 X R1 (3.2.64)

y=Qy+f2(x,y;lZ)

with f1, f2 = 0 (1x12+1 y12) and Cr, r > 2 and where µ is regarded as a parameter.We make the following hypotheses on (3.2.64).

H1. a<0, 3>0, and a+,6540.H2. At p = 0 (3.2.64) possesses a homoclinic orbit connecting the hyperbolic fixed

point (x, y) = (0, 0) to itself, and on both sides of p = 0 the homoclinic orbitis broken. Furthermore, the homoclinic orbit breaks in a transverse mannerin the sense that the stable and unstable manifolds have different orientations

on different sides of it = 0. For definiteness, we will assume that, for µ < 0,the stable manifold lies outside the unstable manifold, for µ > 0, the stablemanifold lies inside the unstable manifold and, for pt. = 0, they coincide seeFigure 3.2.4.

200 3. Homochnic and Heteroclinic Motions

The hypothesis H1 is of a local nature, since it concerns the nature of the eigenvalues

of the vector field linearized about the fixed point. The hypothesis H2 is globalin nature, since it supposes the existence of a homoclinic orbit and describes thenature of the parameter dependence of the homoclinic orbit. Such hypotheses will

be typical of our higher dimensional analyses; also, the global hypothesis will be

more intricate and harder to check in examples. Now an obvious question is, whythis scenario? Why not stable inside unstable for p < 0 and unstable insidestable for µ < 0? Certainly this could happen; however, this is not importantto us at the moment. We only need to know that, on one side of the bifurcationvalue, the stable manifold lies inside the unstable manifold and, on the other side

of the bifurcation value, the unstable manifold lies inside the stable manifold. Ofcourse, in applications, you will want to determine which case actually occurs and,

in Chapter 4, we will learn a method for doing this (Melnikov's method); however,

now we will just study the consequences of a homoclinic orbit to a hyperbolic fixed

point of a planar vector field breaking in the manner described above.

Let us remark that it is certainly possible for the eigenvalues a and f3 to depend

on the parameter p. However, this will be of no consequence provided Hl is satisfied

for each parameter value, and this is true for p sufficiently close to zero.

The question we ask is the following: What is the nature of the orbit structure

near the homoclinic orbit for u near p = 0? We will answer this question bycomputing a Poincare map near the homoclinic orbit as described in Section 3.2a

and studying the orbit structure of the Poincare map.

From 3.2a the analysis will proceed in several steps.

Step 1. Set up the domains for the Poincare map.Step 2. Compute PD .

Step 3. Compute Pl .Step 4. Examine the dynamics of PL = Pl o PO .

We begin with Step 1. Set up the domains for the Poincard map.

For the domain of PO we choose

II0={(x,y)ECE Ix=e>0, y>0}, (3.2.65)

and for the domain of Pl we choose

Hl={(z,y)ECE I x>0,y=e>0} (3.2.66)

3.2. Orbits Homoclinic to Hyperbolic Fixeci Points of O.D.E.s 201

Figure 3.2.5. Ho and Ill.

where CE and CE are defined in (3.2.11), see Figure 3.2.5.

Step 2. Compute PD .

The flow defined by the linearization of (3.2.64) about the origin is given by

x(t) = xoeat(3.2.67)

y(t) = yoel"

The time of flight, T, needed for a point (E, Yo) E Ho to reach III under the actionof (3.2.67) is given by solving

E = yoeRT

to get

T Rlogyo

Thus, PO is given byPO : 11 o - Ill

E a/Q(E,yo) H E ,EI/o

(3.2.68)

(3.2.69)

(3.2.70)

Step 3. Compute Pl .From Step 5, part b of 3.2a, by smoothness of the flow with respect to initial

conditions and the fact that it only takes a finite time to flow from H1 to no alongthe homoclinic orbit, we can find a neighborhood U C Ill which is mapped onto

no under the flow generated by (3.2.64). We denote this map by

P1(x,y;/1) = (PI1(x,y;A),P12(x,y;A)):U C 11 --* 110 (3.2.71)


where PI (0, q 0) = (E, 0). Taylor expanding (3.2.71) about (x, y; A) = (0, q 0)

gives

PI (x, y; A) = (E, ax + bA) + 0 (2) . (3.2.72)

So we havePI UCHi->II0

(3.2.73)

(x, c) '--a (E, ax + bµ)

where a>0 and 6>0.

Step 4. Examine the dynamics of PL = Pl o pLWe have

PL=P1 oPOL:VCll - H0

(E, Yo) '-' (E, aE (-E-) aIp

+ bit)Y0

where V = or

(3.2.74)

PL (y; A): y - Aykk/QI + bA

where A - ae1+(a/Q) > 0 (we have left the subscript "0" off the yo for the sake ofa less cumbersome notation).

Let 6 = la/Q1; then a +,3 # 0 implies 6 # 1. We will seek fixed points ofthe Poincare map, i.e., y E 110 such that

PL(y;A)=Ay6+bA=y. (3.2.75)

The fixed points can be displayed graphically as the intersection of the graph ofPL(y;A) with the line y = PL(y;A) for fixed A.

There are two distinct cases.

Case 1. lal > IQI or 6 > 1.

For this case DyPL(0; 0) = 0, and the graph of PL appears as in Figure 3.2.6

for A>0, A=O,and A<0.So for A > 0 and small (3.2.75) has a fixed point. The fixed point is stable and

hyperbolic, since 0 < DyPL < 1 for a sufficiently small. Appealing to Proposition

3.2.9 we can conclude that this fixed point corresponds to an attracting periodicorbit of (3.2.64), see Figure 3.2.7. We remark that if the homoclinic orbit were to


Figure 3.2.6. Graph of PL for µ > 0, µ = 0, and µ < 0 with 6 > 1.

/I<o

µ=0

µ>o

Figure 3.2.7. Phase Plane of (3.2.64) for 6 > 1.

break in the manner opposite to that shown in Figure 3.2.7, then the fixed point of

(3.2.75) would occur for jL < 0.

Case 2. lal < lal or 6 < 1.


For this case, DyPL(0; 0) = no, and the graph of PL appears as in Fig-ure 3.2.8.

PL

Y

Figure 3.2.8. Graph of PL for µ > 0, /.c = 0, and µ < 0 with b < 1.

So for µ < 0, (3.2.75) has a repelling fixed point. Appealing to Proposi-tion 3.2.9 we can conclude that this corresponds to a repelling periodic orbit for(3.2.64), see Figure 3.2.9. We remark that if the homoclinic orbit were to break inthe manner opposite to that shown in Figure 3.2.9, then the fixed point of (3.2.75)

would occur for µ > 0.

We summarize our results in the following theorem.

Theorem 3.2.11. Consider a system where H1 and H2 hold. Then we have, for y

sufficiently small, 1) If a+/3 < 0, there exists a unique stable periodic orbit on one

side of 1L = 0; on the opposite side of ti there are no periodic orbits. 2) If a+/3 > 0,

the same conclusion holds as in 1), except that the periodic orbit is unstable.

We remark that if the homoclinic orbit breaks in the manner opposite thatshown in Figure 3.2.4, then Theorem 3.2.11 still holds except that the periodicorbits occur for µ values having the opposite sign as those given in Theorem 3.2.11.

Theorem 3.2.11 is a classical result which can be found in Andronov et al. [1971].

Additional proofs can be found in Guckenheimer and Holmes [1983] and Chow and

Hale [1982].

An interesting situation arises if (3.2.64) is invariant under the coordinatechange (x, y) -+ (-x, -y). In this case (3.2.64) is symmetric with respect to 180°


L<0

µ=0

µ>0

Figure 3.2.9. Phase Plane of (3.2.64) for 6 < 1.

rotations about the origin and therefore must possess an additional homoclinic or-

bit. Then H2 would be modified as shown in Figure 3.2.10.

Similar conclusions as those in Theorem 3.2.11 hold with the provision that two

periodic orbits may be formed, one for each homoclinic orbit. There is an important

additional effect due to the symmetry. The symmetry enables us to compute aPoincare map outside of the homoclinic orbit which consists of the composition of

four maps, two maps through a neighborhood of the saddle point and one maparound each homoclinic orbit. In this case a periodic orbit may bifurcate from thehomoclinic orbit which completely surrounds the stable and unstable manifolds of

the fixed point. We leave the details to the interested reader, but in Figure 3.2.11we show the scenario supposing that H1 and H2 hold with [a[ > [(t[.

We end our study of planar systems with the following remarks.

1. The Case a + )3 = 0. In this case, it should be clear that our methodsfail. Andronov et al. [1971] state that, in this case, multiple limit cycles will


L<0

/L 0

A>0

Figure 3.2.10. Behavior of the Symmetric Homoclinic Orbits as µ is Varied.

bifurcate from the homoclinic orbit and present some results for special cases.

Dangelmayr and Guckenheimer [1987] have developed techniques which can be

used in this situation.

2. Multiple Homoclinic Orbits without Symmetry. See Dangelmayr and Gucken-

heimer [1987].


A<0

µ=o

µ>0

Figure 3.2.11. Bifurcations to Periodic Orbits in the Symmetric Case.

3.2c. Third Order Systems

Now we will consider 3-dimensional vector fields possessing an orbit homoclinic to

a fixed point and study the orbit structure in the neighborhood of the homoclinicorbit. We will see that the nature of the orbit structure depends considerably ontwo important properties:

1) The nature of the eigenvalues of the linearized vector field at the fixed point.

2) The existence of symmetries.

Regarding condition 1) above, it should be clear that the three eigenvalues of the


linearized vector field at the fixed point can be of two possible types for saddle type

hyperbolic fixed points:

1) Saddle Al, A2, A3, Ai real, Al, A2 < 0, A3 > 0-

2) Saddle-focus p ± iw, A; p < 0, A > 0.

All other cases of hyperbolic fixed points may be obtained from 1) and 2) by time

reversal. We begin our analysis by considering the saddle with purely real eigenval-

ues.

i) Orbits Homoclinic to a Saddle-Point with Purely Real Eigenvalues

Consider the following:

i = A1x± f1(x,y,z; u)

A2y+f2(x,y,z;u)

i = A3z + f3(x, y, z; A)

(x, y, z, µ) E R1 x R1 x R1 X R1 (3.2.76)

where the fi are C2 and vanish along with their first derivatives at (x, y, z,lc) _

(0, 0, 0, 0). So (3.2.76) has a fixed point at the origin with eigenvalues given by A1,

A2, and A3. We make the following assumptions.

H1. A1, A2 < 0, A3 > 0-

H2. At it = 0 (3.2.76) possesses a homoclinic orbit t connecting (x, y, z) = (0, 0, 0)

to itself. Moreover, we assume that the homoclinic orbit breaks as shown in

Figure 3.2.12 for µ > 0 and µ < 0.

We remark that Figure 3.2.12 is drawn for the case of A2 > Al so that the homo-clinic orbit enters a neighborhood of the origin on a curve which is tangent to the

y axis at the origin. We assume that (3.2.76) has no symmetries, i.e., the system isgeneric.

We will analyze the orbit structure in a neighborhood of r in the standard way

by computing a Poincare map on an appropriately chosen cross-section. We choose

two rectangles transverse to the flow which are defined as follows:

110 = { (x, y, z) E R3 11xI < E,

Ill = {(x, y, z) ER3 11xI <E,

0<z<E}z=E}

(3.2.77)

for some c > 0, see Figure 3.2.13.

3.2. Orbits Homoclinic to Hyperbolic Fixes: Points of O.D.E.s 209

Figure 3.2.12. Behavior of the Homoclinic Orbit Near IL = 0.

Computation of PO .

The flow linearized in a neighborhood of the origin is given by

x(t) = xoeAlt

y(t) = y0ea2t

z(t) = zoel3t

and the time of flight from Ho to H1 is given by

t= 1 log e .

A3 z0

µ> o

µ=o

(3.2.78)

(3.2.79)


Z

Figure 3.2.13. Cross-Section to (3.2.76) Near the Origin.

So the mapPO:110,111

is given by (leaving off the subscript 0's)

E

(3.2.80)

Computation of P1 .

From Step 5, part b of Section 3.2a, and the definition of 111 on some open set

UC1I1, we have

Pi:UCH1--'HOz 0 (a b 0 z eµ

y E + 0 0 0 y F 0

E 0 c d 0 0 fA

(3.2.81)

where a, b, c, d, e, and f are constants. Note from Figure 3.2.12 that we havef > 0, so we may rescale the parameter it so that f = 1. Henceforth, we will

assume that this has been done.

3.2. Orbits Homoclinic to Hyperbolic Fixec Poir_ts of O.D.E.s 211

The Poincare Map pL = Pl o pLForming the composition of Pp and Pl, we obtain the Poincare map defined

in a neighborhood of the homoclinic orbit having the following form.

PLP10pL.VCnono

fix)ax(Z) a3 +bc(Z) A3 +eµ (3.2.82)

Z cx(Z) a3 + de(Z) A3 +

where V = (Pp)-1(U).

Calculation of Fixed Points of PL.

Now we look for fixed points of the Poincare map (which will correspond to

periodic orbits of (3.2.76) ). First some notation; let

sA = aE , B = bEl+a3, C = CE A3 , D = de -\3 .

Then the condition for fixed points of (3.2.82) is

1.111 lx = Axz A3 + Bz A3 + eµ

z=Cxz "3 +Dz +µ.

(3.2.83a)

(3.2.83b)

Solving (3.2.83a) for x as a function of z gives

x=Lal

Bz a3 + etc(3.2.84)ill

1-Az A3

We will restrict ourselves to a sufficiently small neighborhood of the homoclinic

orbit so that z can be taken sufficiently small in order that the denominator of(3.2.84) can be taken to be 1. Substituting this expression for x into (3.2.83b) gives

the following condition for fixed points of (3.2.82) in terms of z and µ only.

a +a i t izlz - tc = CBz A3 + Ce/cz A3 + Dz a3 (3.2.85)

We will graphically display the solutions of (3.2.85) for µ sufficiently small and near

zero by graphing the left hand side of (3.2.85) and the right hand side of (3.2.85)and seeking intersections of the curves.


z

Figure 3.2.14. Graphs of the Right Versus the Left Side of (3.2.85)

for j < 0,1z = 0, and 1z > 0. The Zero Slope Situations.

First, we want to examine the slope of the right hand side of (3.2.85) at z = 0.

This is given by the following expression:

d

dz (a '3 1x2.1)

'\I ICBz a3 + CeEtz 3 + Dz a3

(a i +1\21 -1 1a_11- U21 Dz 3 -1= Ial + a2I CBz A3 + I 1I Ceµz a31 + (__A3_

'A3 A3

Now recall that PL is invertible so that ad-be 0. This implies that AD-BC 0

so that C and D cannot both be zero. Therefore, at z = 0, (3.2.86) takes the

valuesoo if (AiI<A3or IA2I<'A3

0 if A1 > A3 and IA2I>A3There are four possible cases, two each for both the oo-slope and 0-slope situations.

The differences in these situations depend mainly on global effects, i.e., the relative

signs of A, B, C, D, e, and p. We will consider this more carefully shortly. Figure

3.2.14 illustrates the graphical solution of (3.2.85) in the zero slope case.


The two zero slope cases illustrated in Figure 3.2.14 give the same result,namely, that for µ > 0 a periodic orbit bifurcates from the homoclinic orbit.

In the infinite slope case the two possible situations are illustrated in Figure3.2.15.

µ<0

µ=0

µ>0

or

µ<0

µ=0

µ>0z

Figure 3.2.15. Graphs of the Right Versus the Left Side of (3.2.85) for

it < 0, Et = 0, and µ > 0. The Infinite Slope Situations.

Interestingly, in the infinite slope case we get two different results; namely, in

one case we get a periodic orbit for µ < 0, and in the other case a periodic orbit for

µ > 0. So what's going on? As we will shortly see, there is a global effect in thiscase which our local analysis does not detect. Now we want to explain this global

effect.

Let r be a tube beginning and ending on IIo and 1 111, respectively, which con-

tains r. Then rf1W 8(0) is a two dimensional strip which we denote as R. Suppose,

without twisting R, that we join together the two ends of R. Then there are twopossibilities: 1) Ws (0) experiences an even number of half-twists inside r, in which

case, when the ends of R are joined together it is homeomorphic to a cylinder or2) Ws(0) experiences an odd number of half-twists inside r, in which case, when


the ends of R are joined together it is homeomorphic to a Mobius strip, see Figure

3.2.16.

We now want to discuss the dynamical consequences of these two situations.First consider the rectangle D c ITO shown in Figure 3.2.17a which has its lower

horizontal boundary in W'(0). We want to consider the shape of the image of D

under P From (3.2.80) PO is given by

21

EE

E

Now consider a horizontal line in D, i.e., a line with z = constant. From (3.2.87)

we see that this line is mapped to a line given by y = E(E/z)'2/a3 = constant.However, its length is not preserved but is contracted by an arbitrarily large amount

as z --> 0 since A2/A3 < 0. Thus, the lower horizontal boundary of D is mappedinto the origin. Next consider a vertical line in D, i.e., a line with x = constant. By(3.2.87), as z -+ 0 this line is contracted by an arbitrarily large amount in the ydirection and pinched so that it becomes tangent to x = 0 as z --> 0. The upshotof this is that D gets mapped into a "half bowtie" shape. This process is illustrated

geometrically in Figure 3.2.17b.

Now under the map Pi the half bowtie PO (D) is mapped back around Pwith the sharp tip of PO (D) coming back near r fl IIO. In the case whereR- is

homeomorphic to a cylinder, Po (D) twists around an even number of times in itsjourney around r and comes back to HO lying above W3(0). In the case where Ris homeomorphic to a mobius strip, PO (D) twists around an odd number of times

in its journey around F and returns to l0 lying below W3(0), see Figure 3.2.18.Now we will go back to the four different cases which arose in locating the

bifurcated periodic orbits and see which particular global effect occurs.

Recall that the z components of the fixed points were obtained by solving1al+A2H-1\3 P111-A3 A -a

z = CBz A3 + Ce/.pz a3 + Dz a3 + (i . (3.2.88)

The right hand side of this equation thus represents the z-component of the firstreturn of a point to TIO. Then, at p = 0, the first return will be positive if we have

a cylinder (C) and negative if we have a mobius band (M). Using this remark, wecan go back to the four cases and label them as in Figure 3.2.19.

(3.2.87)


No Twist

(Case 1)1/2 Twist(Case 2)

Figure 3.2.16. The Global Geometry of the Stable Manifold.

215


To

D

(a)

7f,

LP0

(b)

Figure 3.2.17. Geometry of the Poincare Map.

We now address the question of stability of the bifurcated periodic orbits.

Stability of the Periodic Orbits.

The derivative of (3.2.82) is given byl l -1 'A P2Az a3 Axz a3 + Bz a3DPL

A'a 1j-1+ V a Dz a 3A3 1'Cxz A3Cz

(3.2.89)

Stability is determined by considering the nature of the eigenvalues of (3.2.89). The

eigenvalues of DPL are given by

'11,2 =trace DPL + (trace DPL) 2 - 4 det (DPL) (3.2.90)

2 2


Figure 3.2.18 The Global Effect Due to the Twisting of the Stable Manifold.

where

1 al+a2-3detDPL = 11 A\231 (AD - BC)z 3

L ixL IaII Ia2Itrace DP = Az 3 + Cxz 3 + -Dz 3A3 A3

(3.2.91)

Substituting equation (3.2.84) for x at a fixed point into the expression for traceDPL

gives

11\1 I

traceDPL = Az + aI CBz -1 + Ia2IDz-1 + IallCeµz-1A3 A3 A3

(3.2.92)

Let us note the following important facts.


µ<oSlope zero (Ixil > X3, 1> 21 >K3)

Slopeoo (1X11 < X3 or 1X21 <A3)

µ<o

µ<o

z Cylinder(C)

z Mobius Band (M)

z Cylinder (C)

z Mobius Band (M)

Figure 3.2.19. The z-components of the Fixed Points

and the Associated Global Effect.

For z sufficiently small

detDPL is

trace DPL is

a) arbitrarily large for IA1 + A21 < A3;Sl 1b) arbitrarily small for IA1 + A21 > A3-

a) arbitrarily large for IAu < X3 or IA2I < X3;(l b) arbitrarily small for lain > A3 and IA2I > A3.

Using these facts along with (3.2.90) we can conclude:

1) For IAu I > A3 and IA2I > A3 both eigenvalues of DPL can be made arbitrarily

small by taking z sufficiently small.

2) For IA1 + X21 > A3 and IA1 < A3 and/or IA21 < A3 one eigenvalue can bemade arbitrarily small and the other eigenvalue can be made arbitrarily large

by taking z sufficiently small.

3.2. Orbits Homoclinic to Hyperbolic Fire Points of O.D.E.s 219

3) For Ial + A2I < A3 both eigenvalues can be made arbitrarily large by takingz sufficiently small.


Theorem 3.2.12. For It 0 and sufficiently small, a periodic orbit bifurcatesfrom IF in (3.2.76). The periodic orbit is a

1) Sink for Iai1 >A3 and IA21>A3;

2) Saddle for IA1 + A2I > A3, IA1I < A3 and/or IA2I < A3;

3) Source for IA1+A2I <A3

We remark that the construction of the Poincare map used in the proof ofTheorem 3.2.12 was for the case A2 > Ai (see Figure 3.2.12); however, the same

result holds for A2 < Al and Al = A2. We leave the details to the reader.

Next we consider the case of two homoclinic orbits connecting the saddle type

fixed point to itself and show how under certain conditions chaotic dynamics mayarise.

Two Orbits Homoclinic to a Fixed Point having Real Eigenvalues.

We consider the same system as before; however, we now replace H2 with H21

given below.

H21 (3.2.76) has a pair of orbits, Fr, I'1, homoclinic to (0,0,0) at Et = 0, and rrand F

1lie in separate branches of the unstable manifold of (0, 0, 0). There are

thus two possible pictures illustrated in Figure 3.2.20.

Note that the coordinate axes in Figure 3.2.20 have been rotated with respectto those in Figure 3.2.12. This is merely for artistic convenience. We will onlyconsider the configuration of case a) in Figure 3.2.19. However, the same analysis

(and most of the resulting dynamics) will go through the same for case b). Our goal

will be to establish that the Poincare map constructed near the homoclinic orbits

contains the chaotic dynamics of the Smale horseshoe or, more specifically, that it

contains an invariant Cantor set on which it is homeomorphic to the full shift ontwo symbols (see Section 2.2).

We begin by constructing the local cross-sections to the vector field near the


(b)

y

yxz

Figure 3.2.20 Possible Scenarios for Two Orbits Homoclinic to the Origin.

origin. We define

ll ={(x,y,z)EIRSIy=E,l1o={(x,y,z)E]R3Iy=E,

IxI<c,

IxI <E,

0<z<e}-E<z<0}

(3.2.93)

111={(x,y,z)E1R3z=E,111={(x,y,z)E]R3Iz=-E,

IxI <e,

IxI <E,

0<y<e}0<y<E}

for E > 0 and small. See Figure 3.2.21 for an illustration of the geometry near theorigin.

Now recall the global twisting of the stable manifold of the origin. We want

to consider the effect of this in our construction of the Poincare map. Let Tr (resp.

T1) be a tube beginning and ending on IT' (resp. Ili) and lI0 (resp. llo) whichcontains r r (resp. r1) (see Figure 3.2.15). Then rr n W 3(0) (resp. T1 n W 1 (0)


Figure 3.2.21. Local Cross-Sections to the Vector Field Near Origin.

is a two dimensional strip which we denote as Rr (resp. RI). If we join togetherthe two ends of )Zr (resp. R1) without twisting Rr (resp. RI), then Rr (resp. Rl) ishomeomorphic to either a cylinder or a mobius strip (see Figure 3.2.15). Thus thisglobal effect gives rise to three distinct possibilities.

1) Rr and Rj are homeomorphic to cylinders.

2) Rr is homeomorphic to a cylinder and R1 is homeomorphic to a Mobius strip.

3) Rr and R1 are homeomorphic to Mobius strips.

These three cases manifest themselves in the Poincare map as shown inFigure 3.2.22.

We now want to motivate how we might expect a horseshoe to arise in these sit-

uations. Consider case 1). Suppose we vary the parameter At so that the homoclinic

orbits break resulting in the images of 1I0 and 110 moving in the manner shown in

Figure 3.2.23. The question of whether or not we would expect such behavior in a

one parameter family of three dimensional vector fields will be addressed shortly.

From Figure 3.2.23 one can begin to see how we might get horseshoe-like dy-

namics in this system. We can choose Fch-horizontal slabs in 11" and IID which are

mapped over themselves in uv-vertical slabs as µ is varied as shown in Figure 3.2.24.

Note that no horseshoe behavior is possible at µ = 0. Of course many things


P (era)

P (701)

(Case 1)

(Case 2)

(Case 3)

Figure 3.2.22. Geometry of the Poincare Map, the Three Cases.

P (ao)

P(r

Figure 3.2.23. Geometry of the Poincare Map for it 0.

need to be justified in Figure 3.2.24, namely, the stretching and contraction rates and

also that the little triangles behave correctly as the homoclinic orbits are broken.


Ah - Horizontal Slabs

o/'

P (xro)

P (v0)

AV - Vertical Slabs

Figure 3.2.24. Horizontal and Vertical Slabs.

223

However, rather than go through the three cases individually, we will settle forstudying a specific example and refer the reader to Afraimovich, Bykov, and Silnikov

[1984] for detailed discussions of the general case. However, first we want to discuss

the role of parameters.

In a three dimensional vector field one would expect that varying a parameter

would result in the destruction of a particular homoclinic orbit. In the case of twohomoclinic orbits we cannot expect that the behavior of both homoclinic orbitscan be controlled by a single parameter resulting in the behavior shown in Fig-ure 3.2.23. We would need two parameters where each parameter can be thought

of as "controlling" a particular homoclinic orbit. In the language of bifurcationtheory this is a global codimension two bifurcation problem. However, if the vector

field contains a symmetry, e.g., (3.2.76) is invariant under the change of coordinates

(x, y, z) - (-x, y, -z) which represents a 180° rotation about the y axis, then theexistence of one homoclinic orbit necessitates the existence of another so that one

parameter controls both. For simplicity we will treat the symmetric case and refer

the reader to Afraimovich, Bykov, and Silnikov [1984] for a discussion of the non-

symmetric cases. The symmetric case is of historical interest, since this is precisely

the situation that arises in the much studied Lorenz equations, see Sparrow [1982].

The case we will consider is characterized by the following properties.

H1'. 0 < -A2 < A3 < -A1, d 0.

H2'. (3.2.76) is invariant under the coordinate transformation (x, y, z) (-x, y, -z)and the homoclinic orbits break for µ near zero in the manner shown in Fig-ure 3.2.25.

The property 111' insures that the Poincare map has a strongly contracting


µ>o

µ=o

µ<o

Figure 3.2.25. Dependence of the Homoclinic Orbits

on the Scalar Parameter µ.

direction and a strongly expanding direction (recall from (3.2.81) that d is an entry

in the matrix defining Pl ).Now the Poincare map PL of IIo U III into 115 U II' consists of two parts

PL: IIO -> II5 U 111 (3.2.94)

with PL given by (3.2.82) and

P! : II0 -. IIo U III (3.2.95)

where by the symmetry we have

PI (x, z) _ -PL (-x, -z) . (3.2.96)


Our goal is to show that, for µ < 0, pL contains an invariant Cantor set on whichit is topologically conjugate to the full shift on two symbols. This is done in thefollowing theorem.

Theorem 3.2.13. There exists µ0 < 0 such that, for 1A0 < it < 0, PL possessesan invariant Cantor set on which it is topologically conjugate to the full shift ontwo symbols.

PROOF: It suffices to show that Al and A2 hold from Section 2.3d. Then theresult will follow from Theorem 2.3.12 and Theorem 2.3.3.

Al. From (3.2.89) with PL(x, z) = (PL (x, z), PL(x, z)) we have

DxpL=Az a3

DPL = I -11I Cxz 3 -1 + Pa 21 Dz -1Z 2

A3 A3

DzPL= I1IAxzas-1+I12IBzA3 1

A3 A3

DzP2 = Cz a3

Now, by H1', 1A1 1 /a3 > 1 and I'A2I /A3 < 1, so we have

lim

limz--'0

limz-0

limz-0 DzPL

__0 since d # 0

(DzPL )-1 B

D<oo, since d54 0.

(3.2.97)

(3.2.98)

So, for z sufficiently small, Al is satisfied.

A. Fix u < 0. We choose µh-horizontal slabs Hr C Ho and H1 C H1 with

"horizontal" sides parallel to the x axis and "vertical sides" parallel to the z axis

such that PL(Hr) and PL(H1) intersect both horizontal boundaries of Hr andH1. This is always possible for z sufficiently small since limo I (DzPL)-1 0,

Z-0see Figure 3.2.26.

By our previous discussion of the image of H0 under PL it should be evidentthat the horizontal and vertical boundaries of Hr and HI satisfy A2. In particular,


PL(H1 )

Figure 3.2.26. Image of HI and Hr under PL.

PL(Hr)

Hr and HI are chosen such that µh = 0 and the vertical boundaries of Hr and H1are µv-vertical slices with µv = 0. Therefore, µh satisfies (2.3.68) and (2.3.74) and,

by Lemma 2.3.11 and Lemma 2.3.8, the Lipschitz constant of the vertical boundaries

of PL(Hr) and PL(H1) satisfy (2.3.69), (2.3.75), and (2.3.78). So A2 holds.

The dynamical consequences of Theorem 3.2.13 are stunning. For µ > 0 there

is nothing spectacular associated with the dynamics near the (broken) homoclinic

orbits. However, for µ < 0 the horseshoes and their attendant chaotic dynamicsappear seemingly out of nowhere. This particular type of global bifurcation hasbeen called a homoclinic explosion.

Observations and Additional References.

We have barely scratched the surface of the possible dynamics associated with

orbits homoclinic to a fixed point having real eigenvalues in a third order ordi-nary differential equation. There are several issues which deserve a more thorough

investigation.

Two Homoclinic Orbits without Symmetry. See Afraimovich, Bykov, and Silnikov

[1984] and the references therein.

The Existence of Strange Attractors. Horseshoes are chaotic invariant sets, yet


all the orbits in the horseshoe are unstable of saddle type. Nevertheless, it should

be clear that horseshoes may exhibit a striking effect on the dynamics of any sys-

tem. In particular, they are often the chaotic heart of numerically observed strangeattractors. For work on the "strange attractor problem" associated with orbitshomoclinic to fixed points having real eigenvalues in a third order ordinary differ-

ential equation see Afraimovich, Bykov, and Silnikov [1984]. Most of the work done

on such systems has been in the context of the Lorenz equations. References for

Lorenz attractors include Sparrow [1982], Guckenheimer and Williams [1980], and

Williams [1980].

Bifurcations Creating the Horseshoe. In the homoclinic explosion an infinite number

of periodic orbits of all possible periods are created. The question arises concerning

precisely how these periodic orbits were created and how they are related to each

other. This question also has relevance to the strange attractor problem.

In recent years Birman, Williams, and Holmes have been using the knot typeof a periodic orbit as a bifurcation invariant in order to understand the appearance,

disappearance, and interrelation of periodic orbits in third order ordinary differ-ential equations. Roughly speaking, a periodic orbit in three dimensions can bethought of as a knotted closed loop. As system parameters are varied, the periodic

orbit may never intersect itself due to uniqueness of solutions. Hence, the knottype of a periodic orbit cannot change as parameters are varied. The knot typeis therefore a bifurcation invariant as well as a key tool for developing a classifica-

tion scheme for periodic orbits. For references see Birman and Williams [1985a,b],

Holmes [1986], [1987], and Holmes and Williams [1985].

ii) Orbits Homoclinic to a Saddle-Focus

We now consider the dynamics near an orbit homoclinic to a fixed point of saddle-focus type of a third order ordinary differential equation. This has become known

as the Silnikov phenomena since it was first studied by Silnikov [1965].

We consider an equation of the following formPx-wy+P(x,y) z)

wx + Py + Q(x, y, z) (3.2.99)

z=.Az+R(x,y,z)where P, Q, R are C2 and 0(2) at the origin. It should be clear that (0,0,0) is afixed point and that the eigenvalues of (3.2.99) linearized about (0,0,0) are given


by p ± iw, A (note that there are no parameters in this problem at the moment;we will consider bifurcations of (3.2.99) later). We make the following hypotheses

on the system (3.2.99).

H1. (3.2.99) possesses a homoclinic orbit F connecting (0,0,0) to itself.

H2. A>-p>0.

Thus, (0,0,0) possesses a 2-dimensional stable manifold and a 1-dimensional un-

stable manifold which intersect nontransversely. See Figure 3.2.27.

z

Figure 3.2.27. The Homoclinic Orbit in (3.2.99).

In order to determine the nature of the orbit structure near F we construct aPoincare map defined near r in the usual manner, see Section 3.2a.

Computation of Po .

Let Ho be a rectangle lying in the x-z plane, and let H1 be a rectangle parallel

to the x-y plane at z = e, see Figure 3.2.27. As opposed to the case of purely real

eigenvalues Ho will require a more detailed description. However, in order to dothis we need to better understand the dynamics of the flow near the origin.

The flow of (3.2.99) linearized about the origin is given by

x(t) = ePt(xocoswt - yosinwt)

y(t) = ePt (xo sin wt + yo cos wt) (3.2.100)

z(t) = zoeat

The time of flight for points starting on Ho to reach H1 is found by solving

e = zoeAT (3.2.101)


Figure 3.2.28. Cross Sections to (3.2.99) Near the Origin.

or1 ET=-logz .

0

Thus, PD is given by (omitting the subscript 0's)

PO : IIp - H1(x\ xlz)P/Acos( log z)

O I '-' x\z)P/AsinlogzzI \ E

(3.2.102)

(3.2.103)

We now consider Hp more carefully. For Hp arbitrarily chosen it is possible for points

on Hp to intersect HO many times before reaching H1. In this case, Po would not

map HO homeomorphically onto PO (IIp). We want to avoid this situation, since

the conditions for a map to possess the dynamics of the shift map described inChapter 2 are given for homeomorphisms. According to (3.2.100) it takes timet = 27r/w for a point starting in the x-z plane with x > 0 to return to the x-z planewith x > 0. Now let x = c, 0 < z < e be the right hand boundary of IIp. Then ifwe choose x = ce2rp/ ', 0 < z < E to be the left hand boundary of IIp, no pointstarting in the interior of Hp returns to Hp before reaching H1. We take this as thedefinition of IIp.

IIp={(x,y,z)E1R3ly=0, ce2rP/ <x<E, 0<z<E} (3.2.104)


111 is chosen large enough to contain Po (110) in its interior.

Now we want to describe the geometry of Po (110) 111 is coordinatized by x

and y, which we will label as x1, y' to avoid confusion with the coordinates of 110.

Then, from (3.2.103), we have

(x1, y1) = (x(E) PIA cos(W log E), x(E)P/A sin(1d log E)) . (3.2.105)Z A z z A Z

Polar coordinates on Hl give a clearer picture of the geometry. Let

r = x12 + y/2yl = tan0x

(3.2.106)

then (3.2.105) becomes

(r' 0) _ x E P/a w log f( (z) A z) . (3.2.107)

Now consider a vertical line in no, i.e., a line with x = constant. By (3.2.107)it gets mapped into a logarithmic spiral. A horizontal line in no, i.e., a line withz = constant, gets mapped onto a radial line emanating from (0, 0, c). Consider the

rectangles

327rp -2s(k+1)a -2,rkARk={(x,y,z)E]R I y=0, Ee W <x<E, Ee W <z<Ee W I-

(3.2.108)

Then we have00

no = U Rk . (3.2.109)k=0

We consider the image of the rectangles Rk by determining the behavior of itshorizontal and vertical boundaries under Po P. We denote these four line segments

as

-2,rkahu(x,y,z)ElR3(y=0,z=Ee W ,

-2w(k+1)ahl={(x,y,z)ER3jy=O,z=Ee W

2-pEe W <x<c}

'LapEe W <x<E}

-2a(k+1)a -2irkAvr(x,y,z) E]R3I y=0, x=E, cc W <z<e W }

22

vl={(x,y,z) E1R.3I y=0, x=Eew , Eexr(w+1)a <x<e-2Wka }.

(3.2.110)


See Figure 3.2.29. The images of these line segments under P are given by

32n k+l p 2nkp

P0(hu)={ (r, 0,z)EIR z=E,0=27 k,ee w <r<Ee w }

l 32n k+2 p 2n k+1 p

)PO (h (r, B, z) E 1R z = e, 0 = 27r(k + 1), ce ' < r < Ee W }

PD (vr)

1

(r, 0, z) E 1R3

3

z = E, 27rk < 0 < 27r(k + 1), r(0) = few }

p2w

PO (v ) (r, 0, z) E 1R z = E, 27rk < 0 < 27r(k + 1), r(0) = Ee }(3.2.111)

so that PO (Rk) appears as in Figure 3.2.29.

PO (vr)-,r. /_Po (h')

V f`k

-27rkXz=Ee

w

-27r(k+1)T

z=Ee W

27rP

x=Ee Wx=E

Figure 3.2.29. Rk and the Geometry of its Image under PD .

The geometry of Figure 3.2.29 should give a strong indication that horseshoes

may arise in this system.

Computation of PL.From Step 5, part b, of Section 3.2a, on some open set U C H1 we have

where

PL:UCH1-+ITOx a b 0 x x

y -+ c d 0 y + 0

E 0 0 0 02,

(.t, 0, 0,) - I' n Hp with E 1e

The Poinearf Map PL = PL o PO .

From (3.2.103) and (3.2.112) we have

PL: PL oP0 : V C Hp __+ Ho

P(Z)x(Za[acos(W-IogZ)+6sinlogZ)]+

x(Z)a [ccos( log Z) +dsin(A log Z)]

hU

(3.2.112)

(3.2.113)

232 3. Homochnic and Heteroclinic Motions

where V = (P0 )-1(U).So, if we choose 110 sufficiently small, then PL(10) appears as in Figure 3.2.30.

P1 0

Figure 3.2.30. The Poincare Map.

We now want to show that PL contains an invariant Cantor set on whichit is topologically conjugate to the shift map. The possibility of horseshoe-likebehavior should be apparent from Figure 3.2.30; however, this needs justification.

In particular, we need to verify Al and A2 of Section 2.3d. First we will need apreliminary result.

Consider the rectangle Rk in Figure 3.2.31. In order to verify the props


PL

Rk

(a)

PL

(b)

Figure 3.2.31. Two Possibilities for the Image of Rk under PL.

behavior of horizontal and vertical slabs in Rk, it will be necessary to verify thatthe inner and outer boundaries of PL(Rk) both intersect the upper boundary of Rkas shown in Figure 3.2.31a. Or, in other words, the upper horizontal boundary ofRk intersects (at least) two points of the inner boundary of PL(Rk). Additionally,

it will be useful to know how many rectangles above Rk that PL(Rk) also intersects

in this manner. We have the following lemma.

Lemma 3.2.14. Consider Rk for fixed k sufficiently large. Then the inner bound-ary of PL(Rk) intersects the upper horizontal boundary of Ri in (at least) twopoints for i > k/a where 1 < a < -A/p. Moreover, the preimage of the verticalboundaries of PL(Rk) fl Ri is contained in the vertical boundary of Rk.

PROOF: The z coordinate of the upper horizontal boundary of Rq is given by

-2n:az = cc W (3.2.114)

The point on the inner boundary of Pp (Rk) closest to (0, 0, c) is given by

4wp 2wkprmin = ce w e w (3.2.115)


Since P1 is a linear map the point on the inner boundary of PL(Rk) = P1°Pp (Rk)

is given by44p 2rrkp

rmin = Kee W e W (3.2.116)

for some K > 0. Now the inner boundary of PL(Rk) will intersect the upperhorizontal boundary of Ri in (at least) two points provided

rmin > 1 .z

Using (3.2.114) and (3.2.116), we compute this ratio explicitly and find

rmin 4rrp 2n(kP+ia)Ke w e w

(3.2.117)

(3.2.118)

Now Ke47rpl' is a fixed constant, so the size of (3.2.118) is controlled by thee(27r/w)(kp+iA) term. In order to make (3.2.118) larger than one, it is sufficientthat kp + is is taken sufficiently large. By H2 we have A + p > 0, so for i > k/a,1 < a < -A/p, kp + is is positive, and for k sufficiently large (3.2.118) is largerthan one.

We now describe the behavior of the vertical boundaries of Rk. Recall Fig-ure 3.2.29. Under PO the vertical boundaries of Rk map to the inner and outerboundaries of an annulus-like object. Now P1 is an invertible affine map. Hence,the inner and outer boundaries of PD (Rk) correspond to the inner and outer bound-

aries of PL(Rk) = P1 o PL (Rk). Therefore, the preimage of the vertical boundary

of PL(Rk) n Ri is contained in the vertical boundary of Rk.

Lemma 3.2.14 points out the necessity of H2 since, if we instead had -p >) > 0, then the image of Rk would fall below Rk for k sufficiently large, as shownin Figure 3.2.31b.

We now want to show that A2 holds at all points of Ho except for possibly on

a countable number of horizontal lines which can be avoided if necessary. If we use

the notation PL = (P1, P2) (note: Pl , PZ stand for the two of components PLand should not be confused with the map P1 along the homoclinic orbit outside of

a neighborhood of the origin), then from (3.2.113) we have

L /DZP1 DzPjlDP L(

/(3.2.119)

\\\ D.,p DZP2


Eaz [acos( log z) -t- bsin -log

A Tz-(1+f){p [acos(A log z) +bsin(A log z)]+ w [-a sin (A log z) + b cos (A log z) ] }

w log E)]DxP2 - Eaz [ccos(a

logz) +dsin(

A z

DzP2 = A eaz-(1+a){p [ccos(A log z) +dsin(A log z)]


+ w [-c sin log z) + d cos (A log z )1 } .

(3.2.120)

Lemma 3.2.15. Al holds everywhere on Ho with the possible exception of a count-

able number of horizontal lines. Moreover, these "bad" horizontal lines can beavoided if necessary.

PROOF: By H2, 1 + p/A > 0 so we have

limz-'0lim

z-*0

limz-0

We need to worry about the term

DzPl

DZP1 II = 0

DP 2 1 = 0 (3.2.121)

(DzP2)-1 1 = 0.

(DZPl)-1 z small. (3.2.122)

We need to show that (3.2.122) is bounded, which may not be the case if

Wp [c cos (- log z) + d sin ( a log z) ] + w [-c sin i log z) + d cos (- log z) ] = 0

A(3.2.123)

or, equivalently, if

c

d

psinlogZ)+wcoslogZ)-pcos ( log z) + w sin (g log x)

(3.2.124)


However, suppose there exists a z value such that (3.2.124) is satisfied. Then, byperiodicity there exists a countable infinity of such z values. Now in practice we

are not interested in (3.2.122) on all of Ho but rather on a countable set of disjointhorizontal slabs contained in Ho. So there would be no problem if the "bad" z values

fell between our chosen horizontal slabs. We can always insure this by changingthe cross-sections Ho and/or HI slightly, which results in a change in c/d. Thus,(3.2.121) and boundedness of (3.2.122) on appropriately chosen ph-horizontal slabs

implies that Al holds.

We now address the issue of the appropriate choice of ph-horizontal slabs and

their behavior under pL, i.e., we must verify A2. We begin with a preliminarylemma.

Lemma 3.2.16. Consider Rk for fixed k sufficiently large. Then PL(Rk) intersects

Ri in two disjoint pv-vertical slabs with pv satisfying (2.3.69), (2.3.75), and (2.3.78)

for i > k/a where 1 < a < -A/p. Moreover, the preimage of the boundaries ofthese pv-vertical slabs lies in the vertical boundary of Rk.

PROOF: By Lemma 3.2.14 PL(Rk) intersects Ri, i > k/a, in two disjoint com-ponents with the preimage of the vertical boundaries of these components lying in

the vertical boundaries of Rk. Therefore, we need only show that these components

are pv-vertical slices with pv satisfying (2.3.69), (2.3.75), and (2.3.78).

By construction Rk is a ph-horizontal slab with ph = 0, and the vertical sides

of Rk are pv-vertical slices with A v = 0. So, by Lemma 2.3.11 and Lemma 2.3.8,

the vertical boundaries of PL(Rk) n Ri are ti,-vertical slices with pv satisfying(2.3.69), (2.3.75), and (2.3.78).

We now use Lemma 3.2.16 to show how we can find two ph-horizontal slabs

in each Rk, k sufficiently large, such that Al is satisfied. Consider PL(Rk) nU Ri]. By Lemma 3.2.16 this consists of two disjoint pv-vertical slabs with p,

i>k/asatisfying (2.3.69), (2.3.75), and (2.3.78). The preimage of these pv-vertical slabs

consists of two disjoint components contained in Rk whose vertical boundaries lie

in the vertical boundary of Rk. Moreover, since the horizontal boundaries of thepv-vertical slabs are ph-horizontal with ph = 0, it follows from Lemma 2.3.10 and

Lemma 2.3.8 that the horizontal boundaries of the two components of the preimage

of the pr-vertical slabs are ph-horizontal with ph satisfying (2.3.68) and (2.3.74).


We label these two ph-horizontal slabs H+k and H_k and associate to each thesymbols +k and -k, respectively. Thus A2 holds on H+k and H_k.

We now put these results together to show that PL contains an invariant Cantor

set on which it is topologically conjugate to the shift map. There are two distinct

possibilities which we will treat separately.

The Full Shift on 2N Symbols.

For k sufficiently large choose N rectangles Rk,... , Rk+N where N is chosen

such that k > (k + N) /a with 1 < a < -A/p. Since k > (k + N) /a is equivalentto k(a - 1) > N with a > 1, for a > 1 and for fixed N it is always possible

to choose k large enough so that this condition is satisfied. Then, as discussed

above, we choose ph-horizontal slabs H+i, H_{ in Ri, i = k__ , k + N, such thatPL(H+i) and PL(H_i) intersects Rk,...,R1,+N in 14v-vertical slabs where A2 issatisfied. Then, since Al and Al are satisfied, it follows from Theorem 2.3.12 and

Theorem 2.3.3 that PL possesses an invariant Cantor set on which it is topologically

conjugate to the full shift on 2N symbols. See Figure 3.2.32 for an illustration ofthe geometry.

The Subshift of Finite Type on an Infinite Number of Symbols.

Consider the space of symbol sequences

E°O,a = { s = {si}°°_- I si E ±k, ,k E 7L - 0 and Isi+1I Iil } .-(3.2.125)

For k sufficiently large, in each Ri, i = k,..., choose two ph-horizontal slabsH+i and H_i to which we associate the symbols +i and -i, respectively. ByLemma 3.2.16,PL(H+i) and PL(H_i) intersect Rk for k > i/a. As above, theH+i, H_i, i = k,..., can be chosen such that Al and Al hold, in which casea simple modification of Theorem 2.3.3 allows us to conclude that PL contains aCantor set on which it is topologically conjugate to the shift map acting on Eoo,`x.

Note that the symbols too correspond to orbits on Ws(0). Hence, in this case,some orbits may "leak out" of the Cantor set. See Figure 3.2.33 for an illustrationof the geometry.


Theorem 3.2.17. a) For each even positive integer N there exists a map

ON: EN _ lIp


Rk

Rk+1Rk+2

Figure 3.2.32. Image of the Rk under pL.

which is a homeomorphism of EN onto ON - cN(EN) such that

pLION - ONoao(ON)

b) For each real a with 1 < a < -A/p, there exists a map

0°0'02: E°°'° --* 110


Figure 3.2.33. Image of the Rk under PL.

which is a homeomorphism of E00 onto Ooo,a - °°'a(E°°'a) such that

000'a o a o (¢0O'a)-1.

Persistence Under Perturbation.

Notice a major difference between the case of purely real eigenvalues at thefixed point and the present case. In the latter case,it was necessary to break


the homoclinic orbit in a specific way in order to get horseshoes. In the presentcase,horseshoes were present in a neighborhood of the homoclinic orbit. It is natural

to ask what happens if this situation is perturbed.Let 110 be the intersection of the unstable manifold with no and ill be the

intersection of the unstable manifold with H1. We consider small C2 perturbations

of the vector field. We denote by Z the z coordinate of the point no = Pl (ill),see Figure 3.2.34.

Figure 3.2.34. Intersections of the Unstable Manifold with 110 and Ill.

Then we have the following theorem.

Theorem 3.2.18. For IZI small enough, one can find M > 1 and, for each N with1<N<M,amap

0N: EN no

which is a homeomorphism onto its image ON = cbN(EN) such that

PL ION=ON o a o (ON)-1

PROOF: We leave the details to the reader but see Tresser [1984].

Thus Theorem 3.2.18 tells us that for sufficiently small C2 perturbations afinite number of horseshoes are preserved, see Figure 3.2.35.

The Bifurcation Analysis of Glendinning and Sparrow.

Now that we have seen how complicated the orbit structure is in the neighbor-

hood of an orbit homoclinic to a fixed point of saddle-focus type, we want to get


Z<0

Figure 3.2.35. Perturbed Horseshoes.

an understanding of how this situation occurs as the homoclinic orbit is created. In

this regard, the analysis given by Glendinning and Sparrow [1984] is insightful.

Suppose that the homoclinic orbit in (3.2.99) depends on a scalar parameter Ec

in the manner shown in Figure 3.2.36.

We construct a parameter dependent Poincare map in the same manner aswhen we discussed the case of a fixed point with all real eigenvalues. This map is


µ<0

µ=0

µ>0

Figure 3.2.36. Behavior of the Homoclinic Orbit with Respect

to the Parameter 1L.

given by

E

C

x E a [a cos a log z + b sin Wlog z] } eµ + x0P -

x (1) A c cos log 1 + d sin log 1] + f µ(3.2.126)

where from Figure 3.2.36 we have f > 0. We have already seen that this mappossesses a countable infinity of horseshoes at µ = 0, and we know that eachhorseshoe contains periodic orbits of all periods. To study how the horseshoes are


formed in this situation as the homoclinic orbit is formed is a difficult (and unsolved)

problem. We will tackle a more modest problem which will still give us a good idea

about some things which are happening: namely, we will study the fixed points of

the above map. Recall that the fixed points correspond to periodic orbits whichpass through a neighborhood of the origin once before closing up. First we put the

map in a form which will be easier to work with. The map can be written in theform v

(xl logz+01) +elt+xpJ s I\ / (3.2.127)

z

where we have rescaled p so that f = 1 (note that f must be positive).Now let

-b= -, a= DE-s. Q=aE-S.We _ -- , 1og6+01, '2 = - logE+02.

(3.2.128)

Then the map takes the form

x axz 6 cos (e log z + 'D1) + eµ + xoll H (3.2.129)

Cz/ ( ,Qxzscos(£logz+4D2)+ti

Now we will study the fixed points of this map and their stability and bifurca-

tions.

Fixed Points.

The fixed points are found by solving

x = axzb cos(E log z + 4b1) + eµ + xp , (3.2.130a)

z=Qxzbcos(elogz+'P2)+/c. (3.2.130b)

Solving (3.2.130a) for x as a function of z gives

eµ + x0X = (3.2.131)

1 - az6 cos(e log z + q1)

Substituting (3.2.131) into (3.2.130b) gives

(z-µ)(1-azacos(elogz+4> 1)) = (eµ+xo)f3z6cos(Clogz+402). (3.2.132)

Solving (3.2.132) gives us the z-component of the fixed point; substituting this into

(3.2.131) gives us the x-component of the fixed point. In order to get an idea about

the solutions of (3.2.132) we will assume that z is so small that

1 - &z5 cos( log z + p 1) - 1 . (3.2.133)

W w


Case 1:6<1

z-Iu

o n

z L<0

z-µ

n

z A =0

VU

z-µ

z fc>0

V U U

Figure 3.2.37. Case 1: 6 < 1.

Then the equation for the z component of the fixed point will be

(z - µ) = (eµ + xo) fJzb cos(e log z + 4) 2 . (3.2.134)

There are various cases shown in Figure 3.2.37.

So in the case 6 < 1 we have

It < 0: finite number of fixed points.

µ = 0: countable infinity of fixed points.

IL > 0: finite number of fixed points.


Case 2: 5>1

z-µ

z k<O

z,u=0

z /L>0

Figure 3.2.38. Case 2: 6 > 1.

The next case is 6 > 1, i.e., H2 does not hold. We show the results inFigure 3.2.38.

So in the case 6 > 1 we have

µ < 0: There are no fixed points except the one at z = µ = 0 (i.e., the homoclinicorbit).

A > 0: For z > 0, there is one fixed point for each A. This can be seen as follows:

the slope of the wiggly curve is of order z6-1, which is small for z small

since 6 > 1. Thus, the z - p line only intersects it once.


Again, the fixed points which we have found correspond to periodic orbits of

(3.2.99) which pass once through a neighborhood of zero before closing up. Our

knowledge of these fixed points allows us to draw the following bifurcation diagrams

in Figure 3.2.39.

(a) Period

µ<0 0 A>0

b>1

0

S<1

µ>0

Figure 3.2.39. Dependence of the Period of the Bifurcated Periodic Orbits

on A. a) 6 > 1, b) 6 < 1.

The 6 > 1 diagram should be clear; however, the 6 < 1 diagram may beconfusing. The wiggly curve in the diagram above represents periodic orbits. It

should be clear from Figure 3.2.37 that periodic orbits are born in pairs and theone with the lower z value has the higher period (since it passes closer to the fixed

point). We will worry more about the structure of this curve as we proceed.


Stability of the Fixed Points.

The Jacobian of the map is given by

(

A CD B)

where

A=az8cos(elogz+4)1)

B = 3xz6-1[bcos(elogz+'p2) - l sin(elogz+42)]

C = axzS-1[bcos(elogz+cb1) - esin(Elogz+4bi)]

D =,3z6

The eigenvalues of the matrix are given by

(3.2.135)

a1,2 = 2 {(A + B) + (A + B)2 - 4(AB - CD) } . (3.2.136)

6 > 1 : For b > 1, it should be clear that the eigenvalues will be small if z issmall (since both z6 and z6-1 are small). Hence, the one periodic orbit existingfor a > 0 for 6 > 1 is stable for it small, and the homoclinic orbit at p = 0 is anattractor.

The case 6 < 1 is more complicated.

6 < 1 : First notice that the determinant of the matrix given by AB - CD onlycontains terms of order z26-1, so the map will be

area contracting 1/2 < 6 < 1,

area expanding 0 < 6 < 1/2,

for z sufficiently small.

So we would expect different results in these two different 6 ranges.

Now recall that the wiggly curve whose intersection with z - to gave the fixed

points was given by

(etc + x0)/3za cos(0og z +

Thus, a fixed point corresponding to a maximum of this curve corresponds to B = 0,

and a fixed point corresponding to a zero crossing of this curve corresponds toD = 0. We want to look at the stability of fixed points satisfying these conditions.

248

D=0

3. Homoclinic and Heteroclinic Motions

In this case Al = A, A2 = B. So for z small, Al is small and A2 is alwayslarge; thus the fixed point is a saddle. Note in particular that, for p = 0, D is very

close to zero; hence all periodic orbits will be saddles as expected.

B=0 The eigenvalues are given by

A1,2=Af A2+4CD,

and both eigenvalues will have large or small modulus depending on whether CD

is large or small, since

A2 ti z25 can be neglected compared to CD - z25-1.

A ^- z E can be neglected compared to CD _ z6-(1/2)

Whether or not CD is small depends on whether 0 < 6 < 1/2 or 1/2 < 6 < 1.So we have

Stable fixed points for 1/2 < 6 < 1.Unstable fixed points for 0 < 6 < 1/2.

Now we want to put everything together for other z values (i.e., for z such that B,D # 0).

Consider Figure 3.2.40 below which is a blow-up of Figure 3.2.37 for various pa-

rameter values and where the intersection of the two curves gives us the z coordinate

of the fixed points.

Now we describe what happens at each parameter value shown in Figure 3.2.40.

IL = pg : At this point we have a tangency, and we know that a saddle-node pair

will be born in a saddle-node bifurcation.

p = p5 : At this point we have two fixed points; the one with the lower z value has

the larger period. Also, the one at the maximum of the curve has B = 0; therefore,

it is stable for 6 > 1/2, unstable for 6 < 1/2. The other fixed point is a saddle.

p = p4 : At this point the stable (unstable) fixed point has become a saddle since

D = 0. Therefore, it must have changed its stability type via a period doublingbifurcation.

p = p3 : At this point B = 0 again; therefore, the saddle has become either purely

stable or unstable again. This must have occurred via a reverse period doublingbifurcation.


Z /L7

Figure 3.2.40. Al >N2>/23>/14>0>/25>u6>/r7

A = µ2 : A saddle-node bifurcation occurs.

249

So finally we arrive at Figure 3.2.41.

Next we want to get an idea of the size of the "wiggles" in Figure 3.2.41 because,

if the wiggles are small, that implies that the 1-loop periodic orbits are only visible

for a narrow range of parameters. If the wiggles are large, we might expect thereto be a greater likelihood of observing the periodic orbits.

Let us denote the parameter values at which the tangent to the curve in Figure

3.2.41 is vertical by

/ci, Ai+1.... , Ai+n, ... ' 0 (3.2.137)

where the µi alternate in sign. Now recall that the z component of the fixed pointwas given by the solutions to the equations

z - µ = (e1 + x0) 3z5 cos( log z +'D 2) . (3.2.138)

So we have

zi - µi = (epi + x0)Qzi cos( log zi + 4) 2) . (3.2.139a)


Period

- -- Saddle OrbitStable (5>1/2) OrbitUnstable (5<1/2) Orbit

- Period DoublingBifurcation

Figure 3.2.41. Stability Diagram for the Bifurcated Periodic Orbits.

zi+1 - ki+1 - (eµi+1 + x0)Qzi+i cos(e log zi+1 + 42) . (3.2.139b)

From (3.2.139a) and (3.2.139b) we obtain

zi - x0/34 cos (e log zi + 41) 2)IL% (3.2.140a)

zi+l - x0/jzi+1 cos(; log zi+1 + (D2)µi+1 - (3.2.140b)

1 + eQ i+1 cos( log zi+1 +'D 2)

Now note that we have

£ log zi+1 - log zi 7r zi+1exp r (3.2.141)

zi

and we assume that z << 1 so that

1 + eQzb(i+l) cos( log zi(i+l) +'D 2) "' 1 . (3.2.142)

So finally we get

µi+1 _ zi+1 + [x0Q cos(e log zi + 'I)2)zs+l(3.2.143)

µi zi - [xOQ cos (e log zi + 2) ] Z6


Now in the limit as z -> 0, (3.2.143) becomes

uQ

1 (ztS

1)5 p(-6) .-exz

(3.2.144)

Recall that 6 = -pla, l; = -w/.1, so we get

lim +1 = - exp p.

i oo pi w(3.2.145)

This quantity governs the size of the oscillations which we see in Figure 3.2.41.

Subsidiary Homoclinic Orbits.

Now we will show that, as we break our original homoclinic orbit (the princi-pal homoclinic orbit), other homoclinic orbits of a different nature arise, and the

Silnikov picture is repeated for these new homoclinic orbits. This phenomena was

first noted by Hastings [1982], Evans et al. [1982], Gaspard [1983], and Glendinning

and Sparrow [1984]. We follow the argument of Gaspard.

When we break the homoclinic orbit, the unstable manifold intersects no atthe point (eµ + xp, µ). Thus, if µ > 0, this point can be used as an initial condition

for our map. Now if the z component of the image of this point is zero, we willhave found a new homoclinic orbit which passes once through a neighborhood of

the origin before falling back into the origin. This condition is given by

0 = /3(eµ + x0)µa cos( log µ + 42) + µ

or

(3.2.146)

-µ=/3(eµ+x0)µscos(Clogp+4) 2). (3.2.147)

We find the solutions for this graphically for 6 > 1 and 6 < 1 in the same manneras we investigated the equations for the fixed points; see Figure 3.2.42.

So for 6 > 1, the only homoclinic orbit is the principal homoclinic orbit which

exists at p = 0.For 6 < 1, we get a countable infinity of µ values

(3.2.148)

for which these subsidiary or double pulse homoclinic orbits exist, as shown inFigure 3.2.43.


A

-k

5>1

/i

5<1

Figure 3.2.42. Graphical Solution of (3.2.147).

Figure 3.2.43. Double Pulse Homoclinic Orbit.

Note for each of these homoclinic orbits, we can reconstruct our original Silnikov

picture of a countable infinity of horseshoes.

For a reference dealing with double pulse homoclinic orbits for the case of real


eigenvalues see Yanagida [1986].

The Consequences of Symmetry.

In the case of a fixed point with real eigenvalues, we saw that the presenceof a symmetry resulted in dramatic dynamical consequences. In particular, thesymmetry implied the presence of an additional homoclinic orbit which resulted in

horseshoes. We now want to examine the dynamical consequences of symmetry in

(3.2.99).

We suppose that H1 and H2 hold and also that (3.2.99) is invariant under the

change of coordinates

(x, y, z) - (-x, -y, -z) . (3.2.149)

This is the only symmetry of (3.2.99) which allows homoclinic orbits, see Tresser

[1984]. In this case, (3.2.99) has a pair of homoclinic orbits Tu, Tl as shown inFigure 3.2.44.

rU

Figure 3.2.44. A Symmetric Pair of Homoclinic Orbits.


Now our previous analysis can be applied to ru and r1 separately with Theo-

rem 3.2.17 applying. However, we are interested in the dynamics near the "figure

eight" ru U r1. The analysis is very similar to that of the nonsymmetric case.We begin by constructing cross sections to the vector field near the origin. We

define2ap

Ho={(x,y,z)E1R3Ee w <x<e, y=0, (zI <E}_ P1 P Spa

p(x,y,z)ER3I-E(e

eWx<-e(e Zeey=0,IzI<e}

HO'+={(x,y,z) EHU I0<z<e}

Ho' _ { (x, y, z) E f1 I -e < z < 0 }IIOl,+ ={(x,y,z)EH0O<z<e}t,-H0 ={(x,y,z) EHo -e<z<0}

(3.2.150)

We construct maps PO 'u, PO '1 on 110 and 11 , respectively, in a manner iden-

tical to the construction of Po in the nonsymmetric case onto Hi = { (x, y, z) I

z = e } and H1 = { (x, y, z) I z = -e }, respectively, where Hi and 111 are chosen

large enough to contain PO and PO '1(HI ), respectively. Maps along the

homoclinic orbits outside of a neighborhood of the origin are constructed also in a

manner identical to that of the nonsymmetric case. Thus, we have

PL,u = Pi o PL'u: Hu,+ -, Hu1 0 0 0

2(3 151). .

PL,1 = PL'1 o H1 ,1 0 0 0


We are now in a position to construct the Poincare map in a neighborhoodof ru U r1. Let denote the linearized flow generated by (3.2.99). Then, by

construction, for each p E Ho (resp. Ho) one and only one of the points Oqw(p)and 0_7rlw(p) belongs to HO (resp. 11u). We denote this point by ¢(p). Let z(p)

denote the z coordinate of any point p E Ho U H. Then the Poincare map isdefined as

P:HOUHD-+HUUHD

( P(P) = PL'u(P)

P

P(P) = PL'l(P)

P() 0(p -P =P(P) = q5(PL'l

())pp))

(

if z(p)z(PL'u(P)) > 0;if z(p)z(PL't(P)) > 0;if z(p)z(PL," (p)) < 0;if z(p)z(PL,1(p)) < 0.

(3.2.152)


Figure 3.2.45. Local Cross-Sections for the Symmetric Case.

We choose a sequence of rectangles Rk E IIo'+, Rk E IIO' such that Ilu'+ _cc 00U Rk and III' = U Rk in exactly the same manner as in (3.2.108). Using the

k=0 k=0same arguments as in the nonsymmetric case, we can show that, for k sufficiently

large, we can choose two µh-horizontal slabs H+k' Huk E Rk and H+k, H1 k E

Rk such that Al and A2 of Section 2.3d hold. See Figure 3.2.46 for an illustrationof the geometry.

We now set up the symbolic dynamics.

Let AN = { ±k, ... , ±(k + N) } for k, N E 7L+, and let SN = AN x {u, 11.Then Eu denotes the set of bi-infinite sequences where each element of the sequence

is contained in SN. For s = (a,u) E SN or s = (a,l) ESN we define Isl = lal


Rtk

7TI0

I

P(H+k)--\ P(H+k)

P(HR k) P(H°°-k)

uHkHuk

Wloc(0)

P(Hpk) P(Huuk)

P(H+k) - P(H+k)

Figure 3.2.46. Horizontal Slabs and Their Images under the

Poincare Map, P.

and we have

uIa={s={s{}°O__ siESooand Isil> lat}.

The main theorem can be stated as follows.

Theorem 3.2.19. a) For each positive integer N there exists a map

ON: EN --, I u,+ U III,u,I 0 0

which is a homeomorphism of Eu I onto ONI =ON(EN

I) such that

P I01v = ON o or ou'l

b) For each real a with 1 < a < -Al p there exists a map

E°o'a -> 11U'+ UII1,-

UPI 0 0

Ruk


which is a homeomorphism of E c'a onto Ooo'a = coo,`(Eoo'a) such thatu,l u,l u,l

P 10.'.: i n o a oU' l

Now suppose that H2 does not hold, i.e., instead we have

-p>A>0. (3.2.153)

In the nonsymmetric case, we saw that there were no horseshoes if (3.2.153) holds,

and the bifurcation analysis of Glendinning and Sparrow showed that the homoclinic

orbit is an attractor and, when it breaks, an attracting periodic orbit is created.However, in the case of a symmetric vector field an interesting effect occurs.

Let E2,+ denote the set of infinite sequences of 1's and 2's. An element ofE2,+ is written as s = {si} o, si E {1, 2}. Choose a point p in a neighborhood of

the origin. We want to consider the forward orbit of p to which we assign an infinite

sequence of 1's and 2's by the following rule. The first entry of the sequence is a 1

(resp. 2) if p moves around Fu (resp. r1) and then re-enters a neighborhood of theorigin. The second entry in the sequence is a 1 (resp. 2) if p subsequently movesaround ru (resp. r1) and re-enters a neighborhood of the origin. We continueconstructing the sequence in this manner. An obvious question is whether any such

forward orbits actually exist. We have the following theorem.

Theorem 3.2.20. If (3.2.153) holds then in each neighborhood of ru U rl thereexist sets of orbits in one to one correspondence with elements of E2,+. The dy-namics encoded in the sequences is such that a 1 corresponds to a circuit aroundru and a 2 corresponds to a circuit around Pl.

PROOF: See Holmes [1980].

Although there are no horseshoes if (3.2.153) holds, Theorem 3.2.20 tells us

that the approach of orbits to Fu U rl is chaotic.

Observations and Additional References

Comparison Between the Saddle with Real Eigenvalues and the Saddle-Focus. Be-

fore leaving three dimensions we want to reemphasize the main differences between

the two cases studied.

Real Eigenvalues. In order to have horseshoes it was necessary to start with twohomoclinic orbits. Even so, there were no horseshoes near the homoclinic orbit until


the homoclinic orbits were broken such as might happen by varying a parameter.

It was necessary to know the global twisting of orbits around the homoclinic orbits

in order to determine how the horseshoe was formed.

Complex Eigenvalues. One homoclinic orbit is sufficient for a countable infinityof horseshoes whose existence does not require first breaking the homoclinic con-

nection. Knowledge of global twisting around the homoclinic orbit is unnecessary,

since the spiralling associated with the imaginary part of the eigenvalues tends to"smear" trajectories uniformly around the homoclinic orbit.

There exists an extensive amount of work concerning Silnikov's phenomenon,

yet there are still some open problems.

Strange Attractors. Silnikov type attractors have not attracted the great amountof attention that has been given to Lorenz attractors. The topology of the spiralling

associated with the imaginary parts of the eigenvalues makes the Silnikov problem

more difficult.

Creation of the Horseshoes and Bifurcation Analysis. We have given part of thebifurcation analysis of Glendinning and Sparrow [1984]. Their paper also contains

some interesting numerical work and conjectures. See also Gaspard, Kapral, andNicolis [1984]. Knot theory has not been applied to this problem.

Nonhyperbolic Fixed Points. There appear to be little or no results concerningorbits homoclinic to nonhyperbolic fixed points in three dimensions.

Applications. The Silnikov phenomenon arises in a variety of applications. See,for example, Arneodo, Coullet, and Tresser [1981a,b]], [1985], Arneodo, Coullet,

Spiegel, and Tresser [1985], Arneodo, Coullet, and Spiegel [1982], Gaspard andNicolis [1983], Hastings [1982], Pilovskii, Rabinovich, and Trakhtengerts [1979],

Rabinovich [1978], Rabinovich and Fabrikant [1979], Roux, Rossi, Bachelart, and

Vidal [1981], and Vyskind and Rabinovich [1976].

3.2d. Fourth Order Systems

We will now study two examples of an orbit homoclinic to a hyperbolic fixed point

of a fourth order ordinary differential equation. The jump from three to four di-mensions brings in a large number of new difficulties and, before proceeding to the

examples, we want to give a brief overview.


1. More Cases to Consider. In three dimensions there were essentially only two

cases to consider (the others could be obtained via time reversal). In four

dimensions there are five distinct cases to consider according to the different

possibilities for the eigenvalues of the linearized vector field at a hyperbolic

fixed point. They are

Real Eigenvalues: 1) \1, A2 > 0, A3, A < 0-2) Al > 0, A2, .A3, A4 < 0.

Complex Eigenvalues: 1) P1 f awl, P2 ± awl; Pi > 0, P2 < 0, w1, w2 # 0.Real and Complex Eigenvalues: 1) P1 ± iwl, al, a2 > 0; P1 < 0, wl 0 0-

2) pl ± iwi, a1<0, a2>0; P1<0,w154 0.

Other cases may be obtained from these via time reversal. If nonhyperbolic

fixed points are considered, then even more cases must be considered.

We will study the example having all complex eigenvalues and the examplehaving real and complex eigenvalues having a one dimensional unstable mani-

fold.

2. More General Horseshoes. In our three dimensional examples we reduced theproblem to the study of a two dimensional Poincare map. The horseshoes con-

tained in a two dimensional map had one expanding direction, one contracting

direction, and one folding "direction" around an axis normal to the plane.For fourth order systems we will be studying a three dimensional Poincaremap. Horseshoes contained in three dimensional maps may have one expanding

direction and two contracting directions or vice versa. Additionally, they may

either have one or two folding directions. The various possibilities are governed

by the nature of the eigenvalues at the fixed point. For the most part, thedifferent possibilities will not be of much concern to us since our main goal is

simply to prove the existence of horseshoes. However, these properties would

be of interest in studying the more "global" aspects of the horseshoes and, in

particular, in finding conditions under which they formed the chaotic hearts of

strange attractors. See Figure 3.2.47 for an illustration of some different types

of three dimensional horseshoes.

3. Computation of the Time of Flight from Ho to Ill. In two and three dimensions

solving for the time of flight was a trivial matter. However (except for the case

of complex eigenvalues), in four dimensions when the unstable manifold is two


Stretch inContract in x-y

T

\Fold Around y

(a)

(c)

(b)

Figure 3.2.47. Examples of Three Dimensional Horseshoes. a) One Expanding, Two

Contracting, and One Folding Direction. b) One Contracting, Two

Expanding, and One Folding Directions. c) One Contracting, Two

Expanding, and Two Folding Directions.


dimensional, the equation for the time of flight is a difficult transcendentalequation which requires a more subtle analysis. We will see an example of this

when we study four dimensional Hamiltonian systems.

4. The Presence of Symmetries. In two and three dimensions it was possible to

more or less guess the possible symmetries of the vector field which allowed the

presence of homoclinic orbits. Moreover, they were all discrete symmetries. In

four dimensions the situation is complicated by the possibility of continuoussymmetries. This would allow for the existence of manifolds of homoclinic

orbits (see Armbruster, Guckenheimer, and Holmes [1987]).

i) A Complex Conjugate Pair and Two Real Eigenvalues

We consider an equation of the following form

i=px-wy+P(x,y,z,w)y=wx+py+Q(x)y,z,w)

(x) y, x, w) a R4 (3.2.154)z=Az+R(x,y,z,w)w=vw+S(x,y,z,w)

where p, ;k < 0, w, v > 0, and P, Q, R, and S are C2 and 0 (2) at the origin. Itshould be clear that (x, y, z, w) = (0, 0, 0, 0) is a hyperbolic fixed point of (3.2.154)

with the eigenvalues of the vector field linearized about the origin given by p±iw, A,

v. Hence, the origin has a three dimensional stable manifold and a one dimensional

unstable manifold. We make the following additional assumptions on (3.2.154).

Hl. Equation (3.2.154) has a homoclinic orbit F connecting (0,0,0,0) to itself.

H2. v>-p>0, -a# v.Our goal is to study the orbit structure of (3.2.154) near r. In order to do this

we will follow our standard procedure of computing a local Poincare map defined

in a neighborhood of r.

Computation of Po P.

We will assume that p > A. In this case the homoclinic orbit is generically

tangent to the x-y plane at the origin. This assumption is merely for geometricalconvenience in constructing cross-sections to the vector field; it will not affect any

of our final results concerning the dynamics of (3.2.154) when HI and H2 hold.


We define the following cross-sections to (3.2.154)

2rp110={(x,y,z,w) EJR.4I E e w <x<E, y=0, 0<w<E, -E<z<E}.

(3.2.155)

As in the case of an orbit homoclinic to a saddle-focus in R3, we choose the xwidth of 110 so that orbits starting on 110 do not reintersect 11 before leaving a

neighborhood of the origin. Additionally, we define

111={(x,y,z, w)EiR4Iw=E}. (3.2.156)

III will just be chosen large enough to contain the image of 110 under the map Pp

which we now describe. See Figure 3.2.48 for an illustration of the geometry.

7f,

Figure 3.2.48. Cross-Sections Near the Origin.


The linearized flow generated by (3.2.154) is given by

x(t) = ePt(xo cos wt - y0 sinwt)

y(t) = ePt(xo sinwt + y0coswt)

z(t) = z0eAt

w(t) = woevt

The time of flight T from 11o to 111 is found by solving

e=woevT

from which we obtain

Using (3.2.157) and (3.2.159)

off the subscript 0's)

(3.2.157)

(3.2.158)

T =

v

log w . (3.2.159)

0

the map Po from 110 to 111 is found to be (leaving

P01: -H1x x(w)p/vcos(v log w)0 x(w)Pl' sin( log w) (3.2.160)

z z(w)A v

w E


We now want to get an idea of the geometry of P (Ho). For this, it will beuseful to consider a foliation of 110 by slabs, as in the case of the saddle-focus in

1R3. We define

ZapRk={(x,y,z,w)EIR4 IEeW <x<e,y=0,0<z<c,

-2ar(k+1)v -2xkvee w < w < ee w }.(3.2.161)

Then we have00

no = U Rk. (3.2.162)k=0

It will be useful to coordinatize the x-y part of H1 by polar coordinates. Denoting

the x, y, z coordinates on 111 by xi, y', z1 in order to- avoid confusion with thecoordinates on 110, we have

r = xi2 + p12 , tan O = y, (3.2.163)X


and, in these coordinates, PP is written

P0:I1p-*ll( Ply

x xl r

0 v log w 0 (3.2.164)

Zz(w A/v = zI

W E E

Now consider an Rk C 110 for fixed k. Using (3.2.164) we make the following

observations concerning PL0 (Rk).

1) The two dimensional sheets w = constant contained in Rk are mapped to 0 =constant under PD .

2) The two vertical boundaries of Rk that are parallel to the w-z plane are mapped

to two dimensional logarithmic spirals.

3) The two dimensional sheet z = 0 contained in Rk is mapped to z' = 0 inii.

4) The ratio z'/w goes to zero as w -* 0 for A < -v and goes to infinity forA> -v.

Using these four remarks the two possibilities for the image of Rk, k fixed, underP' are shown in Figure 3.2.49.

Computation of Pl .

For some open set U C II1 we have

P1:UCII1-110x a b c

y d e fz g h i

f 0 0 0

0 x X

0 y+

0

0 z 0

0 0 0

(3.2.165)

where (y = E((1 + e27rP)/2),0,0,0) = r n 110.


(a)

Po (RK)

(b)

Figure 3.2.49. Geometry of PD (Rk). a) A > -v. b) A < -v.

The Poincare Map PL = PL o PoUsing (3.2.160) and (3.2.165) we have

PL:PL 0 P0:V CHo -*IIbaax( )' cos( log w) +bx(w)" sin (v log --w) czw +

l J(,z)

x

dx(w) cos( log w) +ew) sin(W-, log w) +cz(W

gx(w) cos(v log w +hx(w) L sin (v log w) +flz(w)(3A

2.166)

where V = (Pp) 1(U).

So, if we choose Ho sufficiently small, then PL(Rk) appears as in Figure 3.2.50.

Now we want to show that pL contains horseshoes. From Figure 3.2.50, it


w

x

(a)

x

(b)

Figure 3.2.50. Geometry of PL(Rk). a) A > -v. b) A < -v.


should be evident that this is a strong possibility; however, certain properties must

be satisfied.. In particular, we are concerned with two main effects.

1) The ability to find horizontal slabs which map over themselves into verticalslabs with proper behavior of the boundaries.

2) The existence of sufficiently large stretching and contraction rates along appro-

priate directions.

These effects are formalized in Properties Al and A2, Al and A3, or Al and A2 of

Chapter 2, which imply Theorem 2.3.3 and the resulting chaotic dynamics associ-

ated with the shift map. In our study of an orbit homoclinic to a saddle-focus in1R3 we verified Al and A2 in great detail. However, in this example, we will only

indicate the key points leading to the verification of Al and A2, since the complete

details are very similar to those given for the saddle-focus in 1R3.

Existence of Horizontal Slabs Mapping to Vertical Slabs with Proper Boundary Be-

havior.

Consider Rk C 110, k fixed. It can then be shown that PL(Rk) intersects Rq

in two disjoint µv vertical slabs with ttv satisfying (2.3.69), (2.3.75), and (2.3.78)for i > k/a where 1 < a < -v/p and k sufficiently large. This relies cruciallyon the properties v > -p > 0 and -A # v, and the argument is very similar tothat given in Lemma 3.2.16.

Stretching and Contraction Rates.

An argument similar to that given in Lemma 3.2.15 can be used to show that Al

holds everywhere on IIp with the possible exception of a countable set of (avoidable)

w = constant sheets.

If -A > v there are two contracting directions and one expanding directionand if -A < v, there is one contracting direction and two expanding directions.

Thus, PL contains a countable infinity of horseshoes and a theorem identical to

Theorem 3.2.17 holds. However, despite the rich dynamics which this describes, we

have barely scratched the surface of the possible dynamics of PL and much remains

to be discovered.

ii) Silnikov's Example in iit

The following system was first studied by Silnikov [1967].


We consider an equation of the following form

xl = -P1x1 - w1x2 + P(xi, x2, y1, y2)

x2 = w1x1 - P1x2 + Q(x1,x2,y1,y2)

yl = P2Y1 + W2y2 + R(xl,x2,yl,y2)

y2 = -W2y1 + P2y2 + S(x1, x2, y1, y2)

(x1, x2, yl, y2) ER4 (3.2.167)

where P1, p2, wl, w2 > 0 and P, Q, R, and S are C2 and 0 (2) at the origin.Thus, (xl, x2, y1, y2) = (0, 0, 0, 0) is a fixed point of (3.2.167) and the eigenvalues

of (3.2.167) linearized about the origin are given by -P1+iwl, P2+iw2. Therefore,

the origin has a two dimensional stable manifold and a two dimensional unstable

manifold. Additionally, we make the following assumptions on (3.2.167).

Hl. Equation (3.2.167) has a homoclinic orbit r connecting (0,0,0,0) to itself.

H2. P1 0 P2

So the two dimensional stable and unstable manifolds of the origin intersect non-

transversely along r. Our goal is to study the orbit structure in a neighborhood ofr.

Computation of Po .

We compute the map near the origin given by the linearized flow. For this it

is more convenient to use polar coordinates. Letting

xl = rlcos01

X2 = rl sin B1

Yi = r2cos02

Y2 = r2 sin 02

the linearized vector field is given by

(3.2.168)

(3.2.169)


The flow generated by (3.2.169) is easily found to be

r1(t) = '10e-Pit

01(t) =wit+010r2(t) = r20cP2t

(3.2.170)

02 (t) = -w2t + 020 .

We define the usual cross-sections to the vector field near the origin

HO = { (rl, 01, r2, 02) rl = E }(3.2.171)

H1 = { (rl, 01, r2, 02) r2 = E

Note that no and H 1 have the structure of three dimensional solid tori with the local

stable manifold, i.e., r2 = 0, being the center circle of no and the local unstablemanifold, i.e., rl = 0, being the center circle of Hl. We let pl = (0,0,e,) =r n Wloc and p0 = (c, 0, 0, 0) = r n W. See Figure 3.2.51 for an illustration ofthe geometry.

The time of flight T from no to Hi is found by solving

E = r20eP2T

to obtain

(3.2.172)

T = 1 log e. (3.2.173)

P2 r20

Using (3.2.170) and (3.2.173), the map P is given by (leaving off the subscript 0's)

P0:HO-. H1E P2

B1 61 + P log *2Hr2 E

02 02 - P log r2

(3.2.174)

We now want to get an idea of the geometry of the image of H0 under PD .Consider an infinite sequence of solid annuli contained in HO defined as follows:

-2,r(k+1)p2 -21rkp2Ak = { (r1, 01, r2, 02) I rl = e, 61-a < 01 < 01+a, Ee W1 < r2:5 Ee W1 ,

0 5 02 5 27r } (3.2.175)


x=(x1, x2)

\WUl- /

LP0

Figure 3.2.51. Geometry of the Flow and the Cross-Sections ITO and Ill

Near the Origin.

for some a > 0 and k=0,1,2 ....; see Figure 3.2.52.We want to study the geometry of PO (Ak) for fixed k > 0. In particular, we

are interested in the behavior of the boundary of Ak under PD P. The boundary of

Ak is made up of the union of the two "endcaps," denoted Ek and Ek, and theinner and outer surfaces, denoted Sk and S. More specifically, we have

Ek = {(r1, 01, r2, 02) 1 rl = c, 01 = B1 - a, Ee(k+1)c < r2 < Eekc, 0 < 02 < 2-7r }

Ek = {(r1, 01, r2, 02) rl = c, 01 = Bi + a, ,(k+l)c < r2 < ekc, 0 < 02 < 2v }

Sk = {(rl, 01, r2, 02) rl = c, B1 - a < 0l < Bl + a, r2 = Ee(k+1)c, o < 02 < 2,r }

Sk = {(r1, 01, r2, 02) I rl = E, Bi - a < B1 < Bl + a, r2 = cekc, 0 < 02 < 27r(3.2.176)

where c = -27rP2/wl. See Figure 3.2.53 for an illustration of the geometry.

M. Orbits Homoclinic to Hyperbolic Fixed Points of O.D.E.s

Figure 3.2.52. Geometry of the Ak C 110-

Now, using (3.2.174), we can conclude the following

-27r(ktl)pl -21rkpPO (E%) (r1, 01, r2, 82) lee Wi < rl < ee W1

,

271

01-cc +27rk<01 <81-a+27r(k+1), r2=e,0<82<27r}-2,r(k+i)pl -2akp1

PO (Ek) = { (rl, 01, r2, 02) ee W1 < rl < ce W1 ,

01+a+27rk<01 <01+a+27r(k+1), r2=e,0< 02<27r}-2+r(k+1)p1

PO (Sk) = { (r1, 01, r2, 82) rl = Ee W1 ,

01-a+27r(k+1) 01 <91+a+27r(k+1), r2=e,0<02<2ir}-2,rkp1

PO (Sk) (r1,81,r2,82) rl = Ee 11 ,

B1-a+27rk<01 <B1+a+2irk, r2=e,0<82<27r}.

Putting these together we see that PO (Ak) appears as in Figure 3.2.54.

The Map Pl P.

(3.2.177)

An expression for the affine map Pl can be computed in the usual way. How-


Figure 3.2.53. Geometry of the Boundary of Ak.

ever, we will not do this but rather will describe the relevant features of the geometry

of the map. In particular, since p1 maps to po under the action of the flow generated

by (3.2.167), by continuous dependence on initial conditions we can find a neigh-

borhood of pl which is mapped onto a neighborhood of po. So, for k sufficiently

large, a part of Po (Ak) is mapped over Ak, as shown in Figure 3.2.55.

Horseshoes in PL.

We now point out the relevant features which insure the presence of horseshoes

in PL. The details are similar to those given in our three dimensional examplesand are left to the reader. We consider Ak for fixed k > 0 and PL(Ak).

Proper Behavior of Boundaries under PL. For k sufficiently large, PL(Ak) com-

pletely cuts through Ak. Moreover, part of the image of the endcaps of Ak underPL intersect Ak essentially parallel to their preimage, see Figure 3.2.55. Thus, µh

horizontal slabs can be found in Ak which map over themselves in ,iv vertical slabs

with proper behavior of horizontal and vertical boundaries and with 1.1v and 'Uhsatisfying the necessary requirements (see Section 2.3).


End View

Figure 3.2.54. Geometry of PL0 (Ak).

273

Expanding and Contracting Directions. For P2 > Pi, PL expands along the r2direction, and for P1 < P2, PL contracts along the r2 direction. Lines parallelto Wloc are contracted, and a glance at Figure 3.2.54 shows that lines connecting

Sk and Sk are stretched under PL. Thus, PL contains two expanding directionsand one contracting direction if P2 > P1, or one expanding direction and twocontracting directions if PI < P2. These growth rates can be made arbitrarily large


Figure 3.2.55. Geometry of PL(Ak).

by taking k large enough.Thus, it follows that PL contains a countable infinity of horseshoes with their

attendant chaotic dynamics. A theorem exactly the same as Theorem 3.2.17 canbe stated for PL.

Observations and References.

Much work remains to be done on homoclinic orbits in four dimensions, par-

ticularly concerning detailed analyses of the Poincare maps and the existence ofstrange attractors.

Our second example can be found in Glendinning and Tresser [1985]. Theexistence of subsidiary homoclinic orbits in this example for parametrized systems

is considered by Glendinning [1987]; see also Fowler and Sparrow [1984].

Finally, we remark that we have treated no four dimensional examples hav-


ing multiple homoclinic orbits. One might guess that a saddle point having realeigenvalues with two homoclinic orbits might possess horseshoes in much the same

manner as the Lorenz equations in 1R3. We work out this example for the Hamil-

tonian case (see 3.2e.ii) ) and remark that the same techniques should work in the

general case. However, the precise details remain to be worked out.

3.2e. Orbits Homoclinic to Fixed Points of 4-Dimensional AutonomousHamiltonian Systems

We now will study orbits homoclinic to fixed points in autonomous Hamiltonian

systems. Since all two dimensional (one degree of freedom) Hamiltonian systems

are integrable, and therefore do not possess complicated dynamics associated with

homoclinic orbits, it is natural to begin our study with four dimensional systems.

Let H be a scalar valued function defined on 1R4 which is at least C3.

H(x, y): 14 __4R1'

Consider the vector field

where

(x, y) = (x1, x2, yl, y2) E R4.

xl2

= JDH(xl, x2, yl, y2)y1

y2

(3.2.178)

(3.2.179)

0 0 1 0

0 0 0 1J=-1 0 0 0

0 -1 0 0

This vector field is a Hamiltonian vector field. It is an easy calculation to see that

H(x,y) = constant is invariant for the flow defined by the vector field.

Suppose that (3.2.179) has a fixed point at (x, y) = (0, 0). Then, by Liouville's

Theorem (Arnold [1978]), the eigenvalues of the vector field linearized about (0,0)

must add up to zero. Therefore, there are two types of hyperbolic fixed points for

Hamiltonian systems. They are

1) ±p f is - saddle-focus, and2) +k, ±1 - saddle.


We will study both cases; however, first we want to make some general remarks

concerning the orbit structure for fourth order Hamiltonian systems.

1) As mentioned above, away from fixed points of (3.2.179), orbits of (3.2.179) lie

on three dimensional invariant manifolds defined by H(x, y) = constant.

2) By Liouville's Theorem, the stable and unstable manifolds of fixed points of

(3.2.179) have the same dimensions.

3) Suppose (3.2.179) has a hyperbolic fixed point having two dimensional stable

and unstable manifolds which intersect along a one dimensional homoclinicorbit. Then, generically, this intersection is transversal (see Robinson [19701)

in the three dimensional surface H(x, y) = constant. This statement is nottrue for non-Hamiltonian systems.

We are going to assume that in the above two cases we have a homoclinicorbit r connecting the fixed point to itself. This homoclinic orbit will lie in thetransversal intersection of the stable and unstable manifolds of the origin. Our goal

is to study the orbit structure in a neighborhood of F. We will follow our usualprocedure of constructing a Poincare map near the homoclinic orbit and studying

the orbit structure of the map. However, some modifications must be made dueto the fact that orbits lie on invariant three dimensional manifolds. We begin ourstudy with the saddle-focus.

i) Saddle-focus

This problem was first studied by Devaney [1976. Suppose (3.2.179) has a fixedpoint at the origin having a homoclinic orbit connecting it to itself and the vector

field linearized about the origin is given by

xl = Pxl - wx2

x2 = wx1 + Px2

with flow

yi = -PY1 +Wy2

y2=-Wy1-PY2

p,w > 0 (3.2.180)

xi (t) = e1t(xlo cos wt - x20 sin wt)

x2(t) = e/t(xi0 sin wt + x20coswt)

yi(t) = e Pt(yl0coswt+Y20sinwt)

y2(t) = e Pt(-yiosinwt+y20coswt) .

(3.2.181)

3.2. Orbits Homoclinic to Hyperbolic Fixe: Points of O.D.E.s 277

Without loss of generality, we can assume that the local stable and unstable mani-

folds of the origin are given by

Wloc(0) (x, y) I x = 0 }(3.2.182)

wloc (0) (z, y) I y = 0 }

(note: (x, y) ° (xl, x2, y1, y2) )

We study the orbit structure in a neighborhood of (3.2.179) in the same manner

as we have done previously. Namely, we compute a Poincare map on some appropri-

ately chosen cross-section to the flow into itself. Normally, for a four dimensional

system a cross section would be three dimensional; however, in the Hamiltonian

case, since we are restricted to remain on a 3-d surface, this cross section will betwo dimensional (this is a major simplification). Our map will consist of the com-

position of two maps; one in a neighborhood of the origin given by the linearized

vector field and a global one along r, which is essentially a rigid motion (just as

we have done previously). We will now describe the geometry in a neighborhood of

the origin.

For small enough a the following surfaces are cross-sections to the vector field

Ho = { (x, y) I IxI < E, 1Y1 E } ,(3.2.183)

H1 ={(-,Y) I IxI = E, 1y1 <0-

These surfaces are solid tori (S1 x ]R2). We also consider the intersection of thesesurfaces with the three dimensional energy surface

E0s =HonH-1(0)

(3.2.184)

Ea=H1nH-1(0).

Let as (resp. au) be the intersection of the local stable (resp. unstable) manifoldwith Eo (resp. Eo). So a9 and au are the center circles of the solid tori 110,Hi. Finally, given a transverse homoclinic orbit r, we denote q3 = F n as andqu = F n au. We will attempt to illustrate the geometry in a neighborhood of theorigin in Figure 3.2.56. In this figure we identify the two ends of the cylinders inorder to get the tori. Eo' and E' are represented as two dimensional surfaces inside

Ilo and Ill, respectively. D9 and Du represent two dimensional neighborhoods ofqs and qu in Eo and Eu, respectively.

Computation of PD .


Y

Figure 3.2.56. Geometry of the Flow and Cross-Sections

Near the Origin.

We now want to construct a mapping, PD , of Ds - as into Du - a" (note:Pp cannot be defined on a' since these points are on Ws(0) ). A priori, there isno reason to expect that points in D'8 - as should map into D" - au; however,we shall see that this does happen as a result of the eigenvalues having nonzeroimaginary part.


First we transform the linearized How into polar coordinates. Letting

x1 = ru cOS Bu

x2 = ru sin Ou

y1 = rs cos Bs

Y2 = rs sin Bs

the linearized vector field becomes

and the flow generated by (3.2.186) is given by

ru(t) = r°yept

Bu (t) = wt + Bu

rs(t) = rse-pt

Bs (t) = -wt + 0°

(3.2.185)

(3.2.186)

(3.2.187)

Now we want to show that P maps curves transverse to as in Ds into curveswhich spiral infinitely often around Eu and are C1 -E close (for some given c > 0 )

to au. This is illustrated in Figure 3.2.57.Now how do you see this? Recall the expression for the flow. The length of

time necessary for points on Eo to reach Eo approaches oo as as is approached(we will compute an exact expression for this time shortly). So, as t increases, rsshrinks, ru grows, and Ou and Bs increase monotonically (mod 27r). So, if we view

Bu as being a coordinate for Ep, we can see that the image of a curve S is wrapped

infinitely often around au.

The time of flight from Eo to Eu is found by solving

E = r°uepT (3.2.188)

for

T = 1 log o (3.2.189)p ru


271

PL0

Figure 3.2.57. The Image of a Curve S under PD .

QU

A point on Eo can be labelled (leaving off the subscript 0's) by Bs, x1, x2, anda point on E' can be labelled by Bu, Y1, Y2. (Note: the "extra" coordinate usedto label points on a 2-d surface results from the fact that we do not have an in-trinsic coordinate system for E'9'"; this will not matter.) So the map in Cartesiancoordinates is given by

0 0 /X1 u (xl cos (p log ru) - x2 sin I p log u)

1

X2 fu (x1 sin (p log u + x2 cos (p log ru)

yl 1 (y1 cos (p log ru) + y2 sin (p logfu

Y2 -r (-y1 sin (P log ru) + 112 cos p log ru )

where ru = x1 + x2 and in polar coordinates by

H

(3.2.190)

(3.2.191)

Now let 0:I H DS be a parametrized curve in DS which intersects as at q, whereI = (-T,T) and Q(0) = q for some r > 0; see Figure 3.2.58 (note: in Figure 3.2.58

3.2. Orbits Homoclinic to Hyperbolic Fixc. Points of O.D.E.s 281

the right hand side of the Figure represents E' and E' removed from ir0 and 7rl,respectively, and "flattened out").

Y-

LP0

Uz0

Figure 3.2.58. The Image of the Curve /3(r) under Po P.

Note from Figure 3.2.58 that /3(r) has components /3q, (r), Px, (r), and #X2 (r)

with /3r, (r) = 0 with respect to the coordinates (xl, x2, 0s, rs), and that PL0(/3(T))has components Po (/3(r))Bu, P (/3(r))yl, and P (/3(T))y2 with Po (/3(r))ru =0 with respect to the coordinates (ru, Bu, y1, y2). This coordinatization will beparticularly useful. We have the following lemma.

Lemma 3.2.21. For r sufficiently small PO (6(T)) is C1 c close to au in E.Furthermore, tangent vectors normal to as are stretched by an arbitrarily largeamount and vectors tangent to as are shrunk by an arbitrarily large amount asr-*0.

PROOF: We already know that /3 is mapped around Eo, accumulating on au as


r -+ 0. We need to show that the tangent vector of PD (Q(r)) is a-close to thetangent vector of a', for some given E > 0, and for DS small enough. This will be

true if(drPO (Q(r))yi,drPO (Q(r))Y2)I

lim =0. (3.2.192)T-0

/

dT 0 (a(r))Bul

Note that dTP0 (a(r)) = DPp (o(r)) d .

d

So /using (3.2.190) and (3.2.191) gives

l ldPO (Q(r)) yl = ru [y1 I cos (p log ruj I

p-sin (P

W

log ru

+ 112 (sin (p

log ru f - p cos (Plog ru / / J

,Qal (r)

+ Euu [yl (cos (p log u + P sin (Plog u

+ y2 (sin (W loge

J- w

cos (W loge))

] i3a2 (r)P roll p p ru

+ (W c

(log+ (sin (log)) age (r)

(3.2.193)

drPO (Q(r))y2 = ru [Yl 1 -sin Plog ru) + P cos \P\lo\g ru))

+ y2 (COS (W loge

+ - sin (Wloge

I I, /321 (r)P ru p P ru1l

+22

[Yl (- sin (W log EJ

+W-cosC(w

log e ) /Eru P11 ru p p ru

+ Y2 (COS (Wlog E

J+

wsin (w log E ))]4X2(7-)

P ru

11

p p ru

11+ ru (-sin ( P log ru )) 4Yl (r) + (rE cos ( P log ru / / QY2 (r)

and

drPO(Q(r))9u =

P+Q(r)8,,,

(3.2.194)

(3.2.195)

where Now, as r --* 0, ru,x1,x2 - 0, and we can assume d /3(r)

is bounded so that 4- PP (Q(r))Yi and 4- PP (,0(r))Y2 are bounded as r -> 0, and

drPO(Q(r))au

> oo as r --+ 0. Therefore,

(d Po (a(r))yl, dPo (a(r))y2))lim = 0 (3.2.196)r-+0

d7-PO (a(r))eu

3.2. Orbits Homoclinic to Hyperbolic Fixed: Points of O.D.E.s 283

which means that the tangent vector to the image of /3(r) approaches the tangentvector of au.

Next, we check the stretching and contraction rates for different directions. Let

C=cos E-log-e ru.J ES=sin (log-P ru

T1 = x1C - x2S

T2 = x1S+x2C

T3 = J1C + y2S

T4 = -y1S+y2C.

Then, in Cartesian coordinates, DP0 is given by

iT1+e T2 + uC r3 T1+e T2 uS1T2 EPr T1 + u S 2T2 - EPr Tl + ssC--eru T3 epru T4 r T -1

Eru 3 epruT4

-T wx1 ??wx2eru 4 + epru T3 er, + epru T4

Now look at

DPp Nx2 _ 16X2

Iyl fly,

oxi

(3.2.198)

\Py2I \Qy2)Using 3.2.197 it is (relatively) easy to see that the vector (0, 0,1 14 2 )4correspondsto a tangent vector in the 0' direction and that the length of the image of thisvector goes to zero as r -+ 0 (or equivalently, ru -> 0). Similarly, ((3x1, /3x2, 0, 0)

corresponds to a tangent vector in the direction perpendicular to as, and we havethat the length of

DPo (3.2.199)

goes to oo as r-+0(ru-+0).

The Map Pl1


We now describe the map Pl which takes qu into qs and, consequently, some

neighborhood of qU into qs. Recall that Ws(0) and Ws(0) intersect transversely in

H-1(0) along F. Therefore, Ws(0) intersects Du transversely at qu, and Wu(O)intersects Ds transversely at qs. This is the main feature we need to describe thefull Poincare map; see Figure 3.2.59.

Figure 3.2.59. The Geometry of Wu(0) n Ds andWs(0) n Du.

The Poincare Map PL = Pl o POWe want to show that PL contains a horseshoe, so we need to find P h horizontal

slabs which behave properly under PL and, to verify the stretching and contraction

conditions (see Section 2.3).

We choose a horizontal slab H in DS with horizontal sides "parallel" to as and

vertical boundaries "parallel" to W'(0) n Ds, as shown in Figure 3.2.60.

By Lemma 3.2.21, PO (H) is stretched in the direction of W"(0), contractedin the direction of Ws(0), and wrapped around IJ many times, as shown in Fig-ure 3.2.61, with the vertical boundaries of PO (H) C1 e close to au.

So, for DS and H appropriately chosen, Pl maps PO (H) over H as shown inFigure 3.2.62, with the vertical boundaries of H C1 e close to Wu(0) n Ds.

Thus, we can choose a sequence of horizontal slabs and conclude that pL

contains an invariant Cantor set on which it is topologically conjugate to a fullshift on a countable set of symbols. In other words, we have proven the following

theorem of Devaney.

Theorem 3.2.22 (Devaney [1976]). Consider a two degree of freedom Hamil-

3.2. Orbits Homoclinic to Hyperbolic Fixcc Points of O.D.E.s 285

Figure 3.2.60. Horizontal Slab in Ds.

Figure 3.2.61. The Image of H under PO .

tonian system having a transverse homoclinic orbit to a fixed point of saddle-focus

type (i.e., the eigenvalues are of the form ±p±iw). Then an associated Poincare map

defined on an appropriately chosen cross-section to the homoclinic orbit contains a


pL(H)

H

Figure 3.2.62. The Geometry of PL(H) fl H.

Smale horseshoe.

Finally, we remark that we have shown the existence of horseshoes on the level

set H-1(0). However, due to the structural stability of horseshoes, they will alsoexist on the level sets H-1 (c) for e sufficiently small.

ii) The Saddle with Real Eigenvalues

This problem was first studied by Holmes [19801. We assume that we are givena 2-degree of freedom Hamiltonian system having a fixed point at the origin with

purely real eigenvalues. The vector field linearized about the origin 0 (after apossible linear transformation) is given by

ii = 1x1

i2 = kx2, I, k > 0 (3.2.200)

yi = -lyi112 = -k112

with flow

W°(0) f1 DS

xi (t) = x10elt

x2 (t) = x20ekt

yi (t) =-10e-It

y2(t) = x20e-kt

(3.2.201)

3.2. Orbits Homoclinic to Hyperbolic Fixu. Points of O.D.E.s 287

So (just as in the saddle-focus case) we have

Wsloc(0) _ { (x, y) I x = 0 }

Wloc(0) ={(x,y) I y=0}(x, y) - (X1, X2, YD Y2) - (3.2.202)

We now make the following assumption.

Assumption 1: There are two homoclinic orbits, ra, r,, connecting 0 to itself. ra

leaves 0 in the first quadrant of Wloc (0) and re-enters in the second quadrant ofWloc(0); rb leaves 0 in the third quadrant of Wloc(0) and re-enters in the fourthquadrant of W(0), see Figure 3.2.63.

Figure 3.2.63. The Geometry of the Homoclinic Orbits

Near the Origin.

Now the method of analysis will be the usual one; namely, we construct a map

in the neighborhood of the origin given by the linear flow and compose it with a"global" map along a homoclinic orbit in order to get a Poincare map on someappropriately chosen cross-section. However, in this situation the Poincare mapwill consist of the union of two maps, one along each homoclinic orbit.

Construction of Po .


First we construct maps in a neighborhood of the origin. It will be necessary

to modify our definitions of I10 and H1 slightly. We define

110={(x,y) I x1E1, 1yi=E2},

U1 ={(x,y) xI-E2, Iyl <- E1}.(3.2.203)

Unlike our previous examples, it will be necessary to choose El and E2 carefully in

order to obtain the desired behavior.

E0, E0, as, and au will be as previously defined in the saddle-focus case. Let

ra, rb intersect au, as, at the points pa, Pb, qa, qb. Let Du, D6 be neighborhoods of

pa, Pb, respectively, in Eo, and let Da, D6 be neighborhoods of qa, qb, respectively,

in E. We must now show that points in Ds, Db are mapped into Du U Duunder the action of the flow. This is by no means obvious and will depend on therelationship between q and E2.

We construct the following maps

P0 L'a: Ds - as E0

PL,b: Ds - as --> EO(3.2.204)

and describe what they do, but first we give our second main assumption.

Assumption 2: 1 > k and ra (resp. rb) leaves 0 in Wloc(0) such that pa E au(resp. Pb E a') lies at an angle Ba E (0, 7r/2) (resp. 0 E (7r, 3ir/2) ) and enters0 in Wloc(0) such that qa E as (resp. qb E as) lies at an angle Ba E (ir/2, 7r)(resp. O E (3ir/2, 27r) ); moreover

(tan Bu) (tan Bs) < -6 + 1/k[exp(k - 1) log E ] (3.2.205)

for 6 > 0 and small, where Bs represents the 9s coordinate of points in Sa (resp.Sb), and Ou the ou coordinates of the image of these same points under PO . Se and

Sb are rectangles in Da and Db, respectively, see Figures 3.2.64 and 3.2.65. (Note:

the right side of Figure 3.2.64 represents the two dimensional sheets, E' and E0,removed from 7ro and i'1, respectively, and "flattened out".)

Let Sa be (resp. S6) the part of Sa (resp. SO above as and Sa, (resp. SC)be the part of Sa (resp. Sb) below as, see Figure 3.2.65.

The following lemma describes the orbit structure of the flow near the origin.

3.2. Orbits Homoclinic to Hyperbolic Points of O.D.E.s

Ora

0 1/2

Zun

F°

4a

10.

ou

Figure 3.2.64. Geometry of the Flow and Cross-SectionsNear the Origin.

Dab(Qb

1

31/2

289

21

Lemma 3.2.23. PO 'e(Sa) consists of two components, Pp 'a(Sa) and Po 'a(Sa-)lying across Da and Db , respectively, with the horizontal boundaries of PO 'a(Sa)

and Pp 'a(Sa) C1 e close to as for Sa sufficiently small. Similarly, Po 'b(Sb)

consists of two components, Po'b(S6) and P0'b(S ) lying across Da and Db,respectively, with the horizontal boundaries of PL '6(S6) and Po 'b(S6) C1 c

close to aw for Sb sufficiently small. See Figure 3.2.65.

PROOF: Step 1. We first find conditions on El and e2 so that the images of Sa' ,

S6 ' lie in E. To do this we choose a curve C C Se transverse to as. Now

Po 'a (C) C Eou implies yl (t) + y2 (t) < el (3.2.206)

where yl(t), y2(t) are the yi, y2 components of the image of a point on C.But, using (3.2.201), we get

yl(t) + y2 (t) = y10e-21t + y20e-2kt < (y10

+ Y202 )e-

Since Cc Eo, we have

(3.2.207)

Y10 + y20 = E2 . (3.2.208)


Sa

71/2 71 371/2 27f

PL0

Sf0

0(Sa)

PO (Sb)

U p (Sa )

Pp (Sb)eu

Figure 3.2.65. The Images of Rectangles.

Combining (3.2.206), (3.2.207), and (3.2.208), we see that (3.2.206) will be satisfied

provided

E2e-2kt < E22 1

or

(3.2.209)

E2< ekt . (3.2.210)

El

Equation (3.2.210) tells us that points on Es actually do map to E' under the action

of the linearized flow provided el and E2 (i.e., HO and 111) are chosen appropriately.

Step p. From (3.2.210) we estimate t, or more particularly, the minimum timerequired for points of C to be mapped to E. This occurs for points at the extremaof C since points close to as take arbitrarily long to get to E. At the extrema of


C we have2 2 2

x10 - x20 = E1 (3.2.211)

So, using (3.2.201), we get the equation

x1(t) +22(t) = x10e21t+x20e2kt = E2(3.2.212)

Using the linearized Hamiltonian, H = lxlyl + kx2y2, on the H = 0 surface wehave

1x10y10 = -kx20y20, (3.2.213)

so we can use (3.2.211) and (3.2.213) to express x10, x20 in terms of El to get

e21t e2kt1

E2= E1 1 + a2 + 1 + a-2

where a = ky10Y20

soE2 e21t e2kt2

El 1+a2 + 1+a 2

(3.2.214)

(3.2.215)

Now a2 is a positive number between 0 and oo and 1 > k, so we have

e2kt e2kt e21t e2kt e21t e21t+ < + < + (3.2.216)

1+a2 1+a_2 1+a2 1+a-2 1+a2 1+a-2

1Note that

1 + a2 + 1 + a-2 = 1 .)

Combining (3.2.215) and (3.2.216) we obtain

ekt <EZ

< elt (3.2.217)El

or

Ilog El < tmin < k log E1

.

(3.2.218)

Step 3. We now find out how far the image of C extends around au. The angle,Bu, of the image of a point is given by tan- 1(x2(t)/x1(t)).

Now points on C arbitrarily close to as take arbitrarily long to reach E0. Thuswe would have

x2(t) _ x20ekt - 0 for t large since 1 > k . (3.2.219)xl(t) xloelt


xl

Figure 3.2.66. An End-Wise View of D6 along the Torus H0.

So these points would arrive on E' with points near Bu = 0 and Ou = 7r. Nowthe question is, which points go where? We will look at this situation more closely.

We want to know what happens to C under Po '°' and, more specifically, which

points go to Ou - 0 and which go to Bu - 7r.

We consider points on C arbitrarily close to as. Now let us examine Da by"looking down" the "cut open" torus and seeing how it projects on the x1 - 22plane, see Figure 3.2.66.

Dd must appear as in Figure 3.2.66 for the following reason: On the H = 0energy surface, using the Hamiltonian, we have seen that the following condition

holds210-k 1/20220 1 1/10 '

(1, k > 0) . (3.2.220)

Now, all points in Da are such that their O values satisfy 0, E (7r/2,7r), butOs = tan-l (y20/y10) implies that y20 and y10 have opposite signs. Therefore,

210 and 220 must have the same sign, either +, + or thus, we have justified

Figure 3.2.66.

Now we can answer our original question, namely, where do the points on C


near as go? So now we know there are two possibilities,

lBu = tan-1 j 220e(k-1)tIL 210 J

(3.2.221)

The points with x10, x20 positive are mapped near Bu = 0, and the points withx10, 220 negative are mapped near Bu = r. Furthermore, since we can assume that

all points on C have essentially the same (yl, y2) values (draw a picture) and the

sign of (Y1, Y2) cannot change during their time evolution (i.e., y1(t) = y20e-kt),

then all points of C at least have the same sign as (Yl, y2) (the y1, y2 values areessentially equal for all points on C if it is chosen correctly). It is also important to

note that, for points further out on C from as (i.e., the extrema of C), t decreases,

which results in Ou increasing (this is seen by examining Bu = tan-1 [(k_1)t]xio

So, finally, we have established that Figure 3.2.67 holds.

Poa (C+)

Figure 3.2.67. The Image of C under PO 'a

Using similar arguments, you can show that a curve in D6, transverse to as, is

mapped by PO 'b as in Figure 3.2.68.


nD;pa P

KY-PO (C+) PO

(C-)

aU

71/2 71 371/2 27

Figure 3.2.68. Image of the Curve C under PO '6

Next we need to know how far around vu the curves extend, and then we need

to fatten them up into strips.

From (3.2.201) and (3.2.220) we obtain

x2(0 - x20ekt _ -1y10e(k-I)t(3.2.222)

x1(t) x10elt ky20

From (3.2.218) we have1 E2train <k log E .

So combining (3.2.222) and (3.2.223) gives

x2y20

x1y10

I> exp E(1 k /

log E2'1

(3.2.223)

(3.2.224)

or

Itan Bn tan Os I > I exp 1(1 - log!2]

. (3.2.225)1


So, since the images of C close to a` are mapped to Bu -r 0 or 7r, by continuity

each component of the image of C will extend from 0 (resp. 7r) to an angle Bu (resp.

Bu+7r) given by the above inequality. - ow, if qa, pa, qb, pb1lie on as, o' satisfying

tan Bu tan Os < 1 exp [(i_ I ) logCl2 J

, (3.2.226)

then we can be sure that the image of C under PP 'a "pushes past" Pa and Pb asshown in Figure 3.2.67.

Furthermore, if for some fixed, small 6 > 0, the following is satisfied (every-

thing is still defined the same)

Itan©u tanes1 < -6+ k exp L(1 - I) log E2] . (3.2.227)1

(This allows us to vary Bs slightly.) Then we can fatten up C into a strip, andFigure 3.2.65 will hold.

The analogous situation holds for a strip in D. This proves the lemma. 11

The map Pl .We now discuss the maps along the homoclinic orbits outside of a neighborhood

of the origin. In the usual manner, for Da, D6 sufficiently small, we define

PI 'a: Du --> Da , with Pi 'a(pa) = qa ,

P1 'b: Du - DL'b , with Pi 'b(pb) = qb(3.2.228)

Recall that Ws(0) intersects Wu(0) transversely along ra and Fb. So the imagesof Wu(O) n Du and Wu(0) n D6 under Pi 'a and PL 'b, respectively, intersectW8(0) transversely at qa and qb, respectively. See Figure 3.2.69.

The Poincare Map PL.We define

pL: (Da U D6) - as -> Eu (3.2.229)

as PL,, U pL,b where pL,a = pL,a o P0 L'a and pL,b = PL'b o P0L'b1

. We argue

that PL contains an invariant set on which it is topologically conjugate to a subshift

of finite type.

Consider the four horizontal slabs Sa , Sa , S6 , and Sb S. By Po ,a and Pp 'b


their vertical boundaries are contracted and their horizontal boundaries are ex-panded by an arbitrary amount depending on the size of Da and D6. Under Pb,a

1

Figure 3.2.69. The Geometry of W u(0) n Da, Wu(O) n D6,

W9(O) n Da, and Ws(O) n D.

and P1 'b these four sets are mapped back over Da and Db with their verticalboundaries C1 e close to Pi 'a(Wu(0) n Da) and Pi 'b(Wu(0) n Dbu). See Fig-ure 3.2.70.


2;u

0

Uu

71/2 71 371/2 271

Figure 3.2.70. The Geometry of Horizontal Slabs under PL.

More specifically, we have

PL(Sa) intersects S.+ and Sa

PL(Sa) intersects Sb and S-

PL(Sb) intersects Sa and Sry

PL(Sb) intersects Sb and S6 .

(3.2.230)


Thus, the transition matrix would be

1 1 0 0

0 0 1 1A= (3.2.231)

1 1 0 0

0 0 1 1

Note that A is irreducible (see Section 2.2c).

We have proven the following theorem.

Theorem 3.2.24. The Poincard map PL has an invariant Cantor set on which itis topologically conjugate to a subshift of finite type with the transition matrix A

given in (3.2.231).

We remark that, as in the saddle-focus case, horseshoes also exist on the level

sets H-1 (c) for c sufficiently small due to the structural stability of the horseshoes.

iii) Devaney's Example: Transverse Homoclinic Orbits in an IntegrableSystem

We have seen that when the eigenvalues at the fixed point are real, then the ex-istence of horseshoes near a homoclinic orbit is subtle. Indeed in Holmes' [1980]

example there were three requirements; 1) existence of two homoclinic orbits, 2) the

homoclinic orbits enter and leave a neighborhood of the origin in specific angular

ranges, and 3) the eigenvalues at the fixed point must satisfy a certain relation.The fact that the horseshoes may not be present if any of these requirements is not

met is dramatically illustrated in an example due to Devaney [1978]. Devaney has

constructed a Hamiltonian system on real projective n-space which has a uniquehyperbolic fixed point having real eigenvalues and 2n transverse homoclinic orbits.

He then proves that the system is completely integrable and hence, by a theorem

of Moser [1973], no horseshoes may exist.

3.2f. Higher Dimensional Results

We now describe two results of Silnikov [1968], [1970] which are valid for arbitrary

(but finite) dimensions. The first result deals with the saddle-focus case.We consider the system

2 = Ax + f (x, y)(x, y) E IRm X IEt (3.2.232)

y=By+g(x,y)


where A is an m x m matrix having eigenvalues Al_., Am with negative realparts, B is an n x n matrix having eigenvalues 'Yi...... n with positive real parts,and f and g are analytic and 0 (2) at the origin. Thus, (3.2.232) has a hyperbolicfixed point at the origin having an m dimensional stable manifold, W'(0), andan n dimensional unstable manifold, W"(0). We make the following additionalassumptions.

Al. (3.2.232) possesses a homoclinic orbit r connecting the origin to itself. More-

over, we assume that W'(0) and Wu(0) intersect simply along r in the sensedim(TpW S (0) flTpW"(0)) = 1 V p E r, where TpW S,"(0) denotes the tangent

spaces at the point p.A2. 'Yl and 12 are complex conjugate and Re(ryl) < -Re(ai), i = 1.... , M.A3. Re(ry1) <Re (ryj), j =3,...,n.A4. A certain matrix is nonsingular.

Silnikov has proven the following theorem.

Theorem 3.2.25. If Al through A4 are satisfied, then an appropriately definedPoincare map near r contains an invariant set on which it is topologically conjugate

to the subshift acting on E°O,6 where 6 = -Re(al)/Re(yl).

We remark on the somewhat mysterious A4. This assumption insures that the

closure of the connected part of W"(0) in a sufficiently small neighborhood of ris locally disconnected. Computation of the matrix is involved, and we refer thereader to Silnikov [1970] for the details.

Next we give a result of Silnikov [1968] describing the bifurcation of periodic

orbits from homoclinic orbits.

We consider the system

z = z E Rm'+n, li E 1R1 (3.2.233)

where Z is analytic. We assume that (3.2.233) has a hyperbolic fixed point at z = 0

for IL E [-lo, iio] and that the eigenvalues of (3.2.233) linearized about z = 0 aregiven by A1(IL)...... m(ii) having negative real parts and yl(µ),...,ryn(µ) havingpositive real parts. We further assume

Al. At Ii = 0 (3.2.233) has a homoclinic orbit r connecting z = 0 to itself.Moreover, dim(TpW S (0) fl TpW u(0)) = 1 V p c r.


A2. y1(0) is real and 'yl (0) < -Re(AZ), i = 1, ... , M.

A3. -yl(0) <Re(-yj(0)), =2,...,n.A4. A certain matrix is nonsingular.

Silnikov has proven the following result.

Theorem 3.2.26. If AI-A4 hold, then only one periodic motion may bifurcatefrom I' as µ is varied. The periodic orbit is stable if n = 1 and a saddle if n > 1.

We remark that, similar to the first result, A4 deals with the geometry of the

closure of the unstable manifold with a sufficiently small neighborhood of the origin.

The computation of the matrix is involved, and we refer the reader to Silnikov [1968]

for details. For both results it should be possible to reduce the differentiability of the

vector field from analytic to Cr, r finite. This would be useful for center manifoldtype applications. Much work remains to be done in higher dimensions.

3.3. Orbits Heteroclinic to Hyperbolic Fixed Points ofOrdinary Differential Equations

We will now examine two examples that show how orbits heteroclinic to fixed points

may be a mechanism for the creation of horseshoes. Recall that, in our examples of

homoclinic orbits, horseshoes were created as a result of the violent stretching and

contraction that a region of phase space experienced as it passed near the saddlepoint, with the homoclinic orbit providing the mechanism for the region of phase

space to eventually return near to where it started. Now a heteroclinic orbit doesnot provide a mechanism for a part of phase space to eventually return near to where

it started. However, two or more heteroclinic orbits may provide this mechanism.

Definition 3.3.1. Let p1, p2, ... , pN be hyperbolic fixed points of an ordinarydifferential equation with r1,2, F2,3, , rN-1,N and rN 1 heteroclinic orbits con-

necting pl and p2, p2 and p3, ..., PN-1 and pN, and PN and pl, respectively.Then {pl} U F1,2 U {p2} U r2,.3 U {p3} U U {pN_1} U FN-1,N U {pN} U I'N,1

is called a heteroclinic cycle if it is a nonwandering set. See Figure 3.3.1.

Our analysis of the orbit structure near heteroclinic cycles will be the same as

our analysis of the orbit structure near homoclinic orbits, with the main differencebeing that the Poincare map is the composition of at least four maps, one for a

3.3. Orbits Heteroclinic to Hyperbolic Fixes Points of O.D.E.s

(a) (b)

Figure 3.3.1. a) Heteroclinic Orbit. b) Heteroclinic Cycle.

301

neighborhood of each fixed point and one for each heteroclinic orbit outside of aneighborhood of the fixed points. We begin with our first example.

i) A Heteroclinic Cycle in ]R3

This example was first studied by Tresser [1984]. Suppose we have a third orderC2 ordinary differential equation which possesses two fixed points, P1, P2, havingeigenvalues of the following type:

P1 : Al > 0, A3 < A2 < 0. saddle,

P2 : A > 0, p ± iw, p < 0. saddle-focus.

So each fixed point is hyperbolic and possesses a two dimensional stable manifold

and a one dimensional unstable manifold. Next, we want to hypothesize that these

coincide in such a way as to form a heteroclinic cycle; however, this can occur in a

variety of ways, not all of which result in nonwandering sets. We will assume theset-up in Figure 3.3.2.

Now, if p1,p2,F12 are such that they lie in the same plane, then it is easy to

see that r = r12 u F21 U {P1} U {P2} is a nonwandering set in this geometrical

configuration; we will assume that this is the case.


Figure 3.3.2. The Heteroclinic Cycle.

We now proceed in the usual way; namely, we construct a Poincare map, PL,

defined in a neighborhood of F and study its properties. Construction of maps

associated with the fixed point having real eigenvalues is similar to those in Sec-

tion 3.2c,i), and construction of maps associated with the fixed point having com-

plex eigenvalues is similar to those in Section 3.2c,ii). We will refer back to these

examples for certain details.

Construction of P O and P102P.

We construct cross-sections 1101 and 1111 in a neighborhood of pl as in (3.2.77)

and cross-sections 1102 and 1112 in a neighborhood of p2 as in (3.2.104), see Fig-

ure 3.3.3.

We assume that in a neighborhood of pl the linearized flow in an appropriate

local coordinate system is given by

x1(t) = xloeAlt

y1(t) = y10eA'-t

z1(t) = zloeA3t .

(3.3.1)

The time of flight from 1101 to 1111 is given by t = al log al . So the map, POl, is

3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of O.D.E.s

Figure 3.3.3. Geometry of Cross-Sections Near the Origin.

given by

x1 ao

E H (E) ai

(Zi)z1 (2 )i

303

(3.3.2)

Likewise, we assume that in an appropriate local coordinate system the linearizedflow in a neighborhood of p2 is given by

x2 (t) = ePt(x20 cos wt - Y20 sin wt)

Y2 (t) = ePt (x20 sin wt + y20 cos wt) (3.3.3)

z2(t) = z20eat

The time of flight from 1102 to 1112 is

P 2 1102-'1112

given by t =f

log z2, so P02 is given by

e

I X21 ( x2 (z2) cos a log z2

0 1 -

P01 :1101-'11,1

z2,x2

Z.sin log z2

(3.3.4)


Construction of the Maps P11 and P.The maps P111 along 1'12 and p L along r21 are constructed in the usual

manner. We have

P2: 1112-'1101x2 (0) (a2 b2 0 x2

Y2 -' 0 + c2 d2 0 Y2 ,

E E 0 0 0 0

and

pL11 : 1111 --' 1102

E E 0 0 0 0

Iyl - 0 + 0 al b1 ylzl 0 0 cl dl zl

The Poincare Map PL.We can now compute the Poincare map

PL-P 1 P2 oPA -P02: 1102-no2,

but first we want to simplify the notation slightly. Let

ea2x2 (z ) *

cos a log

z

+ b2x2 (z ) sin-

logz2 2 2 2

P_ PE W E

c2x2z2

cos - logz2

+ d2x2Z12

) sin - logZ2

(3.3.5)

(3.3.6)

(3.3.7)

= klx2 (E) cos(O+q51)Z2

= k2x2 (E) sin(O + 02)2G2

where 0 = log z2 , _ - tan-1 ^2 , kl = a2 + b2, and k2 = c2 + d2.

Then, we get

k1x2 (Z2 cos(8+01)P2 o P02(x2,z2) = P

k2x2 (z- I sin(O + q52)

and

P Z o P2 o P2 (x2, z2) _

E klx2 z cos(8 + q51)

rgE"1 {k2x2 (z2) sin(O+02)] [k1x2 (z'),

a,

1

I

(3.3.8)

(3.3.9)

3.3. Orbits Heteroclinic to Hyperbolic Fixes Points of O.D.E.s 305

and, finally,

PL=P 2oP ioP 2oP02(x2,z2) _A

1 E1+ [kl cos(B + O1)]

c1E1+ [k1 cos(O + 01)]

where

E ak2 = k2x2

Z2

The analysis of pL is very similar to the analysis given in the case of an orbithomoclinic to a saddle-focus in 1R3. We will leave the details to the reader and only

show the relevant features which give rise to horseshoes.

For z2 small, since IA31 > IA21, this map is essentiallyq nay a9

(Z2)

X2k1x2z2A1 cos Al +01)' ,a A,

(3.3.11)

k2x2 z cos Al (B + 1)

since 1A31 > IA21 where kl = alel+(a2/A1)+(pa2/aa1)kia2/all and k2 =clE1+(A2/A1)+(pa2/aa1)k1I a2/1\11. This looks somewhat similar to the map whichwe derived in the original Silnikov situation for homoclinic orbits in 1R,3. Figure 3.3.4

should make clear the similarities.

From Figure 3.3.4 it can be seen that a rectangle is rolled into a logarithmicspiral by the saddle-focus and then pinched into two pieces by the saddle; however,

we still can find horizontal strips which are mapped into vertical strips, so thesituation is essentially the same as the Silnikov situation for homoclinic orbits inIR3

This map can be analyzed in exactly the same way as the map obtained for a

homoclinic orbit to a saddle-focus in iRe, and essentially all of the same conclusions

will hold. (We leave out the details.) Notice that the quantity pA2/AA1 plays the

same role as the quantity -p/A in the case of an orbit homoclinic to a saddle-focusin 1R.3.

We have the following results.

A9Al

a

+ b1 EA1 [k2 sin(8 + 02)] [k1 cos(O + 01)] a

+ d1 E [k2 sin(8 + m2)] [k1 cos(B + 02)](3.3.10)

Al

PE a

kl = k1x2Z2

P



Theorem 3.3.1. The Poincare map PL possesses a countable infinity of horseshoes

provided

P1\2 < 1 .

PROOF: Left to reader.

Tresser [1984] discusses in more detail the coding of orbits in the horseshoesvia symbolic dynamics. If (pa2/AA1) > 1 then PL no longer contains horseshoes;

however, a theorem similar to Theorem 3.2.20 can be proven indicating that theheteroclinic cycle is an attractor with chaotic transients nearby. If the heterocliniccycle is broken via perturbations (such as may occur by varying a parameter inparametrized systems), then a finite number of horseshoes are preserved for suffi-

ciently small perturbations.

ii) A Heteroclinic Cycle in ]R4

This example can be found in Glendinning and Tresser [1985]. We consider anautonomous ordinary differential equation in iR4 which possesses two fixed points,

P1,P2, having eigenvalues of the following type:

P1: (A1, -pi ± iwi, -vi)(3.2.12)

P2: (A2, -P2 ± i-2, -V2)

3.3. Orbits Heteroclinic to Hyperbolic Fixed Points of O.D.E.s 307

with vj > Aj > pj >0, j=1,2.

We suppose that in appropriate local coordinate systems near pl and P2 thevector field takes the form

xi = -pixi `iyiyi = -Wixi - piyi

zi = Aizi

tbi = -viwi

1 = 1, 2. (3.3.13)

So p1 and P2 possess 3-dimensional stable manifolds (with spiralling in the xi, yi

directions) and 1-dimensional unstable manifolds. Furthermore, we suppose that

there exist two heteroclinic orbits, r12 connecting pl to P2 and r21 connecting P2

to Pl. r12 leaves p1 along the z1 axis and r21 leaves P2 along the z2 axis. Wegive a rough illustration of the geometry by suppressing the y1,Y2 directions inFigure 3.3.5.

Figure 3.3.5. Geometry of the Heteroclinic Orbits and Cross-Sections

Near the Origin.


We construct the following 3-dimensional cross-sections to the flow,

1101 = { (x1, y1, z1, w1) Yl = 0 }

rill = { (x1,y1,zl,wl) zl = E}

1102 = { (x2,y2,z2,w2) I Y2 = 0}

1112 = { (x2, y2, z2, w2) I z2 = E }

(3.3.14)

where these cross-sections are defined as in the example of an orbit homoclinic to

a fixed point with eigenvalues p ± iw, A, v (p, A < 0) in Section 3.2d,i).

Now we will construct a Poincare map in a neighborhood of the heteroclinic

cycle I' r12 U I'21 U {pl} U {p2} (note: it should be checked that r is anonwandering set) and study its properties in the usual way. The Poincare mapwill be the composition of four maps

Ppl:1101 - rill

P 1:1111 -4 1102

PL02: H02 ` 1112

PL12: H12 - II01

(3.3.15)

where POi and POi are given by the linearized flow. Note that the fact that theunstable manifold is 1-dimensional for both fixed points (and hence coincides with

the heteroclinic orbits) guarantees that IIiO maps into rill and 1120 maps into 1121

under the action of the linearized flow. We remark that the domain of the P may

not be all of the riij (as you can see from the pictures), but rather an appropriately

chosen subset (as in the Silnikov case in 1R3).

Since we have computed similar maps many times previously, this time we will

just state the results.

POl : 1101 rill

X1 X

P0 x1 ( } al sin \ i log zi )zi

E

wl wi(

(3.3.16)


A2 B2 0 0 x2

0 0 0 0 Y2 (3.3.19)

0 0 0 a2 0

C2 D2 0 0 W2

where we have chosen 1102 and 110, in such a way that r12 and r21 intersect these

hypersurfaces at (e,0,0,0), respectively. We assume that our coordinate systemis such that the matrices assume the above given block diagonal form. This formexpresses the following two geometrical assumptions:

P11 maps the x1-y1 plane ni11 into the z2-w2 plane f1H02;

P12 maps the x2-y2 plane f1II12 into the x1-w1 plane (11101

Now we want to give a step by step geometrical picture of what the maps do.

Step 1.

PL02 : 1102 H12(ln

X2 x2 (zE ) A2 Cos ( 2 log Z2 )

0 H x2 _ A2 sin log (3.3.17)( A2 Z2

Z2

W2 w2 (z-' 1 2

P1 . 1111 -' H02xl E 0 0 0 al x1

yl 0 0 0 0 0 yl (3.3.18)

E 0 1Al B1 0 0 0

wl 0 C1 Dl 0 0 w1

PL01 : 1101 -' 17111

//1

lalX1 x1 ( cos (2 og zl)1 0 (3 3 20)H X1 E al sin log z(1 i

. .

z1C

W1 LiW1

(1)

Assume POi is defined on a piece of 1110 as shown in Figure 3.3.6.

Note that P01 is not defined on zl = 0, since these points are on Ws(p1). Byexamining the expression for PDi given above, the x1 and zl directions are twisted


z,

xl

Figure 3.3.6.

W1

xl

Figure 3.3.7.

into logarithmic spirals with the wl coordinate shrinking to zero as zlFigure 3.3.7.

-3 0, see

Step 2. P1i:1111 I101 maps P01(1101) as shown in Figure 3.3.8.

Step 3. P02:1102 -' 1112 twists P11 0 POl (1101) around in much the same manner


Z2

X2

Figure 3.3.8.

W2

X2

Figure 3.3.9.

as p L deforms ITO,, see Figure 3.3.9.


Step 4. Finally, P2 transports P2 o P1 o P210 1 0(1101) back to 1101 as shown in

Figure 3.3.10.

Z1

X1

Figure 3.3.10.

From Figure 3.3.10 you should be able to imagine that you can find horizontal

slabs which map into vertical slabs with proper behavior of the boundaries. Thenecessary stretching and contraction conditions will be satisfied provided

P1v2 < 1 (3.3.21)' 1'A 2

and2v1 < 1 . (3.3.22)

1\ 11\ 2

(3.3.21) and (3.3.22) imply that the horseshoes have two expanding directions and

one contracting direction.

Remarks.

1) Parametrized Systems. In both of our examples horseshoes were found in a

neighborhood of the heteroclinic cycle. However, if there is no spiralling (i.e.,

only real eigenvalues) it may be necessary to break one or more of the het-eroclinic orbits in order to obtain horseshoes (see 3.2c,i) ). This is possible

3.4. Orbits Homoclinic to Periodic Orb'- Invariant Tori 313

in parametrized systems. However, if there are no symmetries present, typi-

cally it is necessary to have the number of parameters equal to the number of

heteroclinic orbits in order to insure the proper behavior.

2) Other Results. Tresser [1984] has generalized our first example to the situa-

tion of multiple hyperbolic fixed points in It3 with one-dimensional unstable

manifolds (or one-dimensional stable manifolds under time reversal). Devaney

[1976] discusses some heteroclinic cycles in Hamiltonian systems. Heteroclinic

cycles frequently arise in applications. For example, they appear to be themechanism giving rise to "bursting" in a model for the interaction of eddies in

the boundary layer of fluid flow near a wall, see Aubry, Holmes, Lumley, and

Stone [1987] and Guckenheimer and Holmes [1987].

3.4. Orbits Homoclinic to Periodic Orbits and Invariant Tori

In Section 3.2 we studied a variety of examples of orbits homoclinic to hyperbolic

fixed points of ordinary differential equations. We will now study the dynamicsassociated with an orbit homoclinic to a hyperbolic periodic orbit or a normallyhyperbolic invariant torus of an ordinary differential equation. Unlike the case oforbits homoclinic to hyperbolic fixed points of ordinary differential equations where

the results depended on a variety of factors such as dimension of the system, thenature of the eigenvalues at the fixed point, the existence of symmetries, etc., we

will derive a general result implying the existence of "horseshoe-like" dynamicswhich is independent of these considerations (though these factors may give rise to

important dynamical effects which are not captured by our theorem).

The spirit of our analysis will be the same as for orbits homoclinic to hyperbolic

fixed points of ordinary differential equations; however, there will be some important

technical differences. The main difference is that we will not deal at all with anordinary differential equation but rather with a map. This causes no difficulty inapplying our results to ordinary differential equations, for recall from Section 1.6that the study of the orbit structure near a periodic orbit of an ordinary differential

equation could be reduced to the study of the orbit structure near a fixed point ofthe associated Poincare map. Similarly, the study of the orbit structure near an l+1dimensional invariant torus of an ordinary differential equation could be reduced to


the study of the orbit structure of an 1-dimensional invariant torus of the associated

Poincare map. Thus, results for maps describing the dynamics of orbits homoclinic

to l-tori have an immediate interpretation for the dynamics of orbits homoclinic to

l + 1-tori in ordinary differential equations (note: a 0-torus is fixed point and a 1-

torus is a periodic orbit for an ordinary differential equation). Technicalities aside,

the spirit of our analysis will be the same in the sense that we will look for a region

near the homoclinic orbit which is mapped back over itself by some iterate of the

map in a "horseshoe-like manner." In particular, we look for horizontal slabs which

are mapped to vertical slabs with proper behavior of the boundaries and sufficient

stretching and contraction. As in the case for hyperbolic fixed points of ordinarydifferential equations, the homoclinic orbit provides the mechanism for the global

folding of the phase space and the invariant set to which the orbit is homoclinic(i.e., the 1-torus) provides the mechanism for the stretching and contraction. Before

giving specific hypotheses we want to give an intuitive description of the ideas.

Orbits Homoclinic to a Hyperbolic Fixed Point.

Suppose we have a diffeomorphism of 1R2, f, possessing a hyperbolic fixedpoint p0 whose stable and unstable manifolds intersect transversely at some point

p, as shown in Figure 3.4.1. We remark that unlike the case of orbits homoclinic

to hyperbolic fixed points of ordinary differential equations it is possible for thestable and unstable manifolds of a hyperbolic fixed point of a map to intersect ina discrete set of points without violating uniqueness of solutions. This is because

orbits of maps are infinite sequences of discrete points whereas orbits of ordinarydifferential equations are smooth curves.

Also, recall the definition of transversal intersection of manifolds in Section 1.4.

The importance of transversality will become important when we rigorously justify

the following heuristic arguments.

Now p lies simultaneously in the invariant manifolds Ws(p0) and Wu(p0);hence, the orbit of p must lie in both Ws(p0) and Wu(p0). Thus, iterating Fig-ure 3.4.1 gives us the hornoclinic tangle part of which is shown in Figure 3.4.2.

p is called a transverse homoclinic point. So one transverse homoclinic point implies

the existence of a countable infinity of transverse homoclinic points due to theinvariance of Ws(p0) and W a(p0). Note from Figure 3.4.2 that Ws(p0) and W'a(p0)

appear to accumulate on themselves. We will justify this analytically shortly. For a

more detailed and careful discussion of Figure 3.4.2, we refer the reader to Abraham


Figure 3.4.1. Intersection of the Stable and Unstable Manifolds of po.

Figure 3.4.2. The Homoclinic Tangle.


and Shaw [1984].

Our goal is to show how horseshoe-like dynamics may arise from this situation.

Consider the domain D shown in Figure 3.4.3 whose left vertical side lies in WU(p0)

and whose right vertical side touches W s (p0). By invariance D must maintain this

contact with W `(p0) and W'(p0) under all iterations by f. This is an importantpoint to remember.

Figure 3.4.3. Geometry of the Domain D.

Next we consider f (D) which appears as in Figure 3.4.4a. Now we deduce the

behavior of f (D) by noting the portions of f (D) which must remain on WU(p0) and

W s (p0), respectively. However, an obvious question is, why can't f (D) appear as in

Figures 3.4.4b, c, d, since these situations still respect invariance of the manifolds?

The answer is that these situations are indeed possible, and we have only chosen

Figure 3.4.4a for definiteness. However, we make the following comments regarding

the remaining figures.

Figures 3.4.4b, d. These situations cannot occur if f preserves orientation. Recall

from Section 1.6 that Poincare maps arising from ordinary differential equations

3.4. Orbits }tomodhik to Par:odir. Orbits aji Invariant Tori 317

F

rV


must preserve orientation.

Figure 3.4.4c. It is certainly possible for the image of a "lobe" formed by pieces

of W3(p0) and W u (p0) to "jump" over many other lobes under iteration. We have

chosen the situation in Figure 3.4.4a where the lobe goes to the nearest possiblelobe under iteration by f while preserving orientation.

Thus other iterates of D appear as in Figure 3.4.5.

Figure 3.4.5. Iterates of D.

From Figure 3.4.5 it is apparent that f 7(D) intersects D in a horseshoe shape.

This should indicate the possibility of the existence of horseshoe-like dynamics;however, much remains to be rigorously justified which we will do shortly. In

particular, we will show that the geometry allows us to find horizontal slabs whose

image intersects them in vertical slabs with proper behavior of the boundaries and

the necessary stretching and contraction rates. Thus, some iterate of f will contain

an invariant Cantor set on which it is topologically conjugate to a full shift on acountable set of symbols.

Orbits Homoclinic to Normally Hyperbolic Invariant Tori.


Now consider the situation of a diffeomorphism of 1R3, f, having a normallyhyperbolic invariant 1-torus, ro, (i.e., a circle) whose stable and unstable manifolds

intersect transversely in a 1-torus, r, see Figure 3.4.6.

Transverse Homoclinic Torus,T

Figure 3.4.6. A Transverse Homoclinic Torus, Cut Away Half View.

We call r a transverse homoclinic torus. By invariance of W8(ro) and Wu(ro)

the orbit of r must always lie in both W'(70) and Wu(ro). Hence, one transverse ho-

moclinic torus implies the existence of a countable infinity of transverse homoclinic

tori. Thus, iterating Figure 3.4.6 gives Figure 3.4.7.

Using arguments similar to those given in the previous case for a hyperbolic

fixed point, we can find a region D which is mapped over itself by some iterate of

f in a horseshoe-like shape as shown in Figure 3.4.8.

However, in this case we will get a circle's worth of horseshoes. The normal

hyperbolicity insures that the dynamics normal to the invariant torus dominate thedynamics on the torus so that the region D does not "kink up" in the direction ofthe invariant torus as it is being mapped back onto itself by some iterate. Thus, we

should be able to find horizontal slabs in D which are mapped over themselves invertical slabs with proper behavior of the boundaries and the necessary stretching


Figure 3.4.7. The Homoclinic Torus Tangle, Cut Away Half View.

Figure 3.4.8. The Region D and Its Iterates, Cut Away Half View.

and contraction rates. Then Theorem 2.4.3 would imply the existence of a chaotic

invariant set in D having the structure of a Cartesian product of a Cantor set with

a torus. We remark that the dynamics along the direction of the torus is unknown.

We now turn to the rigorous justification of these examples for arbitrary (but

finite) dimension.

We now state our assumptions precisely.

Let P M -- M be a C'r diffeomorphism (r > 2) of the Coo manifold M,where dim M = N, leaving the compact boundaryless submanifold V invariant(i.e., f (V) = V). We make the following "structural" assumptions:

1) V is diffeomorphic to the 1-dimensional torus, T1,

2) V is normally hyperbolic,

3) V has an I + n dimensional stable manifold, W s (V ), and an 1 + m dimensional

3.4. Orbits Homoclinic to Periodic Orbit Invariant Tori 321

unstable manifold, W u(V ), with 1 - rn _ n = N, and4) W5(V) and W u (V) intersect transversely in an l dimensional torus r, i.e.,

dim(TpW3(V) "T"ll`'`(V)) =l V pET,

where TpW's,u(V) denotes the tangent space of Ws,u(V) at p c T. We re-mark that T is an 1-dimensional torus which is both forwards and backwards

asymptotic to V under the action of f ; we refer to T as a transverse hornoclinic

torus.

Before stating our main theorem, we need to make precise the notion of Vbeing normally hyperbolic. Roughly, it means that the directions normal to V are

expanded or contracted more sharply than directions tangent to V under the action

of f. We denote the derivative of f at p E M by Dp f , and we assume that Mis equipped with a Riemannian metric, I I. Let TV M be the tangent bundle of M

over V with the D f invariant splitting (with respect to I 1)

TV M=NS®.Nu®TV

with NS ® TV tangent to WS(V) at V and Nu E) TV tangent to Wu(V) at V.With respect to this splitting, the derivative of f at p E .1 can be written as

Dpf=E,fe)JPf(D Vpf

where

V

N,fDfNPf Dfljtp ,

Vpf = DfIT9V

We assume the V is normally hyperbolic in the following sense:

l0<A<1 such that for vEN, , uEN7('VPsP f)'f'vI < au

(Np f)'- < an lul

,NP f Vp f Np f VpEV, n>0.

(3.4.1)

We now state our main result.


Theorem 3.4.1. Let f satisfy assumptions 1) through 4) given above. Then, ina neighborhood of r, f 1t has an invariant Cantor set of tori, A, for some n > 1.Moreover, there exists a homeomorphism, 0, taking tori in A to bi-infinite sequences

of N symbols such that the following diagram commutes

A

where E denotes the space of bi-infinite sequences of N symbols and or is the shift

mapping on this space.

Before beginning the proof of Theorem 3.4.1 we need to get some preliminary

considerations out of the way.We assume that in a tubular neighborhood B of V, there exists a local coordi-

nate system in which f takes the following form:

x " A(0)x + 91(x, y, 0)

f: y'-'B(0)y+92(x,y,0) (x,y,O) E ]R'i x Km x Tl , (3.4.2)

0 - 93(0)

where g1 =912 =91y=910 =g2 =92x=92y=929 =0 at x=y=0, seeSamoilenko [1972] or Sell [1979]. (Note: for notational compactness we will denote

partial derivatives such as D291 by glx, etc.)Furthermore, the normal hyperbolicity hypotheses given in (3.4.1) are sufficient

for the existence of the splitting

Tp M = NP ®NP ®TpV , p E V

for all points p E V. Thus, WS(V) and W u (V) can be represented in B as graphs

of the functions y = GS(x,0) and x = Gu(y,0), respectively, with G'(0,0) =G2 (0, 0) = G0 "(0,O) = Gu (0, 0) = GY u(0,0) = GB (0, 0) = 0 in B. (Note: thenormal hyperbolicity assumptions 3.4.1 also guarantee that Gu,s are at least C2;this follows from the Cr-section theorem of Hirsch, Pugh and Shub [1977].)

Thus, if we construct the map

0:B-iMs ®N"®TV

(x, y, 0) '-' (x - Gu(y, 0), y - Gs (x, 0), 0)(3.4.3)

3.4. Orbits Homoclinic to Periodic Orbit= and Invariant Tori 323

where q(0,0,0) = (0, 0, 0) and Do (0. 0, 0) = identity, and use (3.4.3) to define

a new coordinate system, we see that locally W8(V) is given by .Ns ® TV andW u (V) is given by M u ® TV (i.e., (3.4.3) "straightens out" W3(V) and W u (V )

in B). Hereafter, we shall assume that we are in this coordinate system and that

910,y,0) = 92(x,0,0) = 0.In this case we can represent the tubular neighborhood B of V as follows

B=BsxBu

whereBs={(x,y,0)EIR°X]R"`xT1I IzI<63i y=0}Bu={(x,y,0)E]R.nx]R"nxT, x=O, IyI<bu}

for some 6s, 6u > 0.

Figure 3.4.9. Geometry of V and V n, Cut Away View.

(3.4.4)

Next we give a preliminary result which describes the dynamics of f near V.

This result is a generalization of the A-lemma (see Palis and deMelo [1982]) fornormally hyperbolic invariant tori. Let V be a µv-vertical slice intersecting Bstransversely in an 1-dimensional torus T. Let V' denote the connected component


of f n (V) n B to which f°(r) belongs, see Figure 3.4.9. Then we have the following

lemma.

Lemma 3.4.2 (Toral A-lemma, Wiggins [1986a]). Let E > 0 be given. Then,for B sufficiently small, there exists a positive integer no such that for n > no V n

is C1 c-close to W''L(V).

PROOF: The proof is very similar to the usual A-lemma for hyperbolic periodicpoints (see Palis and deMelo [19821). The only difference is that we must takeaccount of the "0 dynamics." However, by normal hyperbolicity, we will see that

the 0 dynamics is dominated by the x-y dynamics. The proof proceeds in severalsteps.

1) Estimate the size of various partial derivatives in B.2) Estimate the rate of growth of vectors tangent to V at r as they approach V

under iteration by f along W 3 (V) .

3) Extend the result of Step 2 to a neighborhood of r via continuity.

4) Estimate the rate of growth of vectors tangent to V as they move away fromV under iteration by f along W'(V).

We begin with Step 1.

Step 1. The expression for f in a neighborhood of V is given by

x -> A(9)x+91(x,y,0)

f: y'-' B (0)y+92(x,y,8) (x,y,0) E R' x Rm x T1, (3.4.5)

9F-->93(8)

with

Df =A(0) + g1x

92x

91y (A(0)x) 0 + g10

930l 0 0

B(O)+92y (B(0)y)0 + 920 (3.4.6)

and by the normal hyperbolicity assumptions given in (3.4.1) for bs, 6u sufficiently

small we haveJA(9)II <A <1,

11

B-1(0)I <A<1,(3.4.7)

JJA(0)Ij < 1193011 < )IB(0)I[ V O E T1 .

Let vo be a unit vector in the tangent bundle of V over r denoted TTV. With respect

to the splitting over V, vo can be written as (vo, v0, vg) where vx E Es, v0 E JJw,

3.4. Orbits Homoclinic to Periodic Orbits ar_d Invariant Tori 325

and v0 E TV. Now, since g1 = 91x = 91y = 919 = 92 = 92z = 92y = 920 = 0 at(0, 0, 0) and g1(0, y, 9) = 92 (x, 0, 0) = 0 in B, by continuity we can choose bs, bu

small and 0 < k < 1 such that

A-k<1,

with

and

k > ax{11g1x

b -kL> 1,1193911 I <a<1, VOETI,IIBII 1 kL

kL< (b-1)24

L= max 11+va,v0ET,V

ve0

vy0 l

We note that v0 is nonzero, since V is transverse to Ws(V) at r and, for notation,we let Dfn(vp,vp)v0) = (vn,vn'vn)Step 2. Now we can consider r E B, and we want to study V and its tangent vectors

at r as they approach V under iteration. Note that 92, = 0 since g2(z,0,8) = 0 in

B. First we look at the ratio By (3.4.6), (3.4.7), and (3.4.8) we have0

V

vy1

V2

vy2

< 939v0 < 193911 I

Bvol-k(v0 (1+ v0°I) B1IBL1

8939V < 1193911 1

Bvi j - k lvl l (1 + IIBIIT7B

1

Bvyn-1 -k

930V0n-1

vyn-1 (1+

v08

vy0

192yII,II(Ax)e+gigll,II(By)9+920111 -

v0e

vy0

V

v1y

< a

< a2

1193811 1

IBIIT7B

(3.4.8)

v09

v0g

y9n-1

vyn-1

< an ve0YV0

(3.4.9)


(Note: this result is not surprising: it just tells us that vectors in WU(V) grow muchB

faster than vectors tangent to V, with lim I - I = 0.)n-3oo vn

Next we look at the ratio Xo - Y and determine how it behaves under70

iteration by f. From (3.4.6) we have

--X1V1

vy1

(A + g1x)v0 + g1yvp + ((Ax)e + 91e)v0

(B + 92y)vo + ((By)e + 926)vo

The numerator is bounded above by

(A + k)

v2z

vy2

and the denominator is bounded below by

So we get

--X1 V1vy

1

(A+k) Iv'I +k(Ivol +

(1 -k) Iv01 -k vg

From (3.4.6) and (3.4.8) it is easy to see that

X2

and, using the estimate of

b=1/A-kL)

vB

vy1

v0I + k(IvOI + No

vgk vg

a - kL

X1+k 11+ IvJI1 /

1 -k 1+ vJIVYI

(3.4.10)

(3.4.11)

(3.4.12)

(3.4.13)

(3.4.14)

given in (3.4.9), we get (recall from (3.4.8) that

X2<_X1+kL<XO+kL+kL

a kL b2 b2 b

Continuing on in this manner, we find that

(3.4.15)

vy n

Xn =n < X + kL V' 1 < X0 kL+

(3.4.16)vn bn bt bn b - 1

i=1

)

voe I) , Xo + kL

3.4. Orbits Homoclinic to Periodic Orbits I. v -_-isrt Tori 327

and, since X0/b' -s 0 as n j oo and kL_ < `z41 , there exists an integer n such

that for any n > n we haveb - 1

Xn <4

(3.4.17)

a

Now, originally v0 could have been so that I , vo were as large asvo vo

possible in TV, so there exists n such that for all n > n, the nonzero vectors ofTf,,,(T)(fn(V)) satisfy

vxn

vny

b-14

V'n

vyn< an

v09

vy0

(3.4.18)

Step 3. By continuity of the tangent spaces of Tfn(T)(fn(V)), we can find a /tv-vertical slice V C f n(V) with f n(T) C V such that the slopes of any vectorstangent to V satisfy

vxn

vyn

b - 1

2< anven

vyn

ve0

vy0

, 0 < a < a < I . (3.4.19)

Step 4. Let v = (vx,vy,v9) E Tfn(T)V. We want to estimate the rate of growthof vectors in T fn(T)V. First we note that, if necessary, choosing bs, bu smaller we

can assume there exists k1 > 0 such that

0 < k1 L < min(c, kL) , (3.4.20)

and, since 91(0, y, 0) = 0 in B,

max{B

Note that by (3.4.9) we have

19 y (Ax)g+g1911}<k1.

rr V9max 11+

vy

We have

(3.4.21)

(3.4.22)

(A + 91x)vx + 91yvy + ((Ax)9 + g19)v9

Df (p) v = j 92xvx + (B + 92y)vy + ((By)9 + 929)v9 (3.4.23)

938 v9

3. Homoclinic and Heteroclinic Motions

xvn+1

vnY+1 (3.4.24)

(A+ 91x)vx + 91yvy + ((Ax)g + g19)vg

92zvx + (B + 92y)vY + ((By)g + 928)v01

where the numerator is bounded above by

(A + k) lvx1 +k1 Iv + k1 vg

and the denominator is bounded below by

(1 -k) IvYI - kIv'I - klvoI .

Xn+1 =vn+1

+11141

(3.4.25)

(3.4.26)

< (A+ k) vxl+k1Ivyl+kjIv8 < (A+k)Xn+k1L(X1 -k)1vY1 - kIvzI - kIvO 3-kL-kXn

< Xn + k1Lb - kXn

<Xn+kiL

b - k( )

G Xn+kiL = Xn+kiLb- 2(b-1) (b+1)

(3.4.27)

Let b1 =2

(b + 1) > 1. Carrying out similar calculations we find that

in! kiLXn+n - b-

+b1 - 1

1

So there exists an n such that for n > n

1 1Xn-f-n < _ E + b11

(3.4.28)

(3.4.29)

Since v could have been such that Xn was as large as possible we see that for n > ft

any nonzero vector tangent to f n (V) n B satisfies vY < E (1 + lb 11) . Thus,

given c > 0, there exists no such that for n > no, all nonzero vectors tangent tof n (V) n B satisfy

v <E.vnx

y

3.4. Orbits Hoinoclinic to Periodic Dr','- ...,4 Invariant Tori 329

Next we want to show that f' (1" _ B is stretched in the direction of W' (V).

We do this by examining ratios of tangent vectors perpendicular to V under iteration

by f.

vxn+1

N Ivxn

From (3.4.27) we see that

Now, since X,+1 and Xn

2

2+'

yn

1

y 2vn

V'+1Ivnl

z yn+1

Ivnl

1+X2n+111Xn

> - - kL - kXn.

(3.4.30)

(3.4.31)

are arbitrarily small, we see that the norms of iterates of

tangent vectors normal to V are growing at a rate approaching a - kL > 1. Thus,f n (V) n B is expanding along W'u (V) . This, together with the fact that vectorstangent to f n'(V) n B satisfy IvZ/vyl < e, shows that there exists an no such thatfor n > no, f n(V) n B is C1 c-close to W"(V).

We are now ready to give the proof of Theorem 3.4.1.

Proof of Theorem 3.4.1.

Now WS(V) and W'(V) intersect transversely at r; hence, there exist integers P1, P2

such that f P1(r) , f -P2 (r) E B, with W5(V) and W'(V) intersecting transverselyat fP1(r) and f-P2(r). We denote fPl(r) and f-P2(r) by (x1,0,0) and (0,y2,0),respectively. Consider the following sets:

Ui = { (x, y, 0)

U2 = { (x, y, 0)

X - x1I < Ei, lyl < E1, 0 E TI }

Ix1 C E2, IY - Y21 C E2, 0 E TI }(3.4.32)

for some c1, c2 > 0. Now the strategy is to show that there exists some set U1 C U1

which is mapped onto itself under some iterate of f and on which conditions Aland A3 of Section 2.4 hold.

By Lemma 3.4.2 we know there exists a positive integer no such that, for every

n>n0, fn(Ul)nU2#o.Let U2 be the connected component of f1(U1)nU2 which contains f'z(x1,0, 0).

Then f (U2) - U1 is a subset of U1 containing (x1,0,0). We choose a foliation

of U1 (i.e., an n-parameter family) of µv-vertical slices Va, a E I where I is some

n dimensional index set, such that the Va are parallel to the tangent bundle over


(x1, 0, 0), 0 E T1, of the component of W I (V) n Ul containing (x1, 0, 0), 0 E T1.

Then, by Lemma 3.4.2, for U1 sufficiently small, fa E I, are C1 E-close toW v (V) for n > no.

Now fP1+P2 (0, y2, 0) _ (x1, 0, 0), so a neighborhood of (0, y2i B) is mapped

onto a neighborhood of (xl, 0, 0) and, by Taylor's theorem, f P1 +P2 can be approxi-

mated C1 c-close by a rigid motion (i.e., translation plus rotation) whose derivative

at (0, y2, B) we write asa12

a22

a32

a13

a23

a33

(3.4.33)

where det a22 0 and det (a22 a231 0 since W8(V) and W ` (V) intersect\a32 a33)

transversely at (0,y2,0). A direct consequence of this is that the f' (KO) C U2,a E I, are mapped to an n-parameter family of /,iv-vertical slices which are C1c-close to the component of W'(V) n U1 containing (xl,0,0), 0 E T1. See

Figure 3.4.10 for an illustration of the geometry.

Figure 3.4.10. Geometry Near the Homoclinic Tangle.

Now we want to show that Al and A3 of Section 2.4 hold for the map

F - fP1+P2 o f n: Ul --} Ul . (3.4.34)

3.4. Orbits Homoclinic to Periodic Orbits :-d Invariant Tori 331

Al. We can construct a countable set of µh-horizontal slabs Hi, i = 1,..., inUl whose vertical boundaries are constructed from the Va, a E 1, such that0 < µvµh < 1 by Lemma 3.4.2. Then, from the previous discussion, F(Hi) C Ulwith the vertical boundaries of F(Hi) Cl e-close to the vertical boundaries of theHi. Moreover, since we can insure that the y direction is expanded by an arbitrary

amount by choosing Ul sufficiently small, then it follows that F(Hi) intersects each

of the Hj, j = 1,..., properly; hence Al holds.

AS. We give the argument for the stable sectors. The argument for the unstablesectors is virtually identical. We must show three things (see Section 2.4c).

1) DF(SI) C Sv,2) 17f (PO) > µ N1Po l+

3) 1 < 17f (PO) / Xf(PO)

for all p0 E Y, 0 < µ < 1 - µvµh - µhAv - 7-th-

First some notation. We have(all a12 a13

DF = I a21 a22 a23 Dfn

a31 a32 a33

For Vp0 = (vP0,vP0,vP0) E SP0, PO E )l, let

Dfn(P0)vP0 = lvfn(PO),V f"(PO),V f"(PO)) .

Then

(3.4.35)

allvfn(Po)+

a12vfn(Po) + axl3vfn(p0)

DF(p0)vP0 = I

a21vf"(PO) + a22vf"(PO) + ax23v fn(Po) = vF(P0) . (3.4.36)

a31vfn(Po) + a32vfn(Po) + ax33vf n(PO)

Also note that /1h and ftv can be chosen arbitrarily small by Lemma 3.4.2. We now

begin the argument.

DF(S') C S. We choose the sectors SP0, p0 E N, to be centered along the Va,a E I, in Ul. Let vp0 be any vector in S. By Lemma 3.4.2, for n sufficientlylarge under Df n(p0), we have

zyfn(Po) < e (3.4.37)v f"(Po)


xv fn(PU)< (3.4.38)

vfn(P0) i

Now under DfP1+P2 the images of the tangent spaces of the f n(V0) are C1 e-close

to the tangent spaces of V0. Hence, DF(S') C Sv.

'If (PO) > u TlPo. From (3.4.36) we have

yVP0

a21vf'L(Po) + a22vfn(Po) + a23vfn(Po)I

(3.4.39)

By transversality a22 is invertible; hence, by Lemma 3.4.2 the denominator of(3.4.39) can be made arbitrarily large by taking n sufficiently large.

1 < '7f(PO) I / I X f (Po) I .

This follows from normal hyperbolicity and estimates like

those obtained in Lemma 3.4.2.


1) For the case l = 0 (i.e., the invariant torus reduces to a fixed point), Theo-

rem 3.4.1 becomes the familiar Smale-Birkhoff homoclinic theorem, see Smale

[1963].

2) Theorem 3.4.1 was proved by Wiggins [1986a], but a similar earlier result was

obtained by Silnikov [1968b]. Meyer and Sell [1986] have studied orbits ho-

moclinic to almost periodic orbits using very different techniques and haveobtained a characterization of the dynamics similar to that given in Theo-rem 3.4.1.

Final Remarks.

1) The complicated dynamics associated with an orbit homoclinic to a hyperbolic

fixed point were first noticed by Poincare [1899] in his studies of the restricted

three body problem. In fact, the term "homoclinic" is due to him. Birkhoff[1927] continued Poincare's studies.

2) Most of the work done for orbits homoclinic to hyperbolic fixed points of maps

has been for the two-dimensional case. Gavrilov and Silnikov [1972], [1973] have

studied parametrized systems and have shown that infinite sequences of saddle-

node and period doubling bifurcations accumulate on the bifurcation values for


the homoclinic orbits. Newhouse 1974, [1979] independently discovered many

of the results of Gavrilov and Silnikov and went much further. In fact, thepresence of "Newhouse sinks" is the main difficulty in proving the existence

of a strange attractor in many systems, e.g., the forced Duffing oscillator (see

Guckenheimer and Holmes [1983, for a discussion). See also Robinson [1985].

3) Orbits homoclinic to non-hyperbolic fixed points and non-transverse homoclinic

orbits in dimensions > 3 have not received much study.

4) For applications of knot theory to the study of the bifurcations associatedwith the formation of horseshoes in two-dimensional parametrized systems, see

Holmes and Williams [1985].

5) Heteroclinic tangles formed via transverse heteroclinic orbits often yield the

same type of chaotic dynamics as described in Theorem 3.4.1. The same tech-

niques should apply, and we leave the details to the reader.

6) Orbits homoclinic to normally hyperbolic invariant tori have only recently been

studied. There are many open questions. For example, how do the dynamicsalong the tori "couple" to the chaotic dynamics normal to the tori? Is itpossible to "entrain" the Cantor set of tori? If so, then there may be "ordinary"

horseshoes within the Cantor set of tori.

CHAPTER 4Global Perturbation Methodsfor Detecting Chaotic Dynamics

In Chapter 3 we saw that orbits homoclinic or heteroclinic to hyperbolic fixedpoints, hyperbolic periodic orbits, or normally hyperbolic invariant tori could often

be mechanisms for producing deterministic chaos. In this chapter we will developa variety of perturbation techniques which will allow us to detect such homoclinic

and heteroclinic orbits.

The term "global perturbation" refers to the fact that we are concerned withperturbing a structure which exists throughout an extended region of the phasespace. The main idea behind the methods can be found in the work of Melnikov[19631. Melnikov considered an "unperturbed" system consisting of a planar ordi-

nary differential equation having a hyperbolic fixed point connected to itself by a

homoclinic orbit. As we saw in Section 3.2b, there are no complicated dynamical

phenomena associated with such a system. He then perturbed this system with atime periodic perturbation. In this case the hyperbolic fixed point becomes a hyper-

bolic periodic orbit whose stable and unstable manifolds may intersect transversely,

yielding Smale horseshoes and their attendant chaotic dynamics (see Section 3.3).

Using a clever perturbation technique, he developed a computable formula for thedistance between the stable and unstable manifolds of the hyperbolic periodic or-

bit, thus allowing him to explicitly determine the presence of chaotic dynamics in

specific systems. The following year, Arnold [1964[ generalized Melnikov's method

to a specific example of a time periodic Hamiltonian perturbation of a two degree

of freedom completely integrable Hamiltonian system. The method enabled Arnold

to demonstrate the existence of a global type of instability for Hamiltonian systems

which has come to be known as Arnold Diffusion. Following these developments of

Melnikov and Arnold, the technique appears to have gone unused (at least in the


west) until being rediscovered and applied by Holmes [1979] in his studies of the

periodically forced Duffing oscillator. Since that time a variety of generalizations

of the method have been developed by various workers, and we will describe these

shortly.

This chapter is organized as follows. In Section 4.1 we will describe, in general

terms, the structure of the three different types of systems we are considering andhow they fit into a general theoretical framework. We will also comment on how

our methods are generalizations of previous work. In Section 4.2 we will discussa variety of examples which illustrate the theory, and in Section 4.3 we will make

some comments regarding generalizations of our methods as well as some additional

applications.

4.1. The Three Basic Systemsand Their Geometrical Structure

In this section we describe the structure of the three types of systems under con-sideration and put them in the context of previous work.

Our goal is to develop perturbation techniques which will allow us to detect the

presence of orbits which are homoclinic and heteroclinic to different types of invari-

ant sets. As with most perturbation theories, we will begin with an unperturbedsystem of which we have considerable knowledge of the global dynamics. In our

case, the unperturbed systems will be completely integrable Hamiltonian systems

or parametrized families of completely integrable Hamiltonian systems which have

a degenerate homoclinic or heteroclinic structure (more specifically, they will con-

tain manifolds of nontransverse orbits homoclinic or heteroclinic to parametrized

families of invariant tori). We will consider arbitrary perturbations of such systems

(i.e., the perturbed systems need not be Hamiltonian), and determine the natureof any invariant sets which might remain in the perturbed systems. We will thenuse our knowledge of the nontransverse homoclinic structure of the unperturbedsystem to develop a measurement of the distance between the stable and unstablemanifolds of certain invariant sets which are preserved in the perturbed system.Thus, we will be able to assert the existence (or nonexistence) of homoclinic and

heteroclinic orbits in the perturbed systems.The three types of systems which we will study have the forms:

336 4. Global Perturbation Methods for Detecting Chaotic Dynamics

System I. i = JDxH(x, I) + Egx(x, I, 0, µ; E)

System II.

I = EgI(x,I,0,p;E)0 = IZ (x, I) + Egg (x, I, 0, µ; E)

i = JDxH(x, I) + Egx(x, I,0,µ; E)

I = EgI(x) I,9,A;E)

9 = [I(x,I) + Egg(x,I,e,u;E)

(x, I, 0) E R2n x R'' x Tl

(x, 1, 0)ER2, xT'nxTl

System III. i = JDxH(x, I) + EJDx H(x, I, 0, µ; E)I = -ED0 H(x, I, 0,µ;E) (x, I, 0) E R2n x R'" x Tm9 = DIH(x, I) + EDIH(x, I, 0, µ; E)

where 0 < E « 1, µ E RP is a vector of parameters, and J is the 2n x 2n matrixgiven by

J=0 Id)

-Id 0)where "Id" denotes the n x n identity matrix and "0" denotes the n x n zeromatrix. More detailed information concerning the structure of Systems I, II, and III

will be given shortly; however, at this point we wish to make some general comments

concerning the differences between the systems.

1) The systems obtained by setting e = 0 will be referred to as the unperturbedsystems. Each of the three unperturbed systems has a very similar structure.However, the unperturbed System III is more special in that the entire vector

field is derived from a Hamiltonian HE(x, I, B, µ; E) = H(x, I) +EH(x, I, 0, µ; E).

This need not be the case for the unperturbed Systems I and II.

2) At first glance it appears that Systems II and III are special cases of System1. This is true; however, there are vast differences in the underlying geometryof the vector fields, and what we prove in each context necessitates a separate

discussion for each case.

3) In all three systems, the main difference in describing their general structuresinvolves the behavior of the I component of the vector field. Specifically, we

will need to have some type of control over the I variables for the perturbedsystem.

In System I we will require the perturbed vector field to have a dissipativenature that results in the existence of a stationary point in the I component of


the vector field in some averaged sense. This will result in certain nonresonance

requirements among the frequencies C.

System II is periodic in each component of the I variable.

In System III the perturbations are Hamiltonian. For this case control over the

I variables is obtained using KAM type arguments. As for System I, this will

result in certain nonresonance conditions among the frequencies Il.

Before proceeding with a general discussion of Systems I, II, and III we want to

comment on previous generalizations of Melnikov's idea and how they fit into our

general framework.

Melnikov's [1963] original work is a special case of System I. Setting n = 1,

m = 0, and l = 1 with B = w =constant gives

JDxH (x) + egx (x, 0; p; e)

B=w(x,0)ER2xT'.

This equation has the form of a periodically perturbed oscillator and is the type ofequation originally studied by Melnikov (note: Melnikov's work was actually more

general in that he did not require the unperturbed system to be Hamiltonian; seeMelnikov [1963] and Salam [1987]). Melnikov's results in a more abstract settingwere later rediscovered by Chow, Hale, and Mallet-Paret [1980].

Holmes and Marsden [1982a] studied homoclinic orbits in dissipative and Ham-

iltonian perturbations of weakly coupled oscillators. For dissipative perturbations

their work is a special case of System I with n = 1, m = l = 1, 172 (x, I) = DIH(x, I)

and with the resulting equations having the form

= JDxH(x, I) + egx(x, I, B, u;

I = eg1(x, I, 0, z; (x, 1, 0) E R2 x R1 x T1

B = DIH(x, I) + ego (x, I, 0, u; e)

For Hamiltonian perturbations their work is a special case of System III with n = 1,

m = 1, 1 = 1, and with the equations having the form

JDxH(x, I) + eJDxH(x, I, B, µ; e)

I = -cDgH(x, I, o, µ; e)

9 = DIH(x, I) + eDIH(x, I, 0, µ;

(x,I,0)ER2xR1xT1.


Lerman and Umanski [1984] studied homoclinic orbits in strongly coupled oscil-

lators subjected to both dissipative and Hamiltonian perturbations. For dissipative

perturbations their work is a special case of System I with n = 2, m = 0, 1 = 0 and

with the equations having the form

i=JDxH(x)+egx(x,p;e), xER4.

For Hamiltonian perturbations their work is a special case of System III with n = 2,

m = 0, 1 = 0, and with the equations having the form

i = JDxH(x) + eJDxH(x, µ; e) , x E R4.

Holmes and Marsden [1982b], [1983] gave sufficient conditions for the existence

of Arnold diffusion in a general class of systems. The systems they considered are a

special case of System III with n = 1, m arbitrary, and with the equations havingthe form

i = JDxH(x, I) + eJDzH(x, I, B, µ; e)

I = -eD9H(x, I, 0, µ; e)

B = DIH(I) + eDIH(x, I, 0, Ec; e)

(x, I, 0) E R2 x R"6 x Tm .

In the context of a problem concerning passage through resonance, Robinson

[1983] studied homoclinic motions in a class of equations which are a special case

of System I for n = 1, m arbitrary, l = 0, and with the equations having the form

i = JDxH(x, I) + egx (x, I, /.z; e)(x, I) E R2 X Rm

I =eg1(x,I,µ;e)Wiggins and Holmes [1987] later studied such systems with n = 1, m = 1,

l = 1 with B = w=constant and with the equations having the form

i = JDxH(x,I) +egx(x,I,0,11;e)

I = egI(x) I, 0, FL; E)

B=w

(x, I, 0) E R2 x R1 x T1 .

The general theory for System I was first given by Wiggins [1986b].


System II is a generalization of the work of Wiggins [1988] concerning homo-

clinic orbits in systems forced at low frequency and large amplitude.

Homoclinic orbits in quasiperiodically forced oscillators were studied by Wig-

gins [1986], [1987]. His work is a special case of System I with n = 1, m = 0, 1arbitrary with j = w = constant and with the equations having the form

= JDxH(x) + egx (x, B, µ; E)(x B)ELt2xTl.,

8=wTechniques for studying homoclinic orbits in almost periodically forced oscilla-

tors have been developed by Meyer and Sell [1986] and Scheurle [1985].

Our methods rely heavily on the geometry of complete integrability associ-ated with the unperturbed systems. However, Melnikov type techniques have been

developed for time periodic perturbations of n dimensional systems possessing ahyperbolic fixed point connected to itself by a homoclinic orbit by Greundler [1985]

and Palmer [1984]. Their methods are less geometrical and more functional analytic

in nature and will not be covered in this book.We will comment on additional applications of these techniques as we go along

and mention further generalizations of the ideas at the end of this chapter. We next

turn to a discussion of the structure of Systems I, II, and III.

4.1a. System I

The first type of system which we will consider has the following form

z = JDxH(x,I) +Egx(x,I,9,µ;E)

I = Eg'(x,I,8,A;E) (4.1.1)

B = n (x, I) + Ego (x, I, B, ii; E)

with 0< E<1, (x, I, 8) E 1R2n x lR'"' x T1, and Et E RP is a vector of parameters.Additionally, we will assume the following.

11. Let V C R2n x ]R"L and W C ]Rp x lR be open sets; then the functions

JDxH: V " R2n

gx:V xTxW"R2n9V xTxW" urn

n:V"lR.1

g8:V xTIxWHIt1


are defined and "sufficiently differentiable" on their respective domains of defi-

nition. By the phrase "sufficiently differentiable" we will mean Cr with r > 6

(the reason for this will be explained later on) and, in many cases, r > 2 will be

sufficient. In any event, specifying the exact degree of differentiability is usually

just a technical nuisance since all of our examples will be analytic. Finally, let

us recall from Section 1.1i what it means for part of the phase space of (4.1.1)

to be the 1-dimensional torus. Regarding x, I, µ, and c as fixed, this means thatgx, g', and g6 are 2ir periodic in each component of their 1-dimensional 0 argu-

ments, e.g., gx (x, I, 01, ... , Oi, ... , 01, µ, e) = gx (x, I, 01,...) Bi +29r, ... , 01, A; E)

for any 1 < i < 1.12. H = H(x, I) is a scalar valued function which can be thought of as an m-

parameter family of Hamiltonians, and J is the 2n x 2n "symplectic" matrixdefined by

J=0

(Id)

-Id 0

where Id denotes the n x n identity matrix and 0 represents the n x n zeromatrix.

We will refer to (4.1.1) as the perturbed system.

i) The Geometric Structure of the Unperturbed Phase Space

The system obtained by setting e = 0 in (4.1.1) will be referred to as the unper-turbed system.

i = JDxH(x,I)

I = 0 (4.1.1)0

B = fl(x,I) .

Notice that since I = 0, the x component of the unperturbed system has the form of

an m-parameter family of Hamiltonian systems. Also, the x component of (4.1.1)0

is independent of 0 and, therefore, we can discuss the structure of the x component

of (4.1.1)0 independently of 0. We have the following two "structural assumptions"

on the x component of (4.1.1)0.

13. There exists an open set U C 1R.'n such that for each I E U the system

± = JDxH(x, I) (4.1.1)o,x


is a completely integrable Hamiltonian system. By "completely integrable" we

mean that there exist n scalar valued functions of (x, I), H = K1, K2,..., Kn,

(the Ki are called the "integrals") which satisfy the following two conditions:

1) The set of vectors DxK1, DZK2...., DxKn is pointwise linearly indepen-

dent VI E U at all points of R2n which are not fixed points of (4-1-1)0,x-

2) We define the Poisson bracket of K1, Kj (denoted {K1, Kj } ) as follows

{K1, Kj} = (JDxK2, DxKj) (4.1.2)

where denotes the usual Euclidean inner product, and we require thatthe pairwise Poisson brackets of the Ki vanish, i.e.,

(JDZKI, DZKj) = 0 Vi, j , I E U. (4.1.3)

Furthermore, we assume that the Ki are at least Cr+l, r > 6.

We remark that our definition of complete integrability does not quite agreewith the classical definition. The classical definition of integrability requires the

integrals to be analytic functions and might also relax our requirement of inde-

pendence of the integrals. For our purposes a finite degree of differentiabilityis sufficient.

(Note: for a more complete discussion of complete integrability (which we will

not need for our purposes), we refer the reader to Abraham and Marsden [1978]

or Arnold 11978].)

We want to emphasize that a background in Hamiltonian systems is not a prerequi-

site for the following material; rather, the geometrical consequences of completely

integrable Hamiltonian systems will be important, and we will comment on thoseshortly.

14. For every I E U, (4.1.1)o,x possesses a hyperbolic fixed point which varies

smoothly with I and has an n dimensional homoclinic manifold connecting the

fixed point to itself. We will assume that trajectories along the homoclinicorbit can be represented in the form xI (t, a) where t E R1, a E R.n-1. Thereason we assume that the homoclinic manifold is n dimensional is related to

the independence of the integrals and will be discussed shortly.


Let us now make the following remarks concerning the geometrical consequences of

13 and 14.

Consequences of 13

Let us consider the system (4.1.1)0 x for fixed I = Io E U as a vector field onR2rz Let x0 be a hyperbolic fixed point of (4.1.1)0,2 and denote the n dimensional

stable and unstable manifolds of z0 by WIo (xo) and W' (x0), respectively (note:the dimensionality of the stable and unstable manifolds of x0 is discussed more fully

under the consequences of 14, below. At this point, we ask that the reader acceptthe above statement regarding the dimensions). We have the following preliminary

lemma.

Lemma 4.1.1. Suppose Kl (x0, I0) = cl,... , Kn(x0, I0) = cn, then Kl (x, I0) _cl,...,Kn(x,10) = cn for all x E WIo(x0) UWIo(x0).

PROOF: This is an immediate consequence of the continuity of the K1.

For our purposes, the important geometrical consequence associated with 13 is

contained in the following two propositions (note: TzWIou(zo) denotes the tangent

space of Ws u(x0) at z).

Proposition 4.1.2. For any x E WIo (xo) (resp. WIo (x0)) TxWIo (x0) (resp.TxWIo (x0)) = span {JDxKl (x, 10),..., JDxKn(x, Io)}. Moreover,

Nx = span {DxKl(x,I0),...,D2Kn(x,I0)}

is orthogonal to TTW jo (x) (resp. TxWIo (x) ) and R2' = TxWIo (xo) + Nx (resp.TxWIo (xo) + N.).

PROOF: From Lemma 4.1.1, for any x E WIo (xo) U W u (x0) we have10

Kl (x, I0) - c1 = 0

(4.1.4)

Kn(x,Io)-en=0.

Henceforth, for definiteness, we will give the argument for WIo(xo); however, the

same argument applies to WIo (x0). Let x E W130(x0) and let ,3(t) be a differentiable

curve in W jo (xo) satisfying 8(0) = x for t contained in some open interval about


the origin. Then ,3(t) satisfies (4.1.4), i.e.,

K1(0(t),Io)-c1=0

(4.1.5)

Kn(Q(t), ID) - cn = 0 .

Differentiating (4.1.5) with respect to t gives

(D.K1 (-,10), Q(0)) = 0

(4.1.6)

(D. K. (x,Io ), P (0)) = 0 .

Geometrically, 4(0) is a vector tangent to WIo(xo) at x, or, in other words, /j(0) E

TxW jo (xo), and analytically (3(0) can be viewed as a solution of (4.1.6). So,

since 0(t) is an arbitrary curve in Wjo(xo), any solution of (4.1.6) is an ele-ment of TxW18 (xo). By 13 the n linearly independent vectors JDxKI (x, I0)1 ... ,

JDx Kn (x, I0) each solve (4.1.6). Therefore, TxWI0 (xo) = span{JDzKl (x, I0), ... ,

JDxKn(x,Io)}.The fact that Nx = span {DxKj (x, Io), ... , DxKn(x, I0)} is orthogonal to

TxWI0 (x0) is an immediate consequence of (4.1.3). pRegarding the dimension of Wlo(xo) and Wlo(x0), it should be noted that the

fact that they are each n dimensional follows from (4.1.4), as does the fact that the

DxKi (x, I0), i = 1, ... , n are linearly independent.Proposition 4.1.2 will be extremely useful later on when we construct "homo-

clinic coordinates." The next proposition indicates that if a homoclinic orbit exists

it must necessarily be n dimensional.

Proposition 4.1.3. Suppose WIo(xo) and Wlo(xo) intersect. Then WI0(xo) andW' (x0) coincide along the n-dimensional components of W' 0 (x0) - {x0} and

10WIo (x0) - {x0} which contain WIo (x0) n WIo (x0) - {x0}. Hence, there exists an

n-dimensional manifold of orbits homoclinic to x0.

PROOF: Consider the map

K(x, 10) = (Kl(x,l0),...,Kn(x,I0)):1R2n -* 1Rn.

By Lemma 4.1.1 K(x, I0) = (cl, ... , cn) - c for all x E WIO (xo) U Wu (xo).

So K-1 (c) is an invariant set containing WI0 (xo) U WIo (x0). Now by 13 the


DyK1(x,Io), 1 < i < n, are linearly independent at each x E Wj(xo) UWIo (xo) - {xo}. Therefore, K is onto (i.e., DzK has maximal rank) for eachx E WIo (xo) U WIo (xo) - {x0}. Hence, by the implicit function theorem (seealso the "submersion theorem" in Guillemin and Pollack [1974]), K-1(c) has the

structure of an n dimensional manifold near each x E W',,, (zp) U WI,(, (xo) - {x0}.

So, if WIo (x0) and WIo (x0) intersect, then they must coincide along the n di-mensional components of WIo (xo) - {x0} and WIo (x0) - {x0}, which containWIo (xo) n WIo (xo) - {xo}. El

This proposition is not true if the integrals are not independent, as can be seen

by the following example. Consider the system

x3=-x33x4 = x2 - x2

(x1,x2,x3,x4)ElRXIItx]RxIt.

This system is just the Cartesian product of two integrable systems; the x1 - x3components represent a linear system with a saddle point at the origin, and thez2 - x4 components are just the unforced, undamped Duffing equation. The

Hamiltonian is given by

2 2 4

H(xl, x2, x3, x4) = 21x3 +2

- 2 + 4

with an additional integral given by either

or

K2(x1,x2,x3,x4) = x1x3

2 2 4

K2(x1)z2,x3,x4) =

2- 2 +2

It is an easy calculation to verify (4.1.3) for H and either choice of K2.

Now the system has a hyperbolic fixed point at (x1, x2, x3, z4) = (0, 0, 0, 0)

having two dimensional stable and unstable manifolds. These manifolds intersect

only along a one dimensional homoclinic orbit given by

T = { (x1, x2, x3, x4) E 1R4Ix1 = x3 = 0,x2 = vsech t, x4 = - f sech t tanh t}.


A simple calculation shows that DxH and DZK2 (for either choice of K2) arelinearly dependent on F.

Consequences of 14

1. First we will show that the assumption that the fixed points of (4.1.1)0 x arehyperbolic VI E U leads to the fact that they can be represented as a Cr smoothfunction of the I variables (note: this will be useful for computations).

Recall that the condition for the existence of a fixed point of (4.1.1)o,x is that

for some x0 E 1R2n, I0 E U, we have

JD,H(xo,10) = 0 (4.1.7)

or, since J is nonsingular,

DxH(xo,Io) = 0 . (4.1.8)

Now the assumption that the fixed points are hyperbolic implies

det [JDyH(x0, IO)] # O' (4.1.9)

or, using the fact that J is nonsingular and the determinant of the product is theproduct of the determinants, (4.1.9) is equivalent to

det [DaH(x0, Io)] 0. (4.1.10)

By the implicit function theorem, (4.1.10) is a sufficient condition for there to

exist an m parameter family of solutions of (4.1.8), ry(I), for I in some neighborhood

of 10. By 14 each of these new solutions -y(I) of (4.1.8) is also a hyperbolic fixedpoint of (4.1.1)0,x so by the global implicit function theorem (see Chow and Hale

[1982]) the function -y(I) exists and is C' for each I E U. (Note: for fixed I, thesolution of (4.1.8) may actually have many disconnected components. In that caseour theory may be applied separately to each component.)

2. The symmetry properties of Hamiltonian systems require that, if A is an eigen-

value of JDzH(xo, I0), then so is -A (see Abraham and Marsden [1978] or Arnold[1978]). This implies that the stable and unstable manifolds of hyperbolic fixedpoints of Hamiltonian systems have equal dimensions.

3. We want to make some comments concerning the analytical expression for thetrajectory along the n dimensional homoclinic manifold which we have assumed


could be written in the form xI (t, a), I E U, t E R1, a E R'c-1. The questionwe want to address is why did we choose this particular form for the homoclinicorbit?

The meaning of the superscript I should be clear. It just indicates the paramet-

ric dependence of the homoclinic trajectory on I. We remark that by Theorem 1.1.4

xI (t, a) depends on I in a C' manner.The meaning of the variable a may be a bit more mysterious. However, it

happens that, for some systems possessing certain symmetries, the existence ofone homoclinic trajectory implies the existence of an entire surface or manifold of

homoclinic trajectories. In this case, varying the a in the homoclinic trajectoryxI (t, a) acts to take us from solution to solution of the homoclinic manifold. We

will give an explicit example of a system exhibiting this behavior in Section 4.2c.

4. We now want to discuss the distinction between the terms "trajectory" and"orbit" (see Section 1.1c). Consider the system (4.1.1)0,x for fixed I. Then, by14, this system has a homoclinic trajectory xI (t, a) connecting a hyperbolic fixedpoint to itself. The set of points through which this homoclinic trajectory passes as t

varies between +oo and -oo is called a homoclinic orbit. We will be interested in the

behavior of the perturbed system (4.1.1)E near this homoclinic orbit and, therefore,

we would like a parametrization of the homoclinic orbit so that we can describepoints along it. This can be accomplished by utilizing the fact that (4.1.1)0,x isautonomous. By Lemma 1.1.7, since (4.1.1)o,x is autonomous, then xI (t -to,to,.",) is

also a homoclinic trajectory of (4.1.1)o x for any to E It. Thus, to is defined to bethe time that it takes for the point xI (-to, a) on the homoclinic orbit to flow to the

point x1(0, a). Now, by uniqueness of solutions, there is only one solution passing

through any given point xI (-to, a). So, for fixed I, each point on the homoclinicorbit is uniquely specified by the coordinates (to, a) E R1 x ]R 1. Therefore,

xI (-to, a), (to, a) E R1 x R'n-1, is a parametrization of the homoclinic manifold.

This completes our discussion of the consequences of 13 and 14 for the system

(4.1.1)0,x. We now want to utilize these results to describe the phase space of the

unperturbed system (4.1.1) o in the full (x, I, O) phase space.

Consider the set of points .M in R2n x R' x T1 defined by

M = { (x, I, 0) E R2n x R'n x T1 I x = 'y(I) where -y(I) solves DxH ('y(I), I) = 0

subject to det [D'H(7(I), I)] 0 , V I E U , B E Tl } . (4.1.11)


Then we have the following proposition.

Proposition 4.1.4. M is a Cr m - I dimensional normally hyperbolic invariantmanifold of (4.1.1)0. Moreover, .M has Cr n+m+l dimensional stable and unstable

manifolds denoted W9(M) and Wu(M), respectively, which intersect in the n+m+l

dimensional homoclinic manifold

r= {a),I,00) ER2nxRmXTl I (to,«,I,00)ER1XRr-1xUxT1} .

PROOF: Using the expression for M in (4.1.11), the vector field (4.1.1)0 restricted

to M is given byx=0I=0 IEUB=S2(7(I),I)

with the flow on M given by

(4.1.12)

x(t) = -y(I) = constant

I(t) = I = constant ICU. (4.1.13)

0(t) = S2(y(I),I)t+00-

From (4.1.12) and (4.1.13) it should be clear that M is an invariant manifold (with

boundary) having the structure of an m parameter family of I dimensional tori.Moreover, the flow on the tori is quite simple. Trajectories either close up (i.e.,are periodic) or wind densely around the torus depending on whether the equation

ml[2l (y(I), I) + + m11]& (I(I), I) = 0 does or does not have solutions for inte-

gers ml,... , ml which are not all zero. The fact that W5(M) and W'(M) havedimension n + m + I is an immediate consequence of 14, and the fact that r isn + m + I dimensional follows from Proposition 4.1.3. We now want to discuss the

hyperbolicity properties of M and compute the generalized Lyapunov type numbers.

In order to simplify certain calculations let us make the coordinate change

u=x-ry(I). (4.1.14)

So in the (u, I, 0) coordinate system the invariant manifold M is given by

M={(u,1,0)ElR2rzx1R."LxT1Iu=0, IEU} . (4.1.15)

The vector field (4.1.1)0 linearized about M is given by

348 4. Global Perturbation Methods for Detecting Chaotic D ynamics

6v JD2 H(0, I) 0 0 6u

6I

(= 0 0 0 6I (4.1.16)

6i DuQ(0,I) DIn(0,I) 0 60

where 6u, 61, and 60 represent variations about orbits on M. From (4.1.16) we can

obtain the flow generated by (4.1.1)0 linearized about M. This is given by

Dgt(0, I, 0) = Dc5t(0, I)

exp[(JDuH(0,I))t] 0 0

= 0 idm 0

Dun(0, I) [JDuH(O, I)] -1 exp [(JDuH(o, I))t] DIn(0, I)t idl

(4.1.17)

where idm and idl denote the m x m and I x I dimensional identity matrices,

respectively.

Note that in the u, I, 0 coordinates, vectors tangent to M have zero u com-ponent. With this in mind notice the second and third columns of (4.1.17). Thesetwo columns span TpM for any p E M. Thus, the projection onto TpM is trivialand is given by

0 0 0

Dcbt(p)IIT = 0 idm 0 (4.1.18)

0 DIn(O,I)t idl

We want to decompose T]R.2n+m+l I M into three subbundles. First consider

the linearized equation (4.1.1)0 x regarding I as fixed. We have

bu = JD'H(O, I)bu , I E U (4.1.19)

where bu is the variation from u = 0. Now by 14, for each I E U, 1R2n splitsinto two n dimensional subspaces E'(I), E"(I), corresponding to the stable andunstable subspaces of (4.1.19). Let "0" denote the zero vector in 1Rm+l and consider

the following two disjoint unions

E3 ° U (E3(I),0)ICU

Eu - U (Eu(1), 0) .

ICU

(4.1.20)


Then we have

7,R2n+m+l I'M = TM ® Es ® Eu (4.1.21)

and if we defineNs =T,MeEs

(4.1.22)

Nu=TM ®Eu

then it should be clear that N3 is a positively invariant subbundle under (4.1.17)

and Nu is a negatively invariant subbundle under (4.1.17).

We now want to compute the generalized Lyapunov type numbers associated

with Nu in the context of Theorem 1.3.6. We let Eu and E3 play the roles of thesubbundles I and J in the geometrical set-up for the theorem. Using (4.1.18) and

the properties of (4.1.19) given in 14 we obtain for any p = (u, I, O) E M

A(P) =t

limes

-Y (P) = t limes

lIEuDOt(p)II-1`t = e-''u(I) < 1

lEODOt(P)II1't = eae(I) < 1 (4.1.23)

log DOt(P)lT IIa(P) = t ' oo log IIRE8Dcbt(P)jj -

0

where Au(I) is the smallest real part of any of the n eigenvalues of JDuH(0, I)which have positive real parts, and As(I) is the largest real part of any of the neigenvalues of JD2H(0, I) which have negative real parts. Recall that by 14 wehave

-au(I), as(I) < 0 V I E U (4.1.24)

and, therefore,

) (P) < 1 , 'Y(P) < 1 , °(P) = 0 VP E M . (4.1.25)

A similar calculation follows for NS under the time reversed vector field. Inthis case E'8 and Eu are interchanged and we obtain

A(P) = ea'(I) < 1

-Y(P) =e_A (I) < 1 (4.1.26)

a(p) = 0


graph -(I)

x

Figure 4.1.1. Unperturbed Phase Space of (4.1.1)0.

and, therefore,

A(p)<1, ry(p)<1, a(p)=0 dpEM. (4.1.27)

So M satisfies the asymptotic stability properties for normally hyperbolic in-

variant manifolds described in Section 1.3. However, the perturbation theorems do

not immediately apply since M is neither overflowing nor inflowing invariant. We

will deal with this technical nuisance when we discuss the geometry of the perturbed

phase space. See Figure 4.1.1 for an illustration of the unperturbed phase space of

(4.1.1)0.

ii) Homoclinic Coordinates

We now want to define a moving coordinate system along the homoclinic manifold

F of the unperturbed system which will be useful for determining the splitting of

the manifolds in the perturbed system.For each I E U, consider the following set of n linearly independent vectors in

R2nx]R'nxT1

{(DxH = DxK1,0), (DxK2,0),..., (DxK,1,O)} (4.1.28)

where "0" denotes the m + I dimensional zero vector. Also, we define a set of mlinearly independent constant unit vectors in ]R.2n x 1R' x Tl

0, ... , 1- 1 (4.1.29)


where the h represent unit, vectors in the I{ directions. As a convenient notation, for

a given (t0, a, I, B0) E RI X 1Rn-1 x lR'n x T', let p = (x,(-to, a), I, 60) denotethe corresponding point on T = W S (M) n WI(M) - M. For any point p c r, letTIP denote the m + n dimensional plane spanned by the vectors in (4.1.28) and(4.1.29) where the DxKi are evaluated at p. Thus, varying p serves to move the

plane 11p along the homoclinic orbit F. See Figure 4.1.2.

graph y(I)

Figure 4.1.2. Geometry of IIp.

We will be interested in how WS(M) and Wu(M) intersect Hp for each p E T.

In particular, we want to know the dimension of the intersection and whether ornot the intersection is transversal (see Section 1.4).

Now W'(M) is n + m + 1 dimensional. Therefore, for any point p E WS(M)the tangent space of W s (M) at p, denoted TpW s (M), is an n + m + 1 dimensional

linear vector space (see Section 1.3). By Proposition 4.1.2, an n dimensional vector

space complementary to TpW3(M) is given by

Np = span{(DxKl(p),o),..., (DxKn(p),0)} . (4.1.30)

So it should be evident that

TpWs(M) + Np = 1R,2n+m+l (4.1.31)

and, therefore, by Definition 1.4.1, WS(M) intersects 1Ip transversely for all p Er, since Np c IIp. Next we want to determine the dimension of the intersection.


Since the intersection is transverse, the formula for the dimension of the sum of two

vector spaces gives

2n+m+l=dim[TpWs(M)]+ dim Tplp-dimTp(W3(M)nllp). (4.1.32)

Now dim[TpWs(M)] = n+m+l and dimTpllp = n+m; therefore, dimTp(Ws(M)nllp) = in, which implies that WS(M) intersects rlp transversely in an m di-mensional surface, which we refer to as S. A similar argument can be appliedto conclude that Wu(M) intersects lip transversely in an m dimensional surfaceSP for each p E F. Moreover, since Ws(M) and Wu(M) coincide along I' _W8(M) n WI(M) - M, we have SP = S' for every p c F.

We remark that the reason it is important to determine the dimensions ofthese intersections and whether or not they are transversal lies in the fact thattransversal intersections persist under perturbations. This will be very importantwhen we discuss the splitting of the manifolds. We refer the reader to Figure 4.1.3for two possible scenarios for the intersection of W3(M) and WU(M) with IIp.

iii) The Geometric Structure of the Perturbed Phase SpaceWe now describe some conclusions of a general nature that we can make concerning

the structure of the phase space of the perturbed system. Recall that in the phasespace of the unperturbed system we are concerned with three basic structures, the

invariant manifold M and the stable and unstable manifolds of M, W8(M) andWu(M), respectively. Let us now comment on the structure of each of these setsand how we might expect the structure to change under perturbation. Afterwardwe will give results describing the perturbed structures.

1) The Invariant Manifold M. From Proposition 4.1.4, M is an m+l dimensionalnormally hyperbolic invariant manifold which has the structure of an in pa-rameter family of l dimensional tori. The flow on the tori is quite simple and

is given by 0(t) = 1l (- (I), I)t + 00.Now we would like to argue that most of this structure goes over for the per-

turbed system. However, there are two delicate points which deserve careful

consideration. The first is that M is an invariant manifold in a very precarious

sense due to the fact that it is a manifold with boundary, yet it is still invariant,

since j = 0, and therefore no orbits may cross the boundary. However, inthe perturbed system, I need not be zero, and therefore we must consider the


sP = sP = r n II,

I2

(a)

s;=sP=rnn,

I,

(D.Kz, 0)

o)

Figure 4.1.3. Geometry of the Intersection of W8(M), Wu(M), and r with Hp.

a) n = 1, m = 2. b) n = 2, m = 1.

boundary more closely. Fortunately, Fenichel's invariant manifold theory will

allow us to conclude that M persists for the perturbed system as a locally in-

variant manifold. The second point to be considered is the flow on M and how

it goes over to the perturbed problem. Now an m parameter family of tori with

rational or irrational flow is a very degenerate situation. Under perturbationwe would expect "most" of these tori to be destroyed, and even complicatedlimit sets could result. We will need to develop some technique for determining


the nature of the flow and the existence of possible limit sets on the perturbed

manifold.

2) W8(M) and W'(M). By definition these are the set of points which approachas t ±oo under the action of the unperturbed flow. However if, in

the perturbed system, orbits can cross the boundary of the manifold, then the

manner in which W8(M) and W"(M) go over to the perturbed problem is notso clear and requires careful consideration.

Persistence of M. The main result concerning the persistence of M is the following.

Proposition 4.1.5. There exists E0 > 0 such that for 0 < e < co the perturbedsystem (4.1.1) 6 possesses a Cr in + I dimensional normally hyperbolic locally in-variant manifold

.ME = {(z,I,0)ElR2nxlRm'>Tl I x =%y(I,B;e) =_y(I)+O(e), ICUCUCIlm,OET1}(4.1.33)

where U C U is a compact, connected m dimensional set. Moreover, ME has local

Cr stable and unstable manifolds, Wloc(M) and Woc(M), respectively.

PROOF: Recall the proof of Proposition 4.1.4. In the proof of that propositionwe showed the existence of a subbundle Nu D M such that N' was negativelyinvariant under the linearized unperturbed vector field with A(p) < 1, y(p) < 1,and a(p) = 0 for all p E M. We also showed the existence of a subbundleNs D M such that Ns was negatively invariant under the linearized time reversed

unperturbed vector field with A(p) < 1, 'y(p) < 1, and a(p) < 1 for all p E M.Moreover, NU n NS = TM. Now we would like to apply Theorem 1.3.6. However,

there is a slight problem due to the fact that M is neither overflowing nor inflowing

invariant, since the unperturbed vector field (4.1.1)0 is identically zero on aM. This

technical detail can be dealt with as follows.

Let U C U be a compact in dimensional set. Choose open sets U0 and Uisuch that U C U0 C Ui C U with U0 C Ui. Next choose a C°° "bump" function

v: ]Rm --> 1R (4.1.34)


such thatw(I)=0 forIEUW (I) = 1 for I E aU0

w(I) = -1 for I E 8U{

w(I)=0 forICRm - U(see Spivak [1979]). Now consider the modified unperturbed vector field

(4.1.35)

i = JDxH(x,I)

I = 6w(I)I (x,I,0) E R2n x Rm x Ti (4.1.36)

B=Sl(x,I)

for some 6 > 0.Let M, .MO, and Mi be subsets of M for which I is restricted to lie in U, UO,

and Ui, respectively. Then M C M0 C M{ C M and the following should beevident.

1) M0 is an overflowing invariant manifold under (4.1.36) satisfying the hypotheses

of Theorem 1.3.6.

2) Mi is an inflowing invariant manifold under (4.1.36) satisfying the hypotheses

of Theorem 1.3.6 under the time reversed vector field.

Now, since (4.1.36) and (4.1.1)0 are identical for I C U, it follows from Theorem

1.3.6 that the perturbed system (4.1.1)6 possesses a Cr locally invariant manifold

M6. Moreover, there exist locally invariant manifolds Wloc(Me) and Wloc('Me)which are Cr close to Wloc(M) and Wu (M), respectively. 0

We make several remarks concerning Proposition 4.1.5.

1. The Nature of WI (Me) and W`Oe(Me). M6 is a locally invariant manifold, i.e.,

points may leave M6 by crossing its boundary. We will refer to Wloc(ME) (resp.

Wu (Me)) as the local stable (resp. unstable) manifold of ME. However, thislocterminology deserves some clarification. Normally, one defines the stable (resp.

unstable) manifold of an invariant set as the set of points which are asymptotic

to points on the invariant set as t -* +oo (resp. -oo). This certainly need notbe the case for points in Wloc(.Me) (resp. Wloc(ME)) since, although points in

Wloc('Me) (resp. Wloc(ME)) approach ME in forward (resp. backward) time,

they need not actually limit on any points on ME as t -+ +oo (resp. -oo),


since all points on ME may leave ME in finite time. However, we will retain the

terminology of stable (resp. unstable) manifolds when referring to Wloc(.M )

(resp. Wl(,c(Me)). We refer the reader to Figure 4.1.4 for an illustration of the

geometry of the perturbed manifolds.

Wia (fv( )

MEW1 (ME)

Figure 4.1.4. Geometry of the Perturbed Manifolds with the Angular Variables

Suppressed for Clarity.

2. Differentiability of the Manifolds with Respect to Parameters. We want toshow that a slight modification of the arguments given in Propositions 4.1.4

and 4.1.5 gives that ME, WO 'and Wloc(ME) are also Cr functions of c

and A. Consider the vector field

i = JDxH(x, I)

I=0B= 12(x, I) (x,I,O,e,µ)ER2nxRmxT1 RxRP (4.1.37)

E0µ=0


where the (x, I, 0) components of (4.1.37) satisfy the same hypotheses as before.

Then the set

{ (x) I, 0, e, µ) E R2n x R' x Tl x R x IR I x = -I(I) where -y(I) solves

D2H(ry(I),I) = 0 subject to det [D2H(7(I),I), 0, `d I E U, 0 E Tl}

(4.1.38)

is an m + 1 + 1 + p dimensional, normally hyperbolic invariant manifold having

n + m + 1 + 1 + p dimensional stable and unstable manifolds denoted Ws (A) and

Wu(.M), respectively. Generalized Lyapunov type numbers can be computed

for M as in Proposition 4.1.4 and are identical to those given for M. Thisis because the addition of c and /t as new dependent variables only adds new

tangent directions to M with rates of growth that are only linear in time. Thus,

the argument given in Proposition 4.1.5 goes through identically the same inthis case with the result. that Mei Ws(ME), and W"(ME) are Cr functions of

x, I, 0, e, and IL.

Dynamics on ME. Recall from our previous comments that it is possible thereare no recurrent motions on Me, i.e., all orbits eventually leave Me by crossing itsboundary. In this case, the dynamical consequences of intersections of W3(ME) and

W'2(ME) are not clear, but they deserve further study. In Chapter 3 we saw thatthere could be dramatic dynamical consequences associated with orbits homoclinic

to fixed points, periodic orbits, or normally hyperbolic tori. With this in mind we

would like to determine the existence of such motions on ME. Using the expression

for ME given in Proposition 4.1.5, the perturbed vector field restricted to ME isgiven by

I=EgI(y(I),I,B,µ;0)+0(e2)(1,0) E U x T'. (4.1.39)

B = SZ (-Y (I), I) + 0 (e)

Let us consider the associated "averaged" equations

I = cG(I) (4.1.40)

27r 27r

where G(I) = f ... f gI (7(I), I, 6, u; o) d01 ... del.

We have the following result.


Proposition 4.1.6. Suppose there exists I = I E U such that1) the equation

mini ('Y(I),I) +...+min&Y(I),I)=0 (4.1.41)

has no solutions for any integers ml,..., ml which are not all zero;

2) the averaged equations (4.1.40) have a hyperbolic fixed point at I = I withthe linearized equation having m - j eigenvalues having positive real partsand j eigenvalues having negative real parts.

Then, for e sufficiently small, the vector field restricted to i.e., equations

(4.1.39), has a Cr, r > 3(s + 1), 1 dimensional normally hyperbolic invariant

torus rE(I) having a C3 j +l dimensional stable manifold and a C$ m - j +1 di-mensional unstable manifold. (Note: by Cr we mean differentiability with respectto I, 0, e, and µ.)

PROOF: See Arnold and Avez [1968] or Grebenikov and Ryabov 11983].

1) The question of smoothness of the stable and unstable manifolds in the averaged

equations needs some clarification. For each fixed e, Me will have Cr stable and

unstable manifolds. However, we will need to differentiate the manifolds with

respect to a at e = 0. In this case, the usual smoothness results do not gothrough. This problem was first studied by Schecter [1986], and the degree of

differentiability of the stable and unstable manifolds with respect to c at e = 0

based on the differentiability of the underlying vector field is due to him (note:

his smoothness result is probably not optimal). This is the reason that, whenaveraging is necessary (i.e., when we have I variables in the problem with one

or more frequencies (T1,1 > 1)), we must take (4.1.1) to be at least C6 inorder to get Cl stable and unstable manifolds for invariant sets on Me foundby averaging.

2) If l = 0 (i.e., there are no angular variables in the problem) then averaging isunnecessary and the flow on M is described by the equation

j= eg1(ry(I), I, A, 0) + 0 (e2) . (4.1.42)

3) If l = 1 there is only one frequency and the nonresonance condition (4.1.41)

is always satisfied.


4) The nonresonance requirement (4.1.41) implies that the flow on rE(I) is dense.

Denseness of the flow on TE(I) will be necessary in order to show that certain

improper integrals converge (see Lemma 4.1.27)-

5) Suppose the perturbation is Hamiltonian as in System III. Then the vector field

restricted to ME and averaged over the angular variables becomes

27r 27r

G(I) =(2 ) 1

f... f D9H(-y(I),I,O,h;O)d81 ...d01 = 0. (4.1.43)

0 0

Therefore, averaging gives no information in this case, and more sophisticated

methods are needed. This is the reason why we give a separate discussion for

System III.

Let us now view the l dimensional torus on ME found via Proposition 4.1.6 in

the context of the full 2n + m + I dimensional phase space.

Proposition 4.1.7. Suppose we have I = I C U such that Proposition 4.1.6is satisfied. Then rE(I) is a Cr, r > 3(s + 1), 1 dimensional normally hyperbolicinvariant torus contained in ME having a CS n+ j +l dimensional stable manifold,W s (TE(I )), and a Cs n + m - j + I dimensional unstable manifold, W'i(re(I )).Moreover, C W'3(ME) and WU(re(I)) C Wu(ME).

PROOF: This is an immediate consequence of Propositions 4.1.5 and 4.1.6 andTheorem 6 in Fenichel [1974].

Now our goal will be to determine whether or not Ws(re(I )) and W"(re(I ))intersect, so it is to this that we turn our attention.

iv) The Splitting of the Manifolds

Suppose we have found an I C U such that rE(I) C ME is an l dimensionalnormally hyperbolic invariant torus having an n+j+l dimensional stable manifold,

W s (TE (I)) , and an n + m - j + I dimensional unstable manifold, W u (rc (I)) . We

want to determine whether or not W' (r6 (1)) and W ' (TE (I)) intersect transversely.

If this is the case, then depending on 1, we can appeal to theorems from Chapter 3

and assert the existence of chaotic dynamics in the perturbed system (4.1.1)E.

Let us first recall the geometry of the unperturbed system (4.1.1)0.

Let (to, a, I, B0) E 1R1 x ]R.rt-1 x U x Tl be fixed and denote the corresponding

point on r = W3(M)nWu(M)-M as p - {x'(-t0,a),I,00}. Let HP be the m+n


dimensional plane as previously defined with W3(M) and WL(M) intersecting lip

transversely in the m dimensional surfaces SP and SP , respectively, for each point

p E I'. Note also that r intersects Hp transversely in an m dimensional surface with

r fl IIp = Sp = S' (see Figure 4.1.3).

Now we will consider the geometry of the perturbed system (4.1.1) along r.

Since W8(M) and WU(M) intersect Hp transversely for all p E r, for sufficiently

small W s (M ) and W u (M ) intersect IIp transversely for each p E F in the mdimensional sets fl IIp = Sp and fl Hp = S. However, in thiscase, the sets Sp and SP need not coincide. Also, it may be true thatand intersect Hp in a countable set of disconnected m dimensional sets(see Figure 4.1.5). In this case, our choice of Sp, (resp. corresponds to the

set of points which is "closest" to M in the sense of positive (resp. negative) timeof flow along W a (M ) (resp. W u (M )) . We will elaborate more on this technical

point when we discuss the derivation of the Melnikov vector.

D.,H

Figure 4.1.5. Sets Spup "Closest" to M (Note: the Figure

Is for the Case n=1, m=1=0).

4.1. The Three Basic Systems and Their GE.--.metrical Structure 361

Let us suppose we have used Proposition 4.1.6 and found an I = I E U C1R"'' such that Te(I) E Me is a normally hyperbolic invariant l-torus having

an n + j + I dimensional stable manifold, W S and an n + m - j + ldimensional unstable manifold, Wu(r( (I )). Henceforth we will regard I = I asfixed. Then, since WS(TE(I)) C WS(ME) and WI(re(I)) C WI(Me), we haveW S (rc(I )) n Sp E - Wp (re(I )) is a j dimensional set and W'(re(I )) n SpWp (rE (I)) is an m - j dimensional set. Let ps (x', IE ), and pE - (xE, IE) bepoints in WP and Wp which have the same I coordinate, i.e., IE _IE . It is always possible to choose two such points due to the fact that T,(1) isnormally hyperbolic, implying that the angle between Wloc (T, (1)) and Wlu

is bounded away from zero independently of e. Due to the importance of thisassertion, we summarize the details of the argument in the following lemma.

Lemma 4.1.8. For fixed p c r, consider Wp and Wp (rc(I )) as definedabove. Then, for e sufficiently small there exist two points pE - (xE, IE) EWp (rE(I )) and pE (xE, IE) E Wp (re(I )) such that IE = I,.

PROOF: Consider rE(I) restricted to Me. Since rE(I) is normally hyperbolic, the

angle between the stable and unstable subspaces of the system restricted to Me(equation (4.1.39)) and linearized about r6(I) is bounded away from zero indepen-

dently of e (note: the fact that the angle is independent of a follows from the factthat the e can be removed in the linearized system by a rescaling of time). Since the

stable and unstable manifolds of TE(I) restricted to Me are, for fixed e, C'-close to

the stable and unstable subspaces of the linearized system, the stable and unstable

manifolds of TE(I) intersect transversely at re(I). Next, let us consider the stableand unstable manifolds of rE(I) in the full 2n + m + I dimensional phase space.We can view W s as the stable manifold of re(I) restricted to Me which hasbeen carried into WS(ME) along trajectories in WS(ME) which are asymptotic to

re(ID, similarly for W"(re(I )).

Choose p E F (with I = I fixed) to be in a sufficiently small neighborhood ofMe such that Wloc(Me) and Wloc(Me) intersect Tip in the disjoint m dimensional

sets Sp c and respectively, see Figure 4.1.6. Now WP (T,(1)) is a j dimensional

set contained in Sic, and Wp(re(I)) is an in - j dimensional set contained inS. Let IIp denote the in dimensional subspace of Tip spanned by the vectorsIi, i = 1,...,m, and let Wp,m(rc(I)) and Wp m,(r,(I)) denote the projections of


Wp(Te(I)) and Wp (TE(I)) onto H'. Since (DzKi,O), i = 1,...,n, span an ndimensional space complementary to TpWp (TE (I)) for each p E Wp (re (I )) , and

to Tp-WP (TE(I )) for each p E Wp (TE(I )), then the projections Wp,m(TE(I )) and

WpUm(TE(I )) are j and m-j dimensional sets, respectively. Moreover, Wp,m(TE(I ))

and Wpum(re(I )) must intersect transversely in at least one point, since the stable

and unstable manifolds of TE(I) restricted to ME intersect transversely at TE(I).

Will. (ME) I Wia(ME)

S"P,E

Figure 4.1.6. Intersection of Wloc(Me) and Wloc(Me)

withllp, n=1, m=1, 1=0,j=1.

This argument only proves the lemma for p near ME. Next, choose any point

p E t outside of a neighborhood of Me. Then SAE and SP,E are finite time images

of Wloc(Me) and Wloc(Me),respectively, under the flow generated by (4.1.1)e for

any p E r. Then the result follows, since the angle between Wloc(re(I)) andWloc(re(I)) will remain bounded away from zero under integration by finite time

for a sufficiently small by simple Gronwall type estimates.


We refer the reader to Figure 4.1.7 for two cases of the geometry of Lemma4.1.8.

S"P'E

(D.H, 0)

(b)

W'(i(I))

I,

D.KZ, 0)(D,rH, 0)

Figure 4.1.7. Geometry of Lemma 4.1.7. a) n = 1, m = 2, j = 1.

b)m=1,n=2,j=0.

We remark that Lemma 4.1.8 need not be true in the case where the pertur-bation is Hamiltonian, since rE(I) would not be normally hyperbolic. This is oneof the reasons why Hamiltonian systems are treated separately as System III.

We are now at the point where we can define the distance between W s (rc(I ))


and W'(7-'(I)). At any point p c F, the distance between W s (re (Il) and W u (re (Il )

might naively be defined as

dI (P, EI - I PE - PEI = IxE - Xs E (4.1.44)

with pE and pE chosen as in Lemma 4.1.8. Although this scalar measurement ofthe distance between W8(rc(I )) and is correct, it fails to utilize theunderlying geometry which we have developed, since, although distance is a scalar,

the measurement of distance must be made with respect to a specific coordinate

system. Our coordinate system is the plane Ilp, and we will see that the components

of pE - pE along the coordinate directions defining Hp can be explicitly computed.

The resulting vector will provide a signed measure of distance between W'("(!))and Wt(re(I)) at the point p.

Before proceeding, let us make some additional comments concerning the ge-

ometry behind equation (4.1.44). To measure the distance between two surfaces at

different points along the surfaces, the idea is to move around on the surfaces (i.e., to

move in directions tangent to the surfaces) and measure the distances between them

along directions that are complementary to directions tangent to the surfaces ateach point. The arguments of the distance function (4.1.44) correspond to variables

representing movement tangent to the manifolds (specifically, the to, 00, and a vari-

ables). Now, for m-j appropriately chosen h vectors, denoted {Ii(i),... ,Ij(m_j)},an m - j + n dimensional space complementary to TpW u would be given by

the span of {I(i)'. .,'i(m-j+1)' (DZK1(p),0),..., (D5Kn(p),0)}. Similarly, forthe remaining j h vectors, denoted {Ii(m-j+1) 'Ii(m)}, a j + n dimensionalspace complementary to TpW3(rs(I)) would be given by the span of {Ii(m_j+1)+... , Ii(m), (DxKi (p), 0), ... , (DxKn (p), 0) }. Lemma 4.1.8 takes care of the neces-

sity to measure along the I directions, since it assures us we can fix the components

of distance equal to zero in these directions due to the normal hyperbolicity of r,(I1.

Therefore, in order to determine the distance between W s and W"(rc(I )),we need only measure along the directions (D5K1,0),..., (DxKn,0).

Now our goal is to develop a computable expression for equation (4.1.44),and we do this by employing Melnikov's original trick (Melnikov [1963]). We

define the signed component of the distance measurement along the directions(DxKl, 0), ... , (DxKn, 0) as follows:


di (p; E) = di (to, 00, a, AL; E)

(DxKi (x' (-t(,, a), I) , xE - zE)

DxKi (xT(-t,,, a), I) 11

(4.1.45)

where we have replaced the symbol p by (to, 00, a), since for fixed I = I, any point

on I' can be parametrized by (to, B0, a) E R1 X T l x Rfi-1, and we have alsoincluded the vector of parameters µ E RP denoting possible parameter dependence

of the perturbed vector field. It should be clear that, if for some (to, 00, a, a; e) we

have da (to, 6oi a, y; E) = 0 for i = 1,.. . , n, then W' (TE (I)) intersects W' (r6 (I ))

at this point. At this stage it is still not clear that (4.1.45) gives us any com-putational advantage. However, if we Taylor expand (4.1.45) about c = 0 weobtain

ad'(t0, 80, «, 1L; 0) + 0 (E2) , I = 1, ... , ndi (to, B0, «,1L; c) = d4 (t0, 00, a, IL; 0) + e

19C

(4.1.46)

where

and

d! (to, 00, a, µ; 0) = 0 , i = 1, ... , n since Sp = SP

OX, a

ad1 ) (DxKi (X1(-to,«),I) , a le=o - Oaxe 1E=o)aE DxKi

W(-to, a)I)

and we will shortly show that

w s

(DxKi (.'(-to, «), I) ,8xE - axE Mi (Bo, «; tL)ae E=0 aE E=o

00 t

f [(DxKi, 9x) + (DxKi, (DIJDxH) fg1)j (qo (t), FL; 0) dt,

-00

z = 1,...,n,

(4.1.47a)

or, equivalently,


00

Ma (00,a;p') = f [(DzKi,9y) + (DIKi,9I)] (g0(t),/z;0) dt- 00

-(DIKi (-y(I), fl f 91(gp(t),jL;o)dt)-oo

(4.1.476)

twhere q (t) = (x t (t, a), I, f 1(xI (s, a) ds + Bo) .

(Note: justification for the Taylor expansion of (4.1.45) follows from the fact

that the manifolds vary with c in a Cs s > 1 manner.)

In order to compute (4.1.47), we only need to know the unperturbed homoclinic

manifold and the perturbed vector field. In particular, we do not need to compute

expressions for solutions in W s (TE(I )) and W" (TE(I )). Since I I D,K{(xI (-to, a),

I) 1154 0 on 1', then Mi = 0 implies a = 0. So we see that M1 is essentially theleading order term in a Taylor series expansion for the distance between W8 (TE(I ))

and W'`L (TE(I )) along the direction (DzK1(z' (-to, a), Il, 0). In honor of V. K.Melnikov we refer to the vector

(oo, a; µ))M'(00, a; µ) = (M1 (oo, a; l'), ... 'M1 (4.1.48)

as the Melnikov vector. We remark that the variable to does not appear explicitly in

the argument for the Melnikov vector. When we discuss the derivation of (4.1.47),

we will show that it can be removed by a change of coordinates. This simply reflects

the fact that, when invariant manifolds intersect, they must intersect along trajec-

tories which are (at least) one dimensional. Therefore, to determine the distancebetween the manifolds at different points along the manifolds there is one direction

in which we need not move. In reality, this fact would allow us to remove any one

component of (to, Bo); however, removal of the to component is most convenient.

More discussion of this point will appear in the section concerning the derivation

of the Melnikov vector.

We are now at the point where we can state our main theorem.

Theorem 4.1.9. Suppose there exists a point (00, a, µ) = (Bo, a, p) E T1 x Rn-1 xRP with 1+n-l+p>n such that


1) MI (#0, a; A) = 0,2) DMI (Bo, a; p) is of rank n.

Then for c sufficiently small W s (rE(I )) and W"(rr(I )) intersect near (80, 6"p).

PROOF: By construction, and W"(re(I )) intersect if and only if

d'(to, B09 a, A; E) = EMl (Bo, a; h) ... M. I(00' ai I) + 0 (E2) = 0 (4.1.49)

( IID2H11 IIDxKnll )where we leave out the arguments of the D5K{ for the sake of a more compactnotation. Now consider the function

dI, a, W; E) - Ml (B0, a, Mn (B0, a;

(00te) + 0(f) . (4.1.50)

= ( IIDxHII IIDxKnll )Then we have

a, p; 0) = 0.j1(00, (4.1.51)

Now DMI (B0, a; p) is of rank n. Therefore, we can find n of the variables (00, a, µ),

which we denote by u, such that D"MI(Bo, a; p) is of rank n. Let v denote theremaining I - 1 + p variables. Then we have

det [Du(i'(Bo, a, p; 0) ] =IIDxHII ' 1 IIDxKnll

det [D"MI(Bo, a, A)] 0. (4.1.52)

So, by the implicit function theorem, we can find a C' function u = u(v, c) withuo = u(v0, 0) such that

(u(v,E),v,E) =0

for (v, f) near (v0, 0), and the result follows, since d' = Ed'.

(4.1.53)

0From Chapter 3 we saw that often it is important to determine whether or not

the intersection of W s (rc(I )) and W" (rE(I )) is transverse. For this we have thefollowing theorem.

Theorem 4.1.10. Suppose Theorem 4.1.9 holds at the point (00,a,µ) = (8o, a, p)E T t x IRn-l x 11 and that D(90,a) M'(B0, a, p) is of rank n. Then, for c sufficiently

small, W s (rE(I )) and W"(r6(I )) intersect transversely near (B0, a).

PROOF: Let p denote a point of intersection of Ws(re(I )) and W"(re(I )). ThenTpW'q(rE(I )) is n + j + t dimensional and TpW"(rc(I )) is n + m - j + 1 dimen-sional. By Definition 1.4.1, W'' (rE(I )) and W"(re(I )) intersect transversely at p


if TpW S (TE (I)) + TpW u (Te (I )) = 1R2rz+m+l. By the dimension formula for vector

spaces we have

dim(TpW8(TE(I)) +TpWu(TE(I))) = dim TTW3(T,(I)) + dim TpWU(TE(I))

dimTp(Ws(TE(I)) f1Wu(rc(I))). (4.1.54)

Thus, if WS(re(I)) intersects Wu(T6(I)) transversely at p, then WS(TE(I)) inter-sects W u (rs(I )) locally in an l dimensional set. Therefore, in order for W S (TE(I ))

and W u (TE(I )) to intersect transversely at p, it is necessary and sufficient forTpW S (TE (I)) and TpW u (TE (I)) to contain n + j and n + m - j dimensional inde-

pendent subspaces, respectively, which have no part contained in Tp (W S (TE(I )) fl

Now let us recall the geometry of the unperturbed system. The invariant torusT(I) has n + I dimensional stable and unstable manifolds and an m dimensionalcenter manifold. These manifolds coincide along an n+m+l dimensional homoclinic

manifold. Utilizing this information in the perturbed system, we need to show that

the perturbation has created new independent n + j and n + m - j dimensionalsubspaces in TpW s (re (I)) (/and TpW u (T,(!)), respectively, which are not contained

in Tp(W8(TE(I)) fl Wu(r5(T ))).

Recall that, in the unperturbed system, each point along r can be parametrized

by (to, a, I, 00) (note: where I is fixed in Systems I and III), so fore sufficientlysmall SP',, and Sp,E, P E r, may also be parametrized by (to, a, I, 00). Now let9E _(xE,IE,O0)=4e=(xIf,00)=p denote a point in Wz(re(I))nWu(re(I)).Consider the (2n + m + l) x (I + n - 1) matrices

AE _( )M-10- aa

The columns of these matrices represent vectors tangent to TpW3(TE(I )) and

TpW u (re (I)) along directions that were coincident in the unperturbed system, i.e.,

Ao = A. Consider the n x (2n + m + l) matrix

DXK1(p) 0

N=D2Kn(p) 0

(4.1.56)


Since the D2Ki are independent along the homoclinic orbit, N has rank n. Consider

the n x (1 + n - 1) matrixCE = N(AE - AE) . (4.1.57)

Taylor expanding CE about e = 0, using the definition of the Melnikov vector(4.1.47), and using the fact that MI (00, a, d) = 0, gives

CE = eD(eo a)M1(B0, a,,a) + 0 (E2) (4.1.58)

where (B0, a) are parameters along the unperturbed homoclinic orbit corresponding

to the point in W u (r6(I )) nW a(re(I )) given by Theorem 4.1.9. Now, by hypoth-

esis, D(B0,0,)MI (Bo, a, p) has rank n. So for a sufficiently small Cr. also has rank n.

Then, since N has rank n, we must have that AE - AE is of rank n. This indicates

that AE and AE each contain n linearly independent columns which correspond to

n independent vectors in TpW u (re (1)) and TpW 8 (,r,(!)). Moreover, these vectors

are not in Tp(W8(r6(I)) n Wu(r6(I))), since the rows of N span an n dimensionalsubspace complementary to Tp(Ws(r6(I1) nW"(re(I-))). The remaining j indepen-dent vectors in TpW S (rE (I)) and m - j independent vectors in TpW u (r6 (1)) which

are not in Tp (W 8 (r6 (Il) n W u (r (ID)) come from the breakup of them dimensional

center manifold of the unperturbed system, cf. Lemma 4.1.8.

We make the following remarks regarding Theorem 4.1.10.

1. Theorem 4.1.10 provides only a sufficient condition for transversality. This can

be seen from (4.1.57) and (4.1.58). The jacobian of the Melnikov vector is the

leading order term in the projection onto a particular complementary n dimen-

sional subspace of part of the difference of TpWB(re(I)) andThus, the rank depends on the particular projection, since AE - AE may beof rank n, but N(AE - AE) may have rank less than n.

2. Note that if l = 0 then transversality is impossible. This is due to the fact that,

in this case, the invariant set is a hyperbolic fixed point, and it is impossiblefor its stable and unstable manifolds to intersect transversely since they areconstrained by uniqueness of solutions to intersect along a one dimensionaltrajectory.


4.1b. System II

The second type of system we will consider has the following form:

i = JDZH(x, I) + egz(x, I, O, Eu; e)

i = Eg1 (x, i, 0, µ; e)

cz(x,i) + Ego(x,I)O,ft;E)

(4.1.59) E

with 0 < e << 1, (x, 1, 0) E R2n x T' x T 1, and µ E RP is a vector of parameters.

Additionally, we will assume:

111. Let V C R2n and W C RP X R be open sets; then the functions

JDzH:V x Tm H R2n

gx:VxTmxTI xWHR2n

91:V xTmxT1xW'-4n: V x Ty" F--* RI

go:V xT'mxT1xWHR1

are defined and CT, r > 2.

112. H = H(x,I) is a scalar valued function which can be thought of as an m pa-rameter family of Hamiltonians that is periodic of period 2ir in each component

of the I variable for x fixed. J is the 2n x 2n symplectic matrix defined by

J= (-0

Id 0

Id)

where Id denotes the n x n identity matrix and 0 denotes the n x n zeromatrix.

We will refer to (4.1.59)c as the perturbed system.

i) The Geometric Structure of the Unperturbed Phase Space

The system obtained by setting e = 0 in (4.1.59)e will be referred to as theunperturbed system.

i = JDZH(x, I)

I = 0 (4.1.59)0

8 = fl (x, I) .


We have the following two structural assumptions on the x--component of(4.1.59)0.

113. For each I C Tm the system

± = JDxH(x, I) (4.1.59)0,x

is a completely integrable Hamiltonian system, i.e., there exist n scalar valued

functions of (x, I), H = K1, . . . , Kn which satisfy the following two conditions.

1) The set of vectors DxK1iD2K2,...,DxKn is pointwise linearly inde-

pendent V I E T' at all points of ]R.2n which are not fixed points of(4.1.59)0 X.

2) (JDxKi, Kj) = 0 V i, j, I E Tm where ( , ) is the usual Euclideaninner product.

Furthermore, we assume that the Ki are C', r > 2.

The reader should compare 113 with 13 in our discussion of System I.

114. For every I C T' (4.1.59)o x possesses a hyperbolic fixed point which varies

smoothly with I and has an n dimensional homoclinic manifold connecting the

fixed point to itself. We will assume that trajectories along the homoclinicmanifold can be represented in the form xI (t, a) where t E ]R., a E ]R.n-1.

At this point, the reader should review the discussion of the geometrical conse-quences of 11-14 in our discussion of System I. Much of the same follows in this

case. In particular, consider the set of points M C 1R2n x T'ri x Tl defined by

M = { (x, I, B) C RIn x T'n x T l I x = -I(I) where -y(l) solves DxH(-y(I), I) = 0

subject to det [D'H(-1 (I), I)] 0, I E Tm, 0 E Tl (4.1.60)

then we have the following proposition.

Proposition 4.1.11. M is a Cr m+l dimensional normally hyperbolic invariantmanifold of (4.1.59)0. Moreover, M has Cr n + m + 1 dimensional stable andunstable manifolds denoted W'3(M) and Wu(M), respectively, which intersect inthe n + m + I dimensional homoclinic manifold

r = { (XI(-to, a),I,Bo) C IFbnxTmxTl I (to,a,I,00) E M1xMn-1xTrnxTll.


PROOF: The proof is identical to the proof of Proposition 4.1.3.

Let us make several comments regarding the structure of M.

1. M is a boundaryless manifold. This eliminates the technicalities which wereencountered in System I when proving that M persists under perturbation.

2. M has the structure of an m + I dimensional torus with the flow on the torusgiven by

I(t) = I = constant

0(t) = SZ(7(I),I)t+00(1,0) E T- x T I. (4.1.61)

See Figure 4.1.8 for an illustration of the geometry of the phase space of (4.1.59)0.

Figure 4.1.8. Unperturbed Phase Space of (4.1.59)0.


We define a moving system of homoclinic coordinates for System II in exactly the

same way as we did for System I.

We consider the following n + m linearly independent vectors

{(DXH = DZK1,0), , (D.K0,O)} (4.1.62)

4.1. The Three Basic Systems and T}_ei- geometrical Structure 373

and

{I1, ... ,ImJ (4.1.63)

where the DxKj, i = 1, ... , n, are linearly independent for each I E Tm by 113(except possibly at fixed points of (4.1.59)0), "0" represents the m+1 dimensionalzero vector, and h, i = 1, ... m, represent constant unit vectors in the Ii, i =1, ... , m directions. For a given (t0, a, I, 0) E K1 X 1R.s-1 X Tr" x Tl we let p =(x'(-t0, a), I,Oo) denote the corresponding point on 1' = W8(M) n Wu(M) - M.Then IIp is defined to be the m + n dimensional plane spanned by (4.1.62) and(4.1.63) where the DxKi are evaluated at p. As in System I, we will be interested

in the nature of the intersection of W8(M) and W'u'(M) with IIp.

Using arguments identical to those given for System I, it is easy to see thatW3(M) and Wu'(M) intersect IIp transversely in an m dimensional manifold foreach p E F. We denote the intersection of W8(M) (resp. WU(M)) with Hp by SP(resp. SP ). Moreover, we have rip n r = Sp = SP .

See Figure 4.1.9 for an illustration of the geometry (note the similarity withFigure 4.1.2). Recall that the importance of determining whether or not the inter-sections are transversal lies in the fact that transversal intersections persist under

small perturbation, and this fact is useful in determining the nature of the intersec-tion of the manifolds in the perturbed system.

iii) The Geometric Structure of the Perturbed Phase SpaceWe now want to describe some general conclusions concerning the structure of the

perturbed phase space which are due to the normal hyperbolicity of M. There are

fewer complications along these lines than in System I due to the fact that M isboundaryless. We will point out this fact as we go along.

The main result is the following.

Proposition 4.1.12. There exists co > 0 such that for 0 < e < Co the perturbedsystem (4.1.59)e possesses a Cr m + I dimensional normally hyperbolic invariantmanifold

M = { (x,1, 0) E 1R2n xTm xT1 I x = %y(I,0; e) = y(I) + 0(e) , I E T', B E TI } .

Moreover, Me has local Cr stable and unstable manifolds, and WUc(ME),

which are of the same dimension and Cr close to Wloc(M) and Woc(M), respec-tively.


span (DsK;, 0)

Figure 4.1.9. Homoclinic Coordinates.

PROOF: Due to the fact that M is boundaryless the technical modifications required

in Proposition 4.1.4 are unnecessary. Thus the result follows immediately fromProposition 4.1.10 and Theorem 1.3.7.

Let us now make several remarks regarding Proposition 4.1.11 and its geomet-

rical consequences.

1. The Structure of Me and the Flow on ME

ME has the structure of an m+l dimensional torus. The vector field restricted

to Me is given byI = e9I ('Y(I),I,0, A; 0) + 0(e2)

(4.1.64)B = 11(ry(I), I) + 0 (e) .

In general, the flow on Me is unknown and could involve complicated limit sets such

as Smale horseshoes or resonance phenomena amongst the different frequencies.

In System I it was necessary for us to first locate an invariant torus on Me,since ME was only a locally invariant manifold and not an invariant torus. Thiswas accomplished via an averaging technique. This technique is unnecessary forSystem II, since Me is itself an invariant torus.

Our analysis is insensitive to the dynamics on Me in that there may be addi-tional dynamical phenomena associated with limit sets or resonance phenomena on


the torus. Such questions deserve further study.

2. Differentiability of the Manifolds with Respect to Parameters

Following the manner described in comment 2 after the proof of Proposition

4.1.5, e and µ can be included explicitly as dependent variables in order to showthat M E, Wloc (M E), and Wloc (M E) are Cr in a and Fe.

Our goal will be to determine whether or not WS(ME) and Wu(ME) intersect.

The motivation for this comes from Chapter 3, where we saw that orbits homo-clinic to tori can often be the underlying mechanism for deterministic chaos. We

emphasize that the term "torus" is used in a general sense. For 1 = m = 0, meis a fixed point (0-torus), for l = 1, m = 0 or m = 1, 1 = 0, Me is a periodicorbit (1-torus), and for m + I > 2, ME is a torus with a nontrivial flow, and thedynamical consequences of homoclinic orbits are different in each case.


We now want to describe the geometry associated with our measure of the splitting

of WS(ME) and Wu(ME). This situation is less complicated than for System I, since

in System I we were measuring the splitting of the stable and unstable manifoldsof an invariant torus on ME which were contained in WS(Me) and WL(ME), respec-

tively. In the present situation the whole of Me is the relevant invariant torus, andthis simplifies the geometry.

Let us first recall the geometry of the unperturbed system (4.1.59)0. Let

(t0, a, I, B0) E 1R1 X R'-' x Tr'x T1 be fixed and denote the corresponding point on

F as p - (x1(-to, a), I, Bo). Let Hp be the m + n dimensional plane as previouslydefined, with W s (M) and W u (M) intersecting Hp transversely in the m dimensional

surfaces S' and SP , respectively, for each point p E IF = WS(M) n Wu(M) - Mwhere S' = S' (see Figure 4.1.10).

Now we will consider the geometry of the perturbed system (4.1.59)E along T.

Since W S (M) and W u (M) intersect Hp transversely for all p E I', for a sufficiently

small WS(Me) and Wu(ME) intersect Hp transversely for each p c F in the mdimensional sets W S (M E) n lip = SP e and W u (M E) n Hp S. However, in thiscase the sets SP E and Sp E need not coincide. Also, it may be true that WS(ME) and

Wu(.M) intersect Hp in a countable set of disconnected m dimensional sets as was

discussed for System I (see Figure 4.1.5). In this case our choice of (resp. SPE)


(a )

(b)

Figure 4.1.10. Intersection of W8(M) and W'L(M) with Hp.

a) m=2,n=1. b) m=1,n=2.

corresponds to the set of points which is "closest" to ME in the sense of positive(resp. negative) time of flow along W3(ME) (resp. W'(ME)). We will elaboratemore on this technical point when we discuss the derivation of the Melnikov vector

for System II.

We are now at the point where we can define the distance between W9(ME)and W"(ME). Let pE _ (xE,IE) and pE = (xE,I,) be points in S' and SpE,respectively, which have the same I coordinate, i.e., IE = I,. We remark that,unlike the situation in System I, such a choice of points is no problem in this case.Then the distance between W3(ME) and Wu(ME) at any point p c F may naively

be defined as

d(p, c) = lxE - xEI . (4.1.65)


However, we will develop a computable expression for (4.1.65) which utilizes

the underlying geometry of the distance between W3(ME) and W"(ME) (cf. the

discussion in 4.1a, iv). So, as in the case of System I, we define the signed com-ponent of the distance along the directions (DzKl, 0), , (DxKn, 0) which are


(b)

I2

1{

(D,H, 0)

(D.K2, 0)

0)

Figure 4.1.11. Intersection of W5(Mc) and with 11p.

a) m=2,n=1. b) m=1,n=2.

complementary to the tangent spaces of the manifolds as follows.

(Dzlfi (xI( t0, a), I) xE - x3)di(p,e) n

II D,Ki (xI(-to, a), I) II(4.1.66)

where we have replaced the symbol p by (to, I, 00, a) E 1R.1 x T m x T' x R'- 1, since

any point on r can be parametrized by (t0, I, B0, a), and we have also included the


vector of parameters y E IRP, indicating the possible parameter dependence of the

perturbed vector field.

Taylor expanding (4.1.66) about e = 0 gives

di (t0, I, 00, a, A; E) = di (t0) I, 00, a, lr; 0) + aE(to, I, 90, a, µ; 0)+0 (E2), i = 1,...,n

(4.1.67)

where

and

di(to,I,00ia,u;0) = 0 , i = 1,...,n since SP = Sp

11aai(to,I) aE E= aE o) 1,...,n.II x i( (- o, ), ) II

(4.1.68)

We will shortly show that

ax,(DxKi(x' (-toi a), I),a E E=0

0xE

(9f E=0) = Mi(I,00,a;ja)

00 t

f [(DxKi, 9x) + (DxKi, (DIJDxH) f gI )] (gp(t), IL; 0)dt,-00

(4.1.69a)

i = 1,...,n

or, equivalently,

-0000

-(DIKi(-I (I), I), f 9I(gO(t),y;0)dt)-oo

(4.1.69b)

/twhere qo (t) - (xI (t, a), I, J fl (xI (s, a), I) ds + 00),

and we define the n vector

M(I,Oo,a;9) = (Ml(I,Oo,a;µ),...,Mn(I,00,a;Fr)) (4.1.70)

00

Mi (I, B0, a; Ii) = f [(DxKi,9x) + (DIKi,9I)] (q (t),,u;0)dt


to be the Melnikov vector.

At this point we want to make several remarks concerning the geometry ofour measurement of the distance between W3(Me) and WU(Me) and make somecomparisons with System I.

1. As in System I, we do not explicitly show the variable to in the argumentof the Melnikov vector; this is because we will eliminate to via a change ofcoordinates when we discuss the derivation of M. Our ability to do this arises

from the fact that M measures the distance between manifolds of trajectories

and, therefore, by uniqueness of solutions, if the manifolds intersect, they must

intersect along (at least) one dimensional orbits. Hence, there is one direction

along the manifolds in which we need not move in order to examine the distance

between the manifolds. More details on this point will be given when we discuss

the derivation of the Melnikov vector.

2. We do not need to solve the perturbed equations (4.1.59)e in order to com-pute M. We only need to know the unperturbed homoclinic manifold and the

perturbed vector field.

3. W 3 (M E) and W '(M e) are codimension n manifolds, and the Mi, i = 1, ... , n,

represent (to O(e2) ) measurements along the n independent directions(DZKi, 0), i = 1, ... , n, complementary to the manifolds.

4. In System II the I variables appear as explicit arguments of the Melnikovvector, as opposed to the situation for System I, where it was necessary to fix

I in order to locate an invariant torus on ME. For System II, Me is entirely an

invariant torus and the I variables are part of the parametrization of the torus.

Now, by construction, if di(p,e) = 0, i = 1,...,n, then W3(Me) and Wu(ME)intersect near p. We now state our main theorem.

Theorem 4.1.13. Suppose there exists a point (I, 00, a, p) = (I, 00, a, ft) E T m xT1x1R7L-lxR with m+l+n-l+p>n such that

1) M(I, 80,a,µ) = Q.2 ) DM(I, 60i e, i1) is of rank n.

Then fore sufficiently small Ws(ME) andW'(ME) intersect near (I,Bo,a,#).

PROOF: The proof is identical to that of Theorem 4.1.9.

A sufficient condition for the intersection of W3(ME) and Wu(ME) is given in

the following theorem.


Theorem 4.1.14. Suppose Theorem 4.1.13 holds at the point (I, 00, a, µ) _(I, go, a, µ) E Tn x T l x 1Rn-1 X 1Rp and that D(I Bo'a)M(I, BO,a, µ) is of rankn. Then fore sufficiently small, W'(.ME) and W'(ME) intersect transversely near

(I,#0'a)PROOF: The proof is very similar to the proof of Theorem 4.1.10, but the geometry

of System II gives rise to some slight differences. Let p E Ws(ME)nWu(M,). Then

TpW s (M e) and TpW a (M E) are n + m + I dimensional and, by Definition 1.4.1,

W8(ME) and WU'(ME) intersect transversely at p if TpW8(ME) + TpWu(ME) =R2n+m+l So we need to show that TTW8(ME) and TpWu(Me) each contain n

dimensional independent subspaces which have no part contained in Tp(Ws(M,) n

Wu(ME)).

Let us recall the geometry of the unperturbed system. In this case, M is anm+1 dimensional invariant torus having n+m+l dimensional stable and unstablemanifolds which coincide along an n+m+l dimensional homoclinic manifold. So we

need to show that, in TpW8(Me) and TpW'(ME), new independent n dimensionalsubspaces are created which are not contained in Tp(W3(M,) n Wu(ME)). Theremaining part of the argument proceeds exactly as in the latter part of Theorem4.1.10.

4.1c. System III

We will now consider Hamiltonian perturbations of completely integrable Hamilto-

nian systems. These systems have the form

i = JDxH(x, I) + cJDxH(x, I, 0,µ;e)

I = -ED9H(x,1, 0,µ;e) (4.1.71),

9 = DIH(x, I) + eDIH(x, I, 0, A; e)

with 0 < e K 1, (x, I, 0) E IR2n x ]Rm x Tm, and µ E RP is a vector of parameters.

Additionally, we will assume

III1. Let V C IR2n X IR'n and W C ]R.2n x ]R.' x IRP x R be open sets; then the

functionsH: V --> IR1

H:W x T' ---.R1

are defined and C'+1, r > 2m + 2.


1112. J is the 2n x 2n symplectic matrix defined by

J=(

0 Id

-Id 0 )where Id denotes the n x n zero matrix.

We will refer to (4.1.71) as the perturbed system.

1) The Geometric Structure of the Unperturbed Phase Space

The system obtained by setting = 0 in (4.1.71) will be referred to as theunperturbed system.

i = JDzH(x, I)

I = 0 (4.1.71)0

9 = DIH(x, I) .

We have the following two structural assumptions on the x-component of(4.1.71)0.

1113. There exists an open set U C R' such that for each I C U the system

i = JDxH(x, I) (4.1.71)0,,

is a completely integrable Hamiltonian system, i.e., there exist n scalar val-ued functions of (x, 1), H = K1, K2, ... , K, , which satisfy the following twoconditions.

1) The set of vectors DxK1,DxK2,...,DxKn is pointwise linearly inde-pendent V I E U at all points of R2n which are not fixed points of(4.1.71)0x.

2) (JDxKi, Kj) = 0 V i, j, I E U C Rm where ( , ) is the usual Euclideaninner product. Furthermore, we assume that the K{ are Cr, r > 2m + 2.

1114. For every I E U, (4.1.71)0,x possesses a hyperbolic fixed point which varies

smoothly with I and an n-dimensional homoclinic manifold connecting the fixed

point to itself. We will assume that trajectories along the homoclinic manifold

can be represented in the form x1(t, a) where t E R1 and a E Rn-l.

At this point, the reader should note that the assumptions on the unperturbedstructure of System III are identical to those for System I. The differences willoccur when we consider the perturbed systems.


Analogous to System I and System II we consider the set of points M in R2n X

Rtm x Tm defined by

M = { (x, 1, 0) E R2n x R' x T'n I x = y(I) where -y(l) solves D2H(.y(I), I) = 0

subject to det [D2H(y(I), I)] # 0 , V I E U, 0 E Tm } (4.1.72)

and we have the following proposition.

Proposition 4.1.15. M is a C' 2m dimensional normally hyperbolic invariantmanifold of (4.1.71)0. Moreover, M has C' 2m+n dimensional stable and unstable

manifolds denoted WS(M) and Wu(M), respectively, which intersect in the n + 2m

dimensional homoclinic manifold

r = {(xI(-to,a),I,00) E IR2nXJRmXTm I (t0,a,I,00) E RiXJRn-1xUxTm}.

PROOF: The proof is identical to the proof of Proposition 4.1.4.

As was the case for System I, eventually we will be interested in the detailed

dynamics on M in the perturbed system. For Hamiltonian perturbations we will see

that much of the structure of the flow on M (in particular certain "nonresonant"motions) goes over for the perturbed system. For this reason we want to discuss in

more detail the structure of the dynamics on M.The unperturbed vector field restricted to M is given by

I=0DIH (-i (I), I)

(1,0) E U x T- (4.1.73)B =

with flow given byI(t) = I = constant

(4.1.74)0(t) = DIH(-y(I), I) t + 00 .

So M has the structure of an m parameter family of m tori with the flow on the tori

being either rational or irrational. Let us denote these tori as follows: for a fixedI E U, the corresponding rn-torus on M is

r(I)-{(x,I,0)EIR.2nxUxTmIx= y(I), 1=7} . (4.1.75)

r(I) has n + m dimensional stable and unstable manifolds denoted W S (r(I )) and

Wh(r(I )), respectively, which intersect along the n + m dimensional homoclinicmanifold given by

I I = { /xI( t0, a), 1,00) E R2n xRm x Tm I (t0, a, 0o) E R1XRn-1 X Tm }

(4.1.76)


Additionally, r(I) has an m dimensional center manifold corresponding to the non-

exponentially expanding or contracting directions tangent to M. See Figure 4.1.12

for an illustration of the geometry of the unperturbed phase space.

W'(r(i)) nW' (r(I))

X

Figure 4.1.12. Geometry of the Unperturbed Phase Space of (4.1.71)0.


We define a moving system of homoclinic coordinates along r for System III in the

same way as we did for System I and System II.

We consider the following n + m linearly independent vectors in R2" X Rm XRm:

{(D,zH = DxK1,0),---,(DxKn,O)} (4.1.77)

where DxKi are linearly independent by 1113 except at fixed points of (4.1.71)0,

and "0" denotes the 2m dimensional zero vector and

{I1+ ... ,Im} , (4.1.78)

where the I{ are constant unit vectors in R2, x Rm x Rm parallel to the Iicoordinate axes. For a given (to, a, I, Bo) E R1 x Rfz-1 x U X T m, we letp = (x,(-to, a), I, Bo) denote the corresponding point on r. Then 11p is defined tobe the m + n dimensional plane spanned by (4.1.77) and (4.1.78), where the DxKi

are evaluated at p and the Ii are viewed as emanating from p. So varying p serves

to move 11p along r.


As for Systems I and II, we will be interested in the nature of the intersection of

W" (M), Wu(M), WS(r(I)), and Wu(r(I)) with Hp. We will only state the results,since the details of the arguments are identical to those given in our discussion of

homoclinic coordinates for System I.

WS(M) and Wu(M). Ws(M) and Wu(M) intersect Hp transversely in an m di-mensional manifold for each p E I. We denote the intersection of WS(M) (resp.Wu(M)) with Hp by Sp (resp. Sp ). Moreover, we have SP = Sp.

WS(r(I)) andWu(r(I)). As mentioned previously, for fixed I=IEU,WS(r(I))and W u (r(I1) are n + m dimensional and intersect along the n + m dimensionalhomoclinic manifold

I'I = { a), I, Bp) E lR2n x1Rm x T' I (t0, a, B0) E 1R1x1Rn-1 x T-

I

Now W s (r (I)) and W u (r (I)) intersect Hp at the point (x1 (-to, a), I, Bo), and we

want to argue that this intersection is transversal. We can take the tangent space of

rip at p to be just Hp, i.e., Tpllp = Hp. Now the tangent space to Ws (r(1)) (resp.W u(r(I))) at p is n + m dimensional and can be viewed as being spanned by then vectors (JD,KE, 0), i = 1,... , n (where the JDxK1 are evaluated at p) and mvectors in the B directions (see Proposition 4.1.2). Hence, Hp+TpWs(r(I)) (resp.Hp + TpWu(r(I)) ) = R2n X 1R2' , and therefore WS(r(I)) (resp. WU(r(I)))intersects Hp transversely in a point.

See Figure 4.1.13 for an illustration of the geometry (note the similarities with

Figures 4.1.2 and 4.1.9). Recall that the importance of determining whether the in-

tersections are transversal lies in the fact that transversal intersections persist under

small perturbation; this fact is useful in determining the nature of the intersectionof the manifolds in the perturbed system.

iii) The Geometric Structure of the Perturbed Phase SpaceWe now describe some general conclusions that can be made about the structureof the perturbed phase space. We will be concerned with M, its local stable andunstable manifolds, and the flow on M.

Persistence of M.

The situation for System III regarding M is exactly the same as for System I:namely, M persists as a locally invariant manifold, ME.

4.1. The Three Basic Systems and Their Geometrical Structure

graph 7(I)

Figure 4.1.13. Homoclinic Coordinates.

385

Proposition 4.1.16. There exists co > 0 such that for 0 < e < co the per-turbed system (4.1.71)f possesses a Cr 2m dimensional normally hyperbolic locally

invariant manifold

ME (x,I,0) E IR2nXRmXTm I x=7'(I,0;E) =-I(I) + O(E),

IEUCUCRTZ,0ETm}

where U C U is a compact, connected m-dimensional set. Moreover, ME has local

Cr stable and unstable manifolds, Wloc(Me) and Wloc(ME), which are of the same

dimension and Cr close to W,'oc(ME) and Wloc(ME), respectively.

PROOF: The proof is similar to the proof of Proposition 4.1.5.

We remark that M is also Cr with respect to c and µ (see the remark following

the proof of Proposition 4.1.5).

Dynamics on ME.

We now want to address the question of whether there are any recurrent mo-tions on Me. In particular, we want to know if any of the m parameter family ofinvariant m dimensional tori survive the perturbation. For System I we used themethod of averaging and, as a result, only considered nonresonant motions. How-

ever, as discussed following Proposition 4.1.6, the method of averaging over angular

variables does not work when the system is Hamiltonian, and a more sophisticatedmethod is required.


Recall that the unperturbed vector field restricted to M is given by

I=09 = DIH(y(I), I)

(1, 0) E U x T- . (4.1.79)

Thus, (4.1.79) has the form of an m degree of freedom completely integrable Hamil-

tonian system, where the m I variables are the integrals of the motion and theentire phase space U x T9z is foliated by an m parameter family of m-dimensional

tori with the m frequencies on the tori given by the m vector DIH(-y(I),I). Thequestion of the nature of the dynamics on Me is then a question of what becomesof this family of invariant tori in a completely integrable Hamiltonian system when

the system is subjected to a Hamiltonian perturbation. Some important resultsalong these lines are provided by the Kolmogorov-Arnold-Moser (KAM) theorem(see Arnold [1978], Appendix 8), which we will now state in a form sufficient forour needs.

Theorem 4.1.17 (KAM). Suppose

det [DIH(ry(I), I),' 0, I E U C R-.

Then "most" of the invariant tori persist in (4.1.79) for sufficiently small E. Themotion on these surviving tori is quasiperiodic, having m rationally incommensurate

frequencies. The invariant tori form a majority in the sense that the Lebesquemeasure of the complement of their union is small when a is small.

Let us now make several remarks concerning this important theorem.

1. An immediate question that arises concerning Theorem 4.1.17 is, what doesthe term "most" mean? Mathematically, "most" means a Cantor set of posi-tive measure. For our purposes the important characteristic of the surviving"KAM" tori is that, since a Cantor set of positive measure persists, givena KAM torus there exists another KAM torus arbitrarily close. However, amore precise characterization of the surviving tori in terms of the unperturbed

frequencies can be found in Arnold [1963] or Moser [1973].

2. Note the radical difference regarding the dynamics on Me in the case of dissi-

pative perturbations in System I and the Hamiltonian perturbations in System

III. For System I, discrete nonresonant normally hyperbolic tori persisted onMe and, for System III, most of the nonresonant non-normally hyperbolic tori


persisted. We thus might expect very different dynamical phenomena in thetwo cases. Note that our methods for determining the resulting motion onME allow us only to find certain nonresonant motions, and that more sophisti-

cated techniques (as yet undeveloped) could reveal interesting dynamics that

are missed with present techniques.

3. We now address the question of how differentiable (4.1.79) must be in orderfor the KAM theorem to hold. Originally, the theorem was announced byKolmogorov [1954], with full details given by Arnold [1963] for the case ofanalytic Hamiltonians. The analogous theorem for vector fields with finitely

many derivatives was first given by Moser [1966a,b]. Moser's result applies to

Cr vector fields of the form of (4.1.79) with r > 2m + 2. For a recent reviewof KAM theory and related results see Bost [1986].

Recall the structure in the unperturbed system. For any I = I E U, r(I) is anm dimensional torus on M having m + n dimensional stable and unstable manifolds

W S (r(I )) and W u (r(I )) which intersect along an n + m dimensional homoclinic

orbit 1'j= {(x1(-to,a),I,00) E R2nxR"'xT"' I (to,a,90) ER1xRn-1xTm}.By Theorem 4.1.17, we know that most of these tori persist on ME. Let us denote

the surviving tori by r6(I). Now, the standard KAM theorem tells us nothingabout WS(TE(I)) and Wu(re(I)). However, a generalization of the KAM theorem

due to Graff [1974] tells us that rE(I) has m + n dimensional stable and unstablemanifolds, which we denote as WS(r5(I)) and Wu(re(I)). By invariance of themanifolds, we have WS(r6(I)) C WS(ME) and Wu(r6(I)) C Wu(ME). Now Graff's

theorem is proven only for analytic vector fields. However, he states that smoothing

techniques developed by Moser [1966a,b] can be used to extend the result to thefinite differentiable case. In any case, this is a technical difficulty that will cause us

little concern, because all of our examples will be analytic.


We now develop a procedure for determining whether or not WS(rE(I)) andWU(rc(I)) intersect. The situation will be different than for Systems I and II, sinceW S (r6(I)) and W u have larger codimension than the corresponding stable

and unstable manifolds in Systems I and II, and the KAM tori are not normallyhyperbolic. Hence, we might expect that a larger dimensional Melnikov vector isnecessary in order to determine whether or not W' (,re(1)) and W u (r6(I)) intersect.


Recall the geometry of the unperturbed system. For any point p E r, the2m + n dimensional manifolds W s (M) and W u (M) intersect the m + n dimensional

plane Hp transversely in the m dimensional surfaces SP and SP , respectively, with

SF = SP S. For any I = I E U, the corresponding invariant m torus r(I) E M hasm + n dimensional stable and unstable manifolds W s (r(I )) and W u (r(I )), which

intersect HP transversely in the point p. See Figure 4.1.14 for an illustration of theunperturbed geometry. Note that each point on SP = SP = r fl lip in the figurerepresents W3(r(I)) nWU(r(I)) for some torus r(I).

Sp = Sp = I' n IIp

Iz

I1

(DzH, 0)

Figure 4.1.14. The Unperturbed Geometry of the Manifolds on llp, n = 1, m = 2.

Next we consider the geometry of the perturbed system along I' = Ws(M) -Wu(M) - M. By transversality, for each point p E r, WS(ME) and W'(ME)intersect IIp in the m dimensional surfaces SP,E and respectively. As mentioned

previously, W3(ME) and Wu(ME) may intersect IIp in a multiple number of mdimensional components; however, we choose S' (resp. SP,E) to be the component

which is closest to ME in the sense of positive (resp. negative) time of flow along

W 3 (M e) (resp. W u (M E) ). More attention will be given to this technical detail when

we discuss the derivation of the Melnikov vector. Also by transversality, for a given

389Three Basic Systems and Their Geometrical Structure

in a pointtransversely

4.1. The

uW S (7E(1) intersects IIP

oint pE _ (x , IE)

M TE(I) E ME,

transversely in a pSeeFigure 4.1.15

surviving torusRsW' TE(I) intersects Iip

u reSpectivelY

E

are 4.1.15 the cloud

p = (x E' E and W

onta S 8 and Sp,E+these points are cined in PeE

try Note that in Fig

unstable)

Moreover,perturbed geom

of the stable (resp

for an illustration of the p

u) represent the intersections

S(resp Sp,E

with UP.

of dots on SP,(:

viving tori on MEmanifolds of the sur

uSP'E

u uIul

IR7_ (XU.

\_(DII,b

0)

Figure 4.1.15. The Perturbed Geometry of the Manifolds on RPM n 1, M 2

Wu -rE(I)distance between W S (?E(I )) and ( atWe might naively define the

168) (4.1.80)( su u - x I

the point P as

gp'

Iv), our goal isussion in 4.1a,

underlying

disc

atible with the

and II (cf theas for Systems I awhichis comp

orients of the dis-

$owever,

expression for (4.1.80)comp

table

expressions for the5 stem III as

velop a comp

developingdifference in y

This will involveimportant

only in(re(T )) intersect IIp

geometry.

notice anand W US (r6(I ))

tance along the coordinates o8nCe

Wd to Systems I and U.oppose


single points we cannot guarantee that IE = IE , as was the case for Systems I and

II. Hence, for System III, we must also measure the distance along these directions.

We define the signed component of the distance between pE and pE along the

n + m directions (D,K{, 0), i = 1, ... , n and Ii, i = 1,... , m as follows:

(DxKi (xI (-t0, a), I ), xE - xE)DxKi(xl (-t0, a), 1) II

1

11

(IE)i-n-(IE)i-n,

i = 1,...,n

n+I,-,n+m(4.1.81)

where (IE )i_n (resp. (IE )i_n) represents the (i-n)th component of the m vector I,"

(resp. 1. 3), and we have replaced the symbol p by (to, 00, a) E 1R1 x T'nx 1Rn-1; we

have also explicitly included the vector of parameters µ E RP to indicate thepossible parameter dependence of the perturbed vector field.

Taylor expanding (4.1.81) about c = 0 gives

dI E = dI t B a e = dI t e a 0) + Each (tp B a 0) + D (E2)

i = 1,...,n + m (4.1.82)

where di (t0, 00, a, µ; 0) = 0 since W S (r(I1) n lip = W1 (r (I)) n IIp = p and

8d1

aE (t0, e0, a, FL; 0) _

(DxKi(xI (-tQ, ax°

E=0

IIDxKi(x'(-to' a),1) 1I

aIeaE (aIE

E=01i-n \ aE

E=0

i=n+1,...,n+m.E=0 i-n

(4.1.83)

We will shortly show that

(D.Ki(x'(-to, a), 1),axE

aE

axE

E=0 ac) = Mg (80, a; li)

E=000

rt

J[(DxKi, JDxH) - (DxKi, (DIJDZH)JDsH)] (go(t), p; 0)dt

-00(4.1.84a)


or, equivalently,00

Mi(6p, a; µ) = f [(DxKi, JDZH) (DIKi, D9.11)] (q0 (t), µ; 0) dt

-0000

+(DIKi('Y(ID,I), f DBH(g0(t),/1;0)dt)-00

(4.1.84b)

and

(

aIEae

l (8IEE=p/ a-n 8e

00

f-00

i = n+1,...,n+m (4.1.85)

where q0 (t) = (XI (t, a), I, f t DIH(x7(s, a), I ds + 6p>. Thus, (4.1.84) and (4.1.85)

represent, to order e, the m+n components of the measurement of distance between

Ws(re(I)) and Wu(rc(I)) at the point p. However, there is a slight subtlety.Because the perturbed system is Hamiltonian in the 2n + 2m dimensional phasespace, orbits are restricted to lie in 2n + 2m - 1 dimensional "energy" surfacesgiven by the level sets of the Hamiltonian He = H + eH. Thus, we would expectto need only n + m - 1 independent measurements to determine whether or not

W a (re (I)) and W' (r6 (I)) intersect rather than n+ m. This problem is resolved in

the following 2n+2m dimensional version of a lemma due to Lerman and Umanskii

[1984].

Lemma 4.1.18. pE = pE if and only if di(p, e) = 0, i = 2, ... , n + m.

PROOF: pE = pE implies d1(p, e) = 0, i = 2, ... , n + m is obvious. We now show

that di (p, e) = 0, i = 2, ... , n + m, implies pE = pE .Let p = (x1(-to,a),I,Bp);then any point p E Hp can be expressed as

(4.1.86)

where (Cl, ... , en+m) represent coordinates along the vectors (DxKi(p), 0), i =1,... , n, and Ij_n, i = n + 1, ... , n + m which define lIp. Then, using (4.1.86),we have

pE - pE = (Cu - es) (DxH(p), 0) + ... + en) (DxKn (p), 0)(4.1.87)

+ (en+1 - n+1)I1 + ... + (fi+m - n+m)I7n


(TE(I )) and W u(re(I )), they must lie on the surfaceSince pE and pE lie on Ws

defined by the equation

R (e1, ... , Cn+m; E) = H (P + C1(D x H (P), 0) + ... + en (DxKn (P), 0)

+ en+l Il + ... + n+mlm, I)

+ eH(p+ e1(DxH(P),0) +... + n(DxKn(P),0) (4.1.88)

+ en+1I1 + ... + Cn+mlm, I, BO, E)

- H(ry(I, 80; e), I) - eH(ry(I, B0; E), I, B0, E) = 0

(note: we have suppressed the possible parameters P since they will not affect the

argument). We have

R(0,...,0;0) = H(x1(-to,a),!) -H(-y(I),I) =0 (4.1.89)

and D£1 R(0, ... , 0; 0) = (DxH (x' (-to, a), I), DxH(xJ (-to, a), I-)) # 0. Hence,by the implicit function theorem, we have

El = O(C2, , Cn+m; e) (4.1.90)

for (e1, ... , En+m, E) sufficiently small and where 4S is as smooth as H + EH. Using

(4.1.87) and (4.1.90), we obtain

PE- Pe= ( (e2,...,Cn+m;E)-O(Q2,...,£m+m;e))(DxH(P),0)+...

+ (fin -fin) (DxKn (P), O) + (en+1- n+1)II +...+ (en.+m- en+m)Im .

(4.1.91)

So from (4.1.90) and (4.1.91) it is clear that for e! = Ci , i = 2, ... , n + m wehave pE = pE. This proves the lemma.

This lemma tells us we do not need to measure along the direction (DxH, 0) in

order to determine whether W e (TE(I)) and W u (Te(I )) intersect. Intuitively, this

should be reasonable, since the energy manifold H+cH is preserved and (DxH,O)is a direction complementary to the energy manifold.

We will define the Melnikov vector as

MI (Oo, a; a) = (M2 (B0, a; A), ... , Mn+rn(Oo, (4.1.92)

where we have left out the explicit dependence on to for the same reasons discussed

for Systems I and II (note: this will be elaborated on when we discuss our derivation

of the Melnikov vector). We can now state our main theorem for System III.


Theorem 4.1.19. Suppose there exists I = I E U C M' such that re(I) is aKAM torus on Me. Let (00i a, i) = (B0, a, µ) E T"' x Mn-1 x RP be a point withm+n-1+p>m+n-1 suchthat

1) M ' ( 0 , , , 1 ) = 0

2) DMI(BO,a,µ) is ofrank m+n-1;then W s (re(I )) and W u (r6(I )) intersect near (BO, a, µ).

PROOF: The proof is similar to the proof of Theorem 4.1.9.

A sufficient condition for the transversal intersection of W' (rE(I )) and

W U (re (I)) is given in the following theorem.

Theorem 4.1.20. Suppose Theorem 4.1.19 holds at the point (BO, a, µ) _ (B0, a, µ)

E T'' x 1Rn-1 x RP and that D(00 a)MI (BO, a, µ.) is of rank m + n - 1. Then, fore sufficiently small, W' (TE(I )) and W u (rc(I )) intersect transversely near (BO, a) in

the 2n + 2m - 1 dimensional energy surface.

PROOF: The proof is similar to the proof of Theorem 4.1.10. Let p E W8 (T,(!)) n

W u(rE(I )). Then TpW s (T,(!)) and are m + n dimensional and, by

Definition 1.4.1, W8 (rs(I )) intersects W u (re(I )) transversely at pin the 2n+2m-1

dimensional energy manifold if TpW s(r5(I ))+TpW 1R2n+2m-1 By the

dimension formula for vector spaces we have

2n + 2m - 1 = dim TpW s (rE (I )) + dim TpW u (r (I ))(4.1.93)

- dim Tp (W -(T, (1)) n W u (r (I)))

Thus, if WS(Te(I)) intersects W' transversely in the 2n + 2m - 1 dimen-sional energy manifold at p, then WS(rE(I)) intersects WU(r6(I)) in a one di-mensional trajectory. Therefore, in order for to intersecttransversely at p, it is necessary for TpW s (rE(I )) and TpW u (re(I )) to each con-

tain n + m - 1 dimensional independent subspaces which have no part contained in

Tp (WS(rE(I)) nWu(r6(I))).Let us recall the geometry of the unperturbed phase space. The m dimensional

KAM torus r(I) has m+n dimensional stable and unstable manifolds which coincide

along an s + m dimensional homoclinic orbit. We need to show that independentm + n - 1 dimensional subspaces are created in TpW s (rE(I )) and TpW u (rE (1))

which are not contained in Tp (W' (rE (I)) n W u (TE (1))). The remainder of theproof is the same as the latter part of the proof of Theorem 4.1.10.


v) Horseshoes and Arnold Diffusion

For Systems I and II some dynamical consequences of the intersection of the stable

and unstable manifolds of a normally hyperbolic invariant torus follow from the

associated theorems given in Chapter 3 (note: the term torus is used in a generalsense and also applies to the case of the 0-torus (fixed point) and the 1-torus (pe-riodic orbit) ). However, the somewhat more subtle geometry associated with the

phase space of perturbed completely integrable Hamiltonian systems is responsible

for exotic dynamics, which we will now discuss separately. There are two distinctcases corresponding to differences in the dimensions of the phase space.

n > 1 , m = 1. In this case the phase space is 2n + 2 dimensional and is foliated

by invariant 2n + 1 dimensional energy surfaces. In the unperturbed system, M is a

1-dimensional normally hyperbolic invariant manifold which has the structure of a

1 parameter family of 1-tori, r(I), I C U C ]R1. Each torus has n+ 1 dimensionalstable and unstable manifolds coinciding along an n + 1 dimensional homoclinicorbit. In the perturbed system, M is preserved (denoted ME), and on ME we have

HE = H(-y(I), I) + 0 (E) = constant . (4.1.94)

Now, since I E U C ]R1, from (4.1.94) we see that on a fixed energy manifold I is

likewise fixed. So, in this case, the full results of the KAM theorem are not needed.

On a fixed energy manifold an isolated 1-torus (i.e., periodic orbit) survives and is

normally hyperbolic on the energy manifold. Thus, the Melnikov vector measuresthe distance between the stable and unstable manifolds of a normally hyperbolicperiodic orbit, and the dynamical consequences associated with their intersectionare ordinary Smale horseshoes.

n > 1, m > 2. In the unperturbed system on M we have

H(-1 (I), I) = constant. (4.1.95)

Thus, on a fixed 2n + 2m - 1 dimensional energy manifold we have an m - 1parameter family of m-tori. Each torus has m + n dimensional stable and unstable

manifolds which coincide along an n + m dimensional homoclinic orbit. Note the

important point that, since m > 2, the tori, along with their stable and unstablemanifolds, are not isolated on the energy manifold.


In the perturbed system "most" of the m - 1 parameter family of tori surviveon each energy manifold by the KAM theorem. In this case, it is possible to choose

a set of KAM tori, r,(I1), r,(I2), 7-,(IN), with the property that r,(Ii) isarbitrarily close to r,(Ii+1) for i = 1,...,N - 1. Now, suppose that for some

i with 1 < i < N - 1, Wu(T,(Ii)) intersects W3(r,(Ii)) transversely. Then,by arguments similar to those given in the proof of the Toral Lambda Lemma inChapter 3, it can be shown that Wu(r,(Ii)) accumulates on r,(Ii), resulting in italso transversely intersecting Ws(r,(Ii+1)) and Ws(r,(Ii_1)) (for i > 1), whichare arbitrarily close. This argument can be repeated, with the ultimate conclusion

being that W u(T,(Ii)) transversely intersects Ws(r,(Ij)) for any 1 < i, j < N. The

resulting tangle of manifolds provides a mechanism whereby orbits may wander in

an apparently random fashion amongst the KAM tori. The sequence of tori r,(Ii),

..., re(IN) is referred to as a transition chain, and the resulting motion is calledArnold diffusion, see Figure 4.1.16 for a heuristic illustration of the geometry.

Figure 4.1.16. The Geometry of Arnold Diffusion.

Despite the ubiquity of Arnold diffusion in Hamiltonian systems having more

than two degrees of freedom, there has been surprisingly little work done in the


area since Arnold's original paper in 1964. Nehoroshev [1971], [1972] provided

estimates on the rate of Arnold diffusion and, as mentioned earlier, Holmes andMarsden [1982b] developed the first general techniques to verify the existence of

Arnold diffusion in specific systems. The previously described geometrical picture

of the dense tangling of the manifolds of the KAM tori has yet to be put on arigorous footing along the lines of Chapter 3, although results of Easton [1978],[1981] regarding certain model problems should go through in the general case and

would provide a good starting point. Numerical simulations, which have yielded

great insight into the global dynamics of one and two dimensional maps, have yet

to be put to extensive use in the study of Arnold diffusion (but see Lichtenberg and

Lieberman [1982]). This is probably due to the fact that at least a four dimensional

volume preserving map would be needed in order to exhibit Arnold diffusion, and

it is not immediately clear how to best display the dynamics of a four dimensionalmap on a two dimensional computer screen.

4.1d. The Derivation of the Melnikov Vector

We will now give the derivation of the Melnikov vectors for Systems I, II, and III.

We will do this for the three systems simultaneously, discussing the differences as

we go along.

Recall that the Melnikov vectors for the three systems are given by:

System I.

MI (Oo, a; A) _ (Ml (Oo, a; iL), ... , Mn (Bo a; A)) ( 0 0 ,a ;/ , ) E T l x R"-1 x RP

(4.1.96)

where we will show that

0/0

fg1>]M i (B0, ; ) J= [(DzK, 9x) + (DK, (DIJDH) (90 (t), Ii; 0) dt,

-00i = 1, ... , n (4.1.97a)

or, equivalently,


00

MI (0o,a;A) = f [(D.Kigz) + (DIKi,9I))(go(t),A;O)dt-00

00

-(DIKi('Y(I), I), f 91(90(t),A;O)dt)- 00

(4.1.97b)

and q01(t) = (xl (t, a), I, f t 1l(x1(s, a), I)ds + 90), I E U C U C Rm being chosensuch that r6(I) is a normally hyperbolic invariant torus on ME (see Propositions4.1.6 and 4.1.7).

System II.

M(I,00,a;A) = (Mi(I,Oo,a;u),...,Mn(I,00,a;µ)),(I, 80, a;.u) E Tm x Tt x Rn-1 x Rp (4.1.98)

where we will show that00 t

Mi(I,Oo,a;Fi)= f [(DzKi,9z)+(DzKi,(DIJD.H) f 97)](q (t),µ;O)dt,-00

or, equivalently,

i = 1,...,n.

(4.1.99a)

Mi(I,BO,a;h) =

00

f [(D.Ki,9z) + (DIKi,9I)](q (t), e;O)dt-00

Co

-(DIKi(7(I),I), f 9I(gO(t),A;O)dt)-00

(4.1.99b)

and q0 (t) _ (xl (t, a), I, f t 1 (xl (s, a), I)ds + 00).

System III.

MI (B0, a; u) _ (nIz

(00, a; µ) E T' x Rn-1 x RP (4.1.100)


where we will show that

Mti(00,a;i)_t

f [(DzK f DoH)] (q (t),µ,0)dt,

(4.1.101a)

i = 2,...,n

or, equivalently,

00

MI (00, a; Ft) = f [(DZKiJDZH) - (DIKi, D9H)](gp(t),1t; 0)dt-00

0/0

+(DIKK(-y(I),I), J DBH(gp(t),u;0)dt)-00

(4.1.101b)

and

00

M%(B0, a;,.) f DB2_nII (q0 (t), µ; 0) dt,-00

i = n + 1,...,n + m (4.1.102)

and qo(t) - (xI (t, a), I, f t DIH(xI (s, a)I)ds + 00), I = I E U C U C 1R- beingchosen such that re (1) is a KAM torus on ME (see Theorem 4.1.17).

The procedure for obtaining (4.1.97), (4.1.99), (4.1.101), and (4.1.102) willinvolve deriving a first order, linear ordinary differential equation that a time de-

pendent Melnikov vector must satisfy, solving the equation for the time dependent

Melnikov vector and, finally, evaluating the solution at the appropriate time so as

to obtain (4.1.97), (4.1.99), (4.1.101), and (4.1.102). At the appropriate point wewill discuss the convergence properties of the improper integrals.

As a preliminary to our derivation of the Melnikov vectors, let us recall thegeometry associated with the splitting of the manifolds. We are interested in theperturbed systems in a neighborhood of the n + m + 1 dimensional unperturbedhomoclinic manifold r = W 8 (M) n W u (M) - M, which is parametrized as follows:

r = {(xI(-to,,,),I,00) E ]R.2n x1RmxTt I (to,a,I,00) E llxIRn-1 xUxTt}


(note: for System II U = Tn' and for System III 1 = m). At each point p E 1' weconstructed the n + m dimensional plane TIP spanned by the n linearly independent

vectors (DzKi (p), 0), i = 1, ... , n, where "0" represents the m + I dimensional

zero vector, and the m linearly independent unit vectors I{, i = 1.... , m, where the

h represent constant vectors in the Ii directions. We then argued that W3(M) and

W u (M) intersected Hp transversely in the m dimensional coincident surfaces SP and

Sp , respectively, for each p c I'. This geometrical structure of the unperturbedsystem was the backbone on which we derived our measurement of the splitting of

the manifolds of certain invariant tori in the perturbed system. In the perturbedsystem, M persisted (denoted by ME) and, by transversality, for c sufficiently small,

W s (M E) and W u (M E) intersected 11P transversely in the m dimensional surfaces

Sp E and Sp E, respectively, for each p E I'. Unlike in the unperturbed systems,it was possible for W3(ME) and Wu(ME) to intersect IIp in multiple disconnectedcomponents. With this possibility in mind, we chose SPIE and Su to be the m

PIE

dimensional components that were closest to ME in the sense of elapsed negative

and positive flow time along WS(ME) and Wu'(ME), respectively. The reasoningbehind this choice will be explained shortly. Now, our interest was not necessarily

in the splitting of the stable and unstable manifolds of ME, but rather in the splitting

of the stable and unstable manifolds of invariant tori that were contained in ME.There were three distinct situations, as follows:

System I. An l dimensional normally hyperbolic invariant torus, r,(I), havingan n + j + I dimensional stable manifold, W9(re(I )), and an n + m - j + Idimensional unstable manifold, Wu(re(I)), was located on ME using averaging.The averaging method required nonresonance conditions resulting in the flow on

being irrational with orbits densely filling the torus. We then argued thatW9(rE(I )) n Sp E - WP (rE(I )) was a j dimensional set, and Wu(r6(I )) n SP,EWP was an m - j dimensional set for each p E r. We chose pointsPEE = (xE, IE) E WP '(7-e(1))(I )) and pE = (xE , IE) E WP (rE(I )) such that IE = If.This was possible due to the normal hyperbolicity of re(I) (see Lemma 4.1.8).Then the signed distance between W'3(re(I)) and Wu(re(I)) along the remainingn independent directions on IIp was given by

di (p; E) = d2 (t0, 60, a, µ; E) _ (D2ID(

K((xI(«t01))

I) I

tee), i = 1, ... , n .

(4.1.103)


Now, since Ws(ME) and WU(ME) are differentiable in E, we Taylor expanded(4.1.103) about E = 0 and obtained

(DzKi (xI (-t0, «), I ),a2E - axEaE aEE=o

E=Q 2,I/ a ... ... ,1 _ .D.Ki(x'(-to, a), 1) 11

i = 1,..., n (4.1.104)

where d4 (t0, 80, a, µ; 0) = 0 since SP = Sp.The Melnikov vector was defined to be

MI (00, «, u) = (Ml ( 0 0 ,a ;9 ) ,-- . , M,Ib (eo, a; u)) (4.1.105)

where

MR (00, a; l.t) _ (D2Ki (xI (-to, a), I ),

a.,- ax" IaE 0 aE E=0

(4.1.106)

System H. In this case M E was itself an m +I dimensional normally hyperbolicinvariant torus having n+m+l dimensional stable and unstable manifolds denoted

W8(.e) and W'U(ME), respectively. At each p c P we choose points pE _(xE, IE) E SP E and pE = (xE , IE) E SP,, such that I,' = IE. Then the signeddistance between W8(ME) and Wu(ME) at the point p along the n independentdirections on IIp was given by

(DZKi(xI(-t0,«), I), XU -xE)di(p,E) =di(to,I,©0, ID,K(xI(-t0,«),I)jjx

(4.1.107)

Taylor expanding (4.1.107) about c = 0 gave

s 1axF)

1_(DxKi(xI(-t0,«),I),

azFaE

E=0aE

E=odi (to) I, 00, a, ju; E) = E Dx Ki(xI (-t0, a), I) II+ O(fl) ,

i = 1,..., n (4.1.108)

where di (to, I, 00, a, u; 0) = 0 since SP = Sp .

The Melnikov vector was defined to be

M(I,©0,a,µ) = (Mt(I,©0,a;9),...,Mn(I,00,«;µ)) (4.1.109)


where

uM%(I,BOAa,y) = (DxKi(xI(-t0,a),I), a2E - axE ), i = 1,...,n.

of E=0 aE E=0(4.1.110)

Note that for System II, I is a variable of the Melnikov vector since it is an angular

variable along the torus Me.

System III. In this case, we located an m dimensional invariant torus on ME,

using the KAM theorem. This invariant torus had n + in dimensional stable

and unstable manifolds denoted Ws(TE(I )) and WU(re(I )), respectively. For each

p E I', W5(re(I)) and Wu(re(I )) intersected Ilp transversely in the points pE =(x£, IE) E Sp E and pE _ (xE , IE) E SP E, respectively. The signed distance betweenW s and W u (rE (I)) at the point p along the n + m directions on ITp wasdefined to be

(D.Ki(x,(-to,a),I),xE -xE) i 2d'! (p, E) = di (to, 00, I I DzKi (xI (-/ to, a), I) I I

= ,. . .,n

(IE)t-n-(IE)i-n+ 1=n+1,...,n+m.(4.1.111)

Taylor expanding (4.1.111) about c = 0 gave

da(to,00,a,,L;E) _

where dE (to, 00, 0) = 0

_(DxKi(xI(-to,«),I), aE aEE=0 E=0 + 0 (E2),

DzK%(x'(-to, a),I)II1

i = 2,...,n

/ aIE a(2),E [\ aE 0)%-n C7E

6=0)i-n]+0 E

i=n+1,...,n+m(4.1.112)

since Ws(r(I )) n llp = W'a(r(I )) f11Ip = p.The Melnikov vector was defined to be

MI (00, a; lz) = (M2 (so, a; li), ... , Mn+m(00, a; iz))

where

(4.1.113)

UMz (00,a;u) _ (D.Ki(x1(-to, a),I),a-E - axEaE E=0 aE E=0

(4.1.114)


u sMI B a;

aIE SIE, i = n + 1, ... , n + m .

E ( o, , ) _ . 8E CO2_) -7L - SE E=0) 1-7E

(4.1.115)

We remark that we have not measured along the direction (DxK1, 0) = (DxH, 0),

since, for System III, the level surfaces of H E = H + EH are preserved under theperturbation and the direction (D5K1, 0) is complementary to these surfaces (see

Lemma 4.1.18).

Our goal now is to show that (4.1.106), (4.1.110), (4.1.114), and (4.1.115) are

given by (4.1.97), (4.1.99), (4.1.101), and (4.1.102), respectively. However, first

we want to establish some shorthand notation that will make the formulas moremanageable.

a) We will denote the perturbed vector fields for Systems I, II, and III by

4=f(q)+Eg(q;A,E)

where

(4.1.116)

f = (JDxH, 0, ul) for Systems I and II; f = (JDxH, 0, DIH) for Sys-tern III; g = (gx, gI, gB) for Systems I and II; and g = (JDxH, -DBH, DIH)for System III.

b) We denote trajectories of the unperturbed system along the homoclinic mani-

fold r by

fq(t - t0) = (xI (t - t0, a), I, 1l (x1(s, a)I)ds + 90) (4.1.117)

where f1 = DIH for System III, and we denote trajectories of the perturbedsystem in Ws,u(.ME) by

gE'c(t) =(xE°u(t),IE'"(t),BE'U(t)) (4.1.118)

i) The Time Dependent Melnikov Vector

We define time dependent Melnikov vectors for Systems I, II, and III as follows:

System I.

Mf (t) _ (DxKi(xJ(t - t0, a), f),8xE (t) - 8xE(t) n.SE

c 0 SE ; E=0(4.1.119)


System II.

Mi(t) = (DaKi(x' (t -to,",), I), a2 (t) _ axE(t)a6 ;E=0 ac

System III.

M;(t) =

(DxKi (xI (t - t0, a), I ), aa (t)

), i=1,...,n.e=0

(4.1.120)

axE(t)

E=0 aE e=0), i = 2,...,n

(4.1.121)

I/d1E (t I\/a1E t\ aE Lo) i-7t- ` 19C a=Q) t-n' i = n + 1, ... , n+m

(4.1.122)

where the trajectories q, '(t) = (xE (t), IE (t), BE (t)) and qE (t) = (xE (t), IE (t), BE (t))

lie in the stable and unstable manifolds of the invariant torus on ME and satisfy

q63(0) = (xE (0), IE (0), BE (0)) = (xE, IE , B0) and q6 u(0) = (x (0), IE (0), BE (0)) _(XU, uEIE ,BO)-

The trajectories qE (t) and qE (t) satisfy the equations

qE = f (qE) + cg (q6', W; E) (4.1.123)

qE = f (qE) + Eg(gE , µ, e) . (4.1.124)

We will be interested in the length of the time interval on which these solutions are

valid. We have the following lemma.

Lemma 4.1.21. For e sufficiently small, q, '(t) and qE (t) are solutions of (4.1.123)

and (4.1.124), respectively, which exist on the semi-infinite time intervals [0,00)and (-oo, 0] for all initial conditions q,'(0) and qE (0) contained in the stable and

unstable manifolds of the invariant torus.

PROOF: The proof of this is obvious since, by definition of the stable (resp. un-stable) manifold, given any point in the stable (resp. unstable) manifold of theinvariant torus, the trajectory through this point exists for all positive (resp. nega-

tive) time and is asymptotic to the invariant torus. It is necessary for c to be taken

small in order for ME, along with its stable and unstable manifolds, to exist.

We remark that Lemma 4.1.21 does not imply that the trajectories gE's(t) ap-proximate q0 (t) to within 0 (E) on the appropriate semi-infinite time intervals. This


fact is not needed and, in general, is not true since, on the semi-infinite time inter-

vals, the angular variables of perturbed and unperturbed trajectories may separate

by an 0 (1) amount.

We will be interested in the time evolution of the quantities

8IE'3tN As a shorthand notation, we define

e=0

xU,3(t) - 8x6'3 (t)1 aE a=0

Il,s(t) = a_E's(t)aE I e=o

(4.1.125)

B1'3(t) = .90'U" (t)

ac Lo

Now, by Theorem 1.1.4, solutions of (4.1.123) and (4.1.124) are differentiable with

respect to e. By Theorem 1.1.5, the solutions (xi'3(t),Ii'3(t),01'3(t)) satisfy thefirst variational equation given by

ii'3 JDyH DIJDZHIi'3 = 0 0

Bi'3 Dx1 Dj1

where the entries of the matrix aret

0 xi'3 gZ (q6I(t -to), u; 0)

0 Ii'3 + (g1(q(t_to),;o)0 81'3 ge (g0 (t - to), µ; 0)

(4.1.126)

evaluated on (x1 (t - to, a), I), q1 (t - to) _

(XI (t - to, a), I, f Il (xI (s, a), I)ds + 00) is an unperturbed homoclinic trajectory,and the vector (gZ, g1, g0) is modified appropriately for System III (i.e., recall in

System III we have (gZ, gI, g8) = (JDxH, -D9H, DIH) ).

ii) An Ordinary Differential Equation for the Melnikov Vector

Consider the expression

(DxK%(x1 (t - to, a), 1), xl (t) - xl (t)), i = 1, ... , n(Ms(t)

(Il (t))%-n - (Il (t))%-n' i = n + 1,...,n + m .(4.1.127)

We will derive a linear ordinary differential equation which (4.1.127) must satisfy.

The solution of this equation evaluated at t = 0 will yield the Melnikov vectors for

Systems I, II, and III. However, it will be necessary to impose conditions at ±oo on

the solutions of the equation, and these conditions will be dictated by dynamicalphenomena that are specific to Systems I, II, and III individually.

axe s' t andE Ie=o


As a shorthand notation we have

( (DzKi(xl(t-to,a),I), xi'3(t)), i= 1,...,ns (t) _{'

where now

(Il'3(t))i-n' i = n + 1, ... , n + m(4.1.128)

Mi(t)=AM1(t)-L(t) , i=1,...,n+m. (4.1.129)

Differentiating (4.1.129) with respect to t gives

Mi(t) = '&Y (t) - '& %q (t) , i = 1, ... , n + m (4.1.130)

where

(DzKi (xl (t - t0, a), I)), xl's(t))+(D,K1(xI(t - to, a), I), il'3(t)),

t'3(t)=

(I1'3(t))i-n, i=n+1,...,n+m.(4.1.131)

Using the chain rule, and the fact that I = 0 in the unperturbed system, we obtain

ddt

(DzKi (xI (t - t0, a), I)) = DzKi (xI (t - t0, a), I) it (t - t0, a) . (4.1.132)

Using the fact that it (t - t0, a) = JDxH(xl (t - to, a), I), (4.1.132) becomes

dt(DzK1(xl (t - t0, a), I)) = D2Ki (xl (t - t0i a), I)JDxH (xI (t - t0, a), I)

(4.1.133)

From the first variational equation (4.1.126), we have

iU'3 = JD2H(x'(t-to,a),I)zi''+DIJDxH(xl(t-to,a),I)h''+9z(90(t-to),µ;0)

Ii '3 = 9l (40 (t - to), /µ; 0) .

(4.1.134)

Substituting (4.1.134), (4.1.131) gives (note: henceforth we will leave out the argu-

ments of the functions for the sake of a less cumbersome notation)

Di '3 (t) _

(DxKi, (JDxH)xi'3) + (DzKi, (DIJDxH)Ii's)r

+(DxKi,9x) + (xl'3,(DzK1(JDxH)), i = 1,...,n

i=n+1,...,n+m.(4.1.135)


The first n components of (4.1.135) simplify considerably with the following

lemma.

Lemma 4.1.22. (DxKi, (JD2H)xl'9) + (xu'8, (DyKi) (JD.H)) = 0, i = 1, ... , n.

PROOF: By 13,113, or 1113 we have

(JDxH,DxKi)=0, i=1,...,n. (4.1.136)

Differentiating (4.1.136) with respect to x gives

Dx(JDxH,DZKi) = (JD2H)TDxKi + (DyKi)JDxH = 0, i = 1,...,n(4.1.137)

where "T" denotes the matrix transpose. Taking the inner product of (4.1.137)with 4'8 gives

((JD'H)T DxKi, x1'9) + ((D'Ki) (JDyH), xi's) = 0 , i = 1, ... , n (4.1.138)

or

(DxKi, (JDyH)xl'9) + (x1'9, (DyKi)(JDxH)) = 0, 1= 1,...,n. (4.1.139)

0Using Lemma 4.1.22, (4.1.135) reduces to

04 ,$ (t) = J (DxKi, 9x) + (DxKi, (DIJDxH)Ir's), £ = 1, ... , n

(9 )i-n, n+1,...,n+m.(4.1.140)

iii) Solution of the Ordinary Differential Equation

Integrating Di (t) from -Tu to 0 and &q(t) from 0 to Ts for some T3, Tu > 0 gives

Di (0) - Ag (-Tu) _0

f [(DxKi, 9x) + (DxKi, (DIJDxH)I? )J (gp(t - to), µ; 0)dt,-Tu

0

f(g1)j_(q(t_to),,i;o)dt, i=n+1,...,n+m-Tu

(4.1.141)


T"

f [(DxKi,9x) + (DxKi, (DIJDaH)Il)] (4p(t -t0),µ;0)dt,0

T'f(g1)j(q(t_to),,i;0)dt,

i = 1,...,n

=n+1,...,n+m.

(4.1.142)

0

We will want to consider the limit of (4.1.142) as T9 -> +oo and (4.1.141) as-T' - -oo. The following lemma will be useful.

Lemma 4.1.23. DxKi (y(I), I) = 0, i = 1, ... , n.

PROOF: y(I) is the surface of hyperbolic fixed points of the x components of the

unperturbed vector fields. Therefore,

JDxH (y(I), I) = JDxK1(y(I), I) = 0 (4.1.143)

and, since J is nondegenerate,

D,H(y(I),I) = DxK1(y(I),I) = 0. (4.1.144)

Now from (4.1.137) we have

DZ(JDxH, DxKi) _ (JDzH)T DxKi + (DZKi)JDxH = 0, i = 1,... , n .(4.1.145)

Evaluating (4.1.145) on (y(I), I) and using (4.1.143) gives

(JDiH(y(I),I))TDxKi(-y(I),I) = 0, i = 1,...,n. (4.1.146)

Now, since (-y(I),I) is a hyperbolic fixed point, det [JDyH(y(I), I), 0, andtherefore DxKi (-/ (I), I) = 0, i = 1, ... , n.

Using (4.1.129) the components of the Melnikov vector are given by

Mt (B0, a, µ) = Di (0) - Di (0) . (4.1.147)


Using (4.1.141) and (4.1.142), we will evaluate (4.1.147) for the three systems indi-

vidually.

System I. From the first variational equation (4.1.126) we have

t-toI1 3(t) = Il (t) = J gI . (4.1.148)

Substituting (4.1.148) into (4.1.141) into (4.1.142) gives

Mf (e0, a; /A) = At (0) - Ai (0)T ° t-to

f [(D.Ki, gZ) + (DzKi, (DIJD.H) f gI )] (4p (t - to), µ; 0)dt

-Tu+Os (-Tu) - Di (T s) , i = 1,... , n. (4.1.149)

Now we want to consider the limit of (4.1.149) as -Tu --> -oo and Ts -. +oo.

Lemma 4.1.24. lim Ot (-Tu) = lim AI (Ts) = 0.-T -oo T -->00

PROOF: We will give the argument for Ai ; the argument for Di is similar.

From (4.1.129) we have

Di (t) = (DzKK (xT (t - to, a), I), XU (t)) . (4.1.150)

Now, xi (t) can grow at best linearly in time and, by Lemma 4.1.23, DXKi(xI (t -

to, a), f) goes to zero exponentially fast as t -r -oo. Therefore, lim i (t) = 0.t-- -oo

So now we have obtained (4.1.97a)

Mt (Bo, a;.a) _00 t-tof [(DaKi, gx) + (D.Ki, (DIJD2H) f gI)] (9p(t - to), A; 0)dt ,

-00i = n. (4.1.151)

Proposition 4.1.25. The improper integrals in (4.1.151) converge absolutely.

PROOF: X-'(t - to, a) -+ ry(I) exponentially fast as t --+ ±oo, since -y(I)

is a hyperbolic fixed point of the x component of the unperturbed vector field.


Therefore, by Lemma 4.1.23, DxKi (x' (t - to, a), I) --> 0 exponentially fast ast-to

t -+ foo. Now, since gx and (DIJD2H) f gI are bounded on bounded subsets of

their respective domains of definition, we can conclude that the integral in (4.1.151)

converges absolutely as Ts - +oo, -T" -. -oo.

We remark that convergence properties of Melnikov type integrals were first

studied in detail by Robinson [1985].

We now show how to obtain the form of equation (4.1.97b).

Lemma 4.1.26. (DxKi, (DIJD2H)Ii'e) _ -(d (DIK{), Ir's) evaluated on theunperturbed homoclinic orbit (x1(t - t0, a), I).

PROOF: On the unperturbed homoclinic orbit we have

dt(DIKi) = (DxDIKi)(JDxH) (4.1.152)

since I = 0. Differentiating the Poisson Bracket we have

DI (JDxH, DxKi) _ (DIJDxH)T D2Ki + (DIDxKi)T JDZH = 0 (4.1.153)

where "T" denotes the matrix transpose. Combining (4.1.152) and (4.1.153), and

using the fact that DxDIKi = (DIDzK{)T, we get the following identity on theunperturbed homoclinic orbit

(DIJDxH)T DxKi = -d (DIKi) . (4.1.154)

Taking the inner product of (4.1.154) with Ii'' gives

((DIJDxH)T Dxifi,ii's) = (-dt(DIKi),Il's) (4.1.155)

but

((DIJDxH)T DxKi, il's) = (DxKi, (DIJDxH)Ii'9) (4.1.156)

which gives the result.

Now, from (4.1.149), we have

M{ (B0, A (0) - Ai (0)

0 V[(DzKi,gz)1

dt[(D.Ki,g') -(DIKi), I )] dt + J - (dt (DIKK), Ii)-Tu

0 J

1,...,n (4.1.157)


where we have left out the argument of the integrand for the sake of a less cumber-

some notation. Integrating the second term in each integrand once by parts, and

using the fact that Il = Il = gI, gives

To

M!(00, a,11) = f [(DxKi,gz)+(DIKi,gI)]dt-(DIKi,Il)

-Tu

0)

-T°(DIKi, If

T'

0

+Di (-T") - Di (T8) , i = 1,... , n . (4.1.158)

As before, we want to consider the limit of (4.1.158) as Te, Tu --> oo. First we give

two preliminary lemmas.

Lemma 4.1.27. For each e sufficiently small there exists monotonely increasingsequences of real numbers {V}, j = 1, 2,..., with 1 m T"U = oo such that

1) lim JqE (Ts) _ q'(-V') = 0,2 400 . 7

2) lim I g'(q (T 3),1,; 0) I = lim IgI(gE (-T u),, ; 0) I = 0.i-00 7 j->oo 7

PROOF: 1) This follows from the fact that orbits in the stable and unstable man-

ifolds of the torus approach the torus with asymptotic phase (see Fenichel [1974],

[1979]). Then we can choose sequences of times so that q8 (t) and qE (t) approach

the same point on the torus along these sequences of times. 2) Recall that gIhas zero average on the torus (see Proposition 4.1.6). Thus, rather than choosingsequences of times such that q, '(t) and qE (t) approach any arbitrary point on thetorus, we can choose the sequences so that qE (t) and qE (t) approach a point on the

torus such that gI vanishes at that point. This uses the fact that trajectories onthe torus are dense, see proposition 4.16.

0 TLemma 4.1.28. jliirn -(DIKi,Il) T° - (DIKi,If) o] = -(DIKi('7(I),I),

f g'(q0 (t - to), µ; 0)dt), where {T }, {T' } are chosen as in Lemma 4.1.27.-00

- (DIKi(x' (-to, a), P), h(0)) + (DIKi(xI (-T - to, a), Il, Il (-Tj ))

-(DIKi(xI(V -to,a),I1,h(Tj))+(DIKi(xI(-to,a))I,Ii(0))(4.1.159)


Since Il (0) = Il (0), (4.1.159) reduces to

(DIKK (x1(-Tj -to, a), IlIj (-Tj )) - (DIKi (x7(T'-to, a), IJ)Ii(T' )). (4.1.160)

Now as -Tj -+ -oo we have

x7(-Tj - to, a) - 'Y(I) ,

and as TS - oo we have

(4.1.161)

XI (Tj - to,-) -, -Y(I) . (4.1.162)

Also, from the first Variational equation (4.1.126) we obtain

0/0

jh00 I1 (T') - Ii (-T?) = J gI (q (t - t0), /c; 0) dt (4.1.163)

-00

Therefore, using (4.1.160), (4.1.161), and (4.1.163), we have

Turn [(DIKi(x'(-Tj -to,a),I-)Il (-Tj )) - (DIKi(x7(T' -to,a),I)Ii(Tj))]00

°-(DIKi(Y(I),I), f 97(gp(t-to),µ;0)dt) . (4.1.164)

-00

So, using Lemmas 4.1.24 and 4.1.28, we obtain

00

Mi (00,a;µ) = f [(DxKi,9y)+(DIKi,9I)] (go(t-to),A;0)dt,

-0000

-(DIKi('Y(I1, I1 f 91(gp (t - to),,u; O)dt)-00i = 1,...,n. (4.1.165)

Now (4.1.165) converges absolutely, since it is just another way of writing (4.1.151)

(note: (DIKi (x7(t - t0, a), I) -DIKi (ry(I), 1)) --> 0 exponentially fast as t -' +0o


by Lemma 4.1.23, and by the fact that xI (t - to, a) --> y(I) exponentially fast ast -> +oo). However, the two terms in the integral

00

f (DIKi (x1(t - to, a), I), 9I (q0 (t - t0),µ; 0))dt (4.2.166)

-00

0/0

(DIKi('YJ 91(gp(t-to),/a;0)dt) (4.2.167)

-00

each individually only converge conditionally. This is expressed in the followingproposition.

Proposition 4.1.29. Let {T? }, {T? }, j = 1, 2,..., be chosen as in Lemma 4.1.27.Then (4.2.166) and (4.2.167) converge conditionally when the limits of integration

are allowed to approach +oo and -oo along the sequences {Tq} and {-T!'), re-spectively.

As j -> oo, the homoclinic trajectory approaches the invariant torus r5(I) ex-

ponentially fast along the sequences of times {T! J, So, by Lemma 4.1.27,

along these sequences of times gI goes to zero exponentially fast along the homo-

clinic trajectory. Recall that DIKi is assumed to be bounded on bounded subsets of

its domain of definition. Therefore, the term (DIKi, gI) goes to zero exponentially

fast on the homoclinic trajectory with the choice of sequences of times satisfyingLemma 4.1.27.

System II. The components of the Melnikov vector for System II are also given by

(4.1.151):

Mi (I, Oo, a; {t) =00 t-/ to

f [(DxKi, gx) + (DxKi, (DIJDxH) J gI)] (qp (t - to), µ; 0)dt, (4.1.168a)

-00

or, using Lemma 4.1.26 and 4.1.28,


00

Ma (1, Oo, a; u) = f-00

Co

-(DIK&y(I),I), f 9I(gp(t - to),//.;0)dt)-00

i = 1,..., n . (4.1.168b)

The absolute convergence of (4.1.168) is established using an argument identical

to that given in Proposition 4.1.25 (note: recall that I is an m vector of angularvariables for System II).

System III. Substituting go = JDxH and gl = -D6H into (4.1.151) gives

Mi (00, a; h) _00 t-tof [(DzKi, JDxH) - (DxKi, (DIJDxH) f D o (qp (t - to), A; 0)dt,

-00(4.1.169a)

or, using Lemma 4.1.26 and 4.1.28,00

M1 (Bo, a; it) = f [(DxKi, JDxH) - (DIKi, D9H)] (qp(t - to),.u; 0)dt-00

00

+(DIKi(0'(I-),I), f D9H(go(t-to),lp;0)dt)-00

(4.1.169b)

i = 2,...,n

for the first n - 1 components of the Melnikov vector. Absolute convergence of(4.1.169) is an immediate result of Proposition 4.1.25. The remaining m components

of the Melnikov vector require a more careful consideration.

From (4.1.141) and (4.1.142) we have

Mi (B0, a;.u) = Da (0) - Di (0)T°

_- f Dga_n I(qp(t-t0),µ;0)dt+Di(-T")-Di(T3),

[(DzKi, g x) + (DIKi, gl)] (qp (t - to),.u; 0) dt

-Tui=n+1,...,n+m. (4.1.170)


Now we want to consider the limit of (4.1.170) as T8, Tu -, oo.

Lemma 4.1.30. For {T1}, {T'} chosen according to Lemma 4.1.27,

lim Ii A (T? )j--oo

=0, i=n+1,...,n+m.

PROOF: From (4.1.128) we have

Oi(-Tj) - i(Ts)-(I1(-Tj))i-n-(Ij(Ts))i-n, i=n+1,...,n+m.(4.1.171)

Since the I variables were chosen to lie in the unstable and stable manifolds of the

KAM torus re(I) the lemma is an immediate consequence of Lemma 4.1.27.

So we have

00

Mt (Bo, «; JL) f ne;_nH(go (t - to),.u; 0)dt ,-00

i=n+1,...,n+m. (4.1.172)

Proposition 4.1.29 applies directly to (4.1.172) and allows us to conclude that(4.1.172) converges conditionally.

iv) The Choice of SP,, and Sp,

Recall our discussions of the splitting of the manifolds for the three systems. In

the unperturbed systems, for each p E I, W3(M) and Wu(M) intersect the planeHp transversely in the coincident m dimensional surfaces Sp and Sp , respectively.

Therefore, in the perturbed system, for c sufficiently small, W8(Me) and

intersect the plane Hp transversely in the m dimensional surfaces Sp, and Sple,respectively, for each p E r. However, it is possible for W8(ME) and Wu(Me) tointersect HP in countably many disconnected components, see Figure 4.1.5.

In our construction of the measurement of distance between the stable andunstable manifolds of the invariant torus we chose points in SPie and that were

defined to be the components of W3(Me) n Hp and Wu(Me) fl Hp closest toMe in terms of positive and negative time of flow along W3(Me) andrespectively. Before proceeding, let us define these sets more precisely.


Definition 4.1.1. Let qE(t) be any trajectory in W$(ME) with qE(0) E Sp,E C

W9(ME) n llp. Then Sp,E is said to be the component of l llp closest toME in the sense of positive time of flow along W9(Me) if for all t > 0 gE(t)f1IIp = 0.

A similar definition holds for SP,E in negative time with the obvious modifications.

Now we want to argue that the procedure utilized in deriving the computable

form of the Melnikov vector given in (4.1.97), (4.1.99), (4.1.101), and (4.1.102)results in the Melnikov vector being a measure of the distance between points in

the stable and unstable manifolds of the invariant torus which satisfy Definition4.1.1.

For a fixed p = (xI (-to, a), I, 6o) E r, let SpIE and be the components of

WS(ME) n lip and WL(ME) n llp which are closest to ME in the sense of positive

and negative time of flow along W8(ME) and respectively. Let Sp,E and

sP,E denote additional components of W8(Me)flllp and W"(ME)flllp. Recall that

llp is defined to be the span of { (DzKK (xI (-to, a), I), 0) }, i = 1,. .. , n, and {I{},

i = 1, ... , m. We will denote the time varying plane used in the construction ofthe time dependent Melnikov vector as Ilp(t), which is the span of the time varying

vectors { (DZKi(xI (t - to, a), I), 0) }, i = 1, ... , n, and the constant vectors {Ii},i = 1,.. ,m.

Now consider the expression D$ (t) defined in (4.1.128). A (0) represents the

0 (e) term of the projection of the point pE _ (xE, IE) in the stable manifold of the

invariant torus along the ith coordinate on Hp. Di (t) represents the evolution ofA (0) on the time interval [0, oo) with the plane llp(t) evolving along a trajectory in

the unperturbed homoclinic orbit r and the point p' (t) evolving along a trajectory

in the perturbed stable manifold of the invariant torus. Now suppose that at t = 0pE(0) pE is contained in rather than S. Then, by Definition 4.1.1, thereexists some T > 0 such that pE(T) E Sp,E C IIp. But, in this case, the plane Ilphas moved to 11p(T) Hp. Therefore, A (T) does not approximate to O(E) theprojection onto the ith coordinate of 1Ip(T) of a point in the stable manifold of theinvariant torus which is contained in Ws(ME) fl llp(T). So we see that the onlyway in which A (t) can be defined for all t E [0, oo) is if the point in the stablemanifold of the torus is contained in SP(t)

Eas defined in Definition 4.1.1 . A similar

argument follows for Ai (t) on the time interval (-oo, 0]. See Figure 4.1.17 for an

illustration of the geometry behind this argument.


E

Figure 4.1.17. Geometry of Di (0-

v) Elimination of to

We now discuss how the parameter to can be eliminated as an independent vari-able of the Melnikov vector. Recall that the unperturbed homoclinic orbit canbe parametrized by (to, a, I, Bo) where I is fixed in Systems I and III. Varying(to, n, I, Bo) corresponds to moving along the unperturbed homoclinic orbit andmeasuring the distance between the perturbed stable and unstable manifolds ofthe invariant torus at each point. By uniqueness of solutions, if the stable andunstable manifolds intersect at a point, they must intersect along (at least) a onedimensional orbit (note: the intersection could be higher dimensional if there aresymmetries in the system). Therefore, since the Melnikov vector measures the dis-

tance between trajectories in the stable and unstable manifolds, then one zero ofthe Melnikov vector should imply the existence of a one parameter family of zeros,

with this parameter being redundant in order to determine whether the stable andunstable manifolds intersect. Now we discuss how this geometric fact is manifested


mathematically in the Melnikov vector.

The arguments of the integrands defining the components of the Melnikov

vector for Systems I, II, and III were

t(xI(t - to, a), I, f 1l (x1(s, a), I) ds + oo, Fc; O) . (4.1.173)

If we make the change of variables t - t + to, the limits of integration of theintegrals defining the components of the Melnikov vector do not change; however,

the arguments of the integrands become

t+to

(XI (t, a), I, J St (x' (s, a), I) ds + Bo, µ; 0) . (4.1.174)

Now to still appears explicitly in the argument of the integrands. However, notethat all functions are periodic in each component of the 0 variable, so for any fixed

component of the 0 variable, say the ith component, varying 8{o is equivalent tovarying to. Therefore we may consider to as fixed, and, for convenience, we take

to = 0. With this choice we arrive at the form of the components of the Melnikovvector given in (4.1.97), (4.1.99), (4.1.101), and (4.1.102). Note that it would beequivalent to let to vary and fix any one component of 00. We will interpret thisgeometrically in terms of a Poincare map in Section 4.1e.

4.1e. Reduction to a Poincare Map

The theorems regarding the dynamical consequences of the intersection of the stable

and unstable manifolds of normally hyperbolic l-tori (l > 1) were stated in thecontext of maps; for l > 1 we had Theorem 3.4.1. The techniques of this chapterwere developed in terms of vector fields; however, it is a simple matter to reduce

the study of Systems I, II, and III to the study of a local Poincare map defined ina neighborhood of r.

A local cross-section E of the phase space is constructed by fixing any one com-

ponent of the angular variables whose time derivative is nonzero (in the perturbed

system) in a neighborhood of 1. Then the Poincare map associates points on thecross-section E with their first return to E under the action of the flow generated by

the perturbed vector field. We remark that, in applying the theorems of Chapter3 to specific problems, it is not so important that the Poincare map is actuallyconstructed, but merely that it can be constructed.


Now we want to describe the elimination of the parameter to or any one com-

ponent of Bp in the argument of the Melnikov vector in the context of the Poincare

map.

Recall that the homoclinic orbit r can be parametrized by the n + m + Iparameters (to, a, I, 0) (note: for System III, 1 = m). Hence, the intersection ofI' with the cross-section can be described by n + m + 1 - 1 parameters where oneangular variable has been fixed corresponding to the one which defines E. So, inthis case, the elimination of the angular variable defining the cross-section from the

argument of the Melnikov vector leads us to the interpretation that the Melnikov

vector is restricted to the cross-section and measures the distance between the stable

and unstable manifolds of an invariant torus of the Poincare map. The elimination

of to from the argument of the Melnikov vector (i.e., setting to = 0) could then be

thought of as measuring the distance between the stable and unstable manifolds of

the invariant torus of the Poincare map only along the a, I, and I - 1 of the Bodirections, and then varying the cross-section E. Mathematically, both points ofview are equivalent.

4.2. Examples

We now give a variety of examples which will serve to illustrate the theory developed

in Section 4.1.

4.2a. Periodically Forced Single Degree of Freedom Systems

We give two examples of the simple pendulum subjected to time periodic external

forcing. The first example involves a forcing function having 0 (e) amplitude and0(1) frequency and is an example of System I. The second example involves aforcing function having 0 (1) amplitude but 0 (e) frequency and is an example ofSystem II. More details on these examples can be found in Wiggins [1988].

4.2. Examples 419

1) The Pendulum: Parametrically Forced at O (c) Amplitude,0 (1) Frequency

We consider a simple planar pendulum whose base is subjected to a vertical, periodic

excitation given by Ey sin ftt, where c is regarded as small and fixed. The equation

of motion for the system is given by

it + soil + (1 - El sin fZt) sin x1 = 0 (4.2.1)

where x1 represents the angular displacement from the vertical and 6 representsdamping. See Figure 4.2.1 for an illustration of the geometry.

Figure 4.2.1. The Simple Pendulum.

Writing (4.2.1) as a first order system of equations gives

x1

x2

8

= x2

= -sinxl + E[-ysin0sinxl

= fl

- 6x21 (xl, x2, B) E T1 x R1 x T1 . (4.2.2)

The unperturbed system is given by

(4.2.3)


and it should be clear that the (xl, x2) component of (4.2.3) is Hamiltonian with a

Hamiltonian function given by

x2H = 2 -cosxl (4.2.4)

The (xl, x2) component of (4.2.3) has a hyperbolic fixed point given by

(x1,-t2) = (-,0) = (-7r,0) . (4.2.5)

Thus, when viewed in the full x1 - x2 - 0 phase space, (4.2.3) has a hyperbolicperiodic orbit given by

M = (x1,.2,B(t)) = (7r,0,flt+00) = Hr,0,flt+00) . (4.2.6)

This hyperbolic periodic orbit is connected to itself by a pair of homoclinic trajec-tories given by

(xih(t),x2h(t),0(t)) = (+2 sin- 1(tanht),±2secht,lit+00) (4.2.7)

where "+" refers to the homoclinic trajectory with x2 > 0 and "-" refers tothe homoclinic trajectory with x2 < 0. We remark that the (xl, x2) componentof (4.2.7) can be found by solving for the level curve of the Hamiltonian given by

H = 1. So the periodic orbit is connected to itself by a pair of two dimensionalhomoclinic orbits, and from 4.1a, i) these homoclinic orbits, denoted re, can beparametrized as

rt = { (x h(-to), x h(-to), Bo) E T1 x R1 x Tl I (to, 00) E R1 x T1 } . (4.2.8)

So, for any fixed (to, 00) E IR1xT1, p± = (xlh(-to), x2h(-to), Bo) denotes uniquepoints on r± and, in this case, the plane Hpt is one dimensional and is the spanof the vector

(DzH(xlh(-to),x2h(-to)),0) = (sinxlh(-to),x h(-t0),0) (4.2.9)

which intersects r± transversely at each p± E rf, i.e., for all (to, 00) E Rl x T1.

See Figure 4.2.2 for an illustration of the geometry of the unperturbed phase space.

4.2. Examples 421

identify

X,=-7f

LI/IX12

Xj

Xq=7r

Figure 4.2.2. Geometry of the Unperturbed Phase Space.

It should be clear that r± = Ws(M) nWI(M) - M.

With the geometry of the unperturbed phase space described, we now ask what

becomes of this degenerate homoclinic structure fore # 0. By Proposition 4.1.5 we

know that the hyperbolic periodic orbit persists, which we denote as Me, and itslocal stable and unstable manifolds, which are denoted Wloc(Me) and Wloc(ME),respectively, are Cr close to Wloc(M) and Wloc(M), respectively. We would like to

determine if W8(Me) and W'L(Me) intersect transversely for, if this is the case, then

we can appeal to the Smale-Birkhoff homoclinic theorem to assert the existence of

horseshoes and their attendant chaotic dynamics in our system.

Recall that fore sufficiently small, for each point p± E r±, Ws(ME) andintersect II ± transversely in the points $st and $ f , respectively.

P P P 1e

This is because ITP± intersects r± transversely for c = 0, and the manifolds varysmoothly with E. See Figure 4.2.3 for an illustration of the geometry of the perturbed

phase space.


7e=Oo+2ir

identify

Figure 4.2.3. Geometry of the Perturbed Phase Space.

Now the distance between W8(ME) and W'i(ME) at the point p± E r± hasbeen shown to be

d±(to, 001 b,'Y, 0) = EM± (to, 00, 6, y, it)

+ 0 (E2) (4.2.10)II DxH (x± (-to), x2h (-t0)) II

where

11 DxH(xlh(-to), x2h(-to)) I =

[D,1H(xlh(-to),x h(-to))]2+[Dx2H(xlh(-to),x2h(-t0))]2 (4.2.11)

and by (4.1.47)0/0

M±(to, Oo, b, ry, Sl) = J{-6 [x2h(t-to)]2+^yx2h(t-to) sinxlh(t-to) sin(stt+0o)}dt.-00

Substituting (4.2.7) into the integral (4.2.12) gives

M+(to, 00, 6,'Y, Il) = M (to, 0o, b,'Y, fl) = M(to, 0o, b, -Y, ci)

= -86+ 277rSZ2cos(f1t0 + 00) .

sinh

(4.2.12)

(4.2.13)

4.2. Examples 423

Before proceeding with an analysis of the Melnikov function and a discussion

of its dynamical implications, let us make some remarks concerning to and 00 in(4.2.13) (cf. 4.1d and 4.1e). Notice that varying either to or 00 in (4.2.13) hasthe same effect; thus, we can view one or the other as fixed. Geometrically, fix-ing 00 corresponds to fixing a cross-section Ego of the phase space (cf. Section

1.6) and considering the associated Poincare map. Then, varying to correspondsto moving along the unperturbed homoclinic orbit r and measuring the distancebetween the perturbed stable and unstable manifolds of the hyperbolic fixed point

of the Poincare map. Alternatively, fixing to and varying 00 corresponds to fixing

a point on EBo and then varying the cross-section Ego. Either point of view ismathematically equivalent as we showed in 4.1.d and 4.1.e.

Using (4.2.13), along with Theorems 4.1.9 and 4.1.10, we can show that for e

sufficiently small there exists a surface in the r - b - [I parameter space given by

ery = 4eb sinh 2 + O(c2) (4.2.14)

above which transverse intersections of the stable and unstable manifolds of thehyperbolic periodic orbit occur.

In order to more easily present the information obtained in equation (4.2.14),

we give two graphs that give the shape of the curves defined by (4.2.14) whenone of the parameters is fixed, one in -y - b space with fl # 0 fixed and theother in -y - b space with S # 0 fixed. In each case, the curves in Figures 4.2.4aand 4.2.4b are such that quadratic homoclinic tangencies occur on the curves, and

transverse homoclinic orbits occur above the curves (note: see Guckenheimer and

Holmes [19831 for bifurcation theorems concerning quadratic homoclinic tangencies).

Notice in Figure 4.2.4b that, along the bifurcation curve as 11 -+ 0, it appears that

-y -+ oo. Of course -y cannot become too large, in which case we would be outside

the range of validity of the theory. Thus, we have no information about the lowfrequency limit; however, from these results we might expect that the amplitude of

the excitation must become large in order for transverse homoclinic orbits to exist.

Our next example will verify this conjecture.

Now our results show that (4.2.2) contains transverse homoclinic orbits to a

hyperbolic periodic orbit. So by the Smale-Birkhoff homoclinic theorem, (4.2.2)

contains an invariant Cantor set on which the dynamics can be described symbol-

ically via the techniques in Section 2.2. However, we want to go a bit further and


EY

V(a)

EY

(b)

ES

n

Figure 4.2.4. a) Graph of (4.2.14), Il 0 fixed.

b) Graph of (4.2.14), b 0 fixed.

describe the dynamical implications of the transverse homoclinic orbits in terms of

the oscillatory motions of the pendulum. For this purpose it will be useful to deform

the pair of homoclinic orbits into a shape which is more amenable to our geometric

arguments. Consider Figure 4.2.5; in 4.2.5a we show the pair of homoclinic orbits

in the unperturbed system on the cylinder. The + sign refers to the upper homo-clinic orbit on which the corresponding motion of the pendulum is clockwise, and

the - sign refers to the lower homoclinic orbit on which the corresponding motion

of the pendulum is counterclockwise. In 4.2.5b imagine that the pair of homoclinic

orbits have been slipped off the cylinder and flattened out in the plane in 4.2.5c.Figure 4.2.5d just represents a convenient rotation and deformation of 4.2.5c.

Now let us consider the time Poincare map of the perturbed system denoted

4.2. Examples 425

x 0 37r2

cX2n

a) b) c) d)

Figure 4.2.5. Geometry of the Unperturbed Homoclinic Orbits.

by P. In this case, the map has a hyperbolic fixed point whose stable and unstable

manifolds may intersect transversely to give the familiar homoclinic tangle shown

in Figure 4.2.6.

P2(H-)

Figure 4.2.6. The Formation of the Horseshoe.

P2 (H+)

Notice the "horizontal" slabs H+ and H_ in Figure 4.2.6. Under P4 H+


and H_ are mapped back over themselves in the "vertical" slabs P4(H+) = V+and P4(H_) = V_, with points in H+ corresponding to clockwise motions ofthe pendulum and points in H_ corresponding to counterclockwise motions of the

pendulum. Now, suppose we have shown that p4 satisfies the conditions Al and A2

or Al and A3 of Section 2.3 on H+ and H_ (where H+ and H_ are appropriatelychosen). Then Theorem 2.3.3 or Theorem 2.3.5 allows us to conclude that, given any

bi-infinite sequence of +'s and -'s, where + corresponds to a clockwise rotation and

- corresponds to a counterclockwise rotation, the pendulum exhibits such a motion.

In a similar manner, the abstract results of Proposition 2.2.7 can be interpreteddirectly in terms of clockwise or counterclockwise rotations of the pendulum. The

verification of the conditions of Theorem 2.3.3 and 2.3.5 we leave as an exerciseto the reader since they are similar to examples in Chapter 3. Also, see Holmes

and Marsden [1982a] for an estimate of the number of iterates of the Poincare map

which are necessary to form the horseshoe in terms of the perturbation parameter

E and the Melnikov function.

ii) The Pendulum: Parametrically Forced at 0(1) Amplitude,0 (E) Frequency

We consider a similar, parametrically forced pendulum as in 4.2a, i), but withthe base subjected to a vertical, periodic excitation given by ry sin EStt, where E is

regarded as small and fixed. The equation of motion for this system is given by

xl + Ebil + (1 - y sin Eflt) sin zl = 0 (4.2.15)

where x1 represents the angular displacement from the vertical and 6 representsdamping. See Figure 4.2.1 for an illustration of the geometry. Writing (4.2.15) as

a system gives

zl = x2i2 = -(1 - -1 sin l) sin z1 - E6z2

I =Eli(zl, z2, I) E T1 x IRl x T1 . (4.2.16)


:ii = z2

i2 = -(1 - ysinI)sinx1 (4.2.17)

I =0

4.2. Examples 427

and it is easily seen that (4.2.17) has the form of a 1-parameter family of Hamiltonian

systems with Hamiltonian given by

2H(x1,x2,I)= 2 sinI)cosxl. (4.2.18)

The unperturbed system has a fixed point at

(x1,22) = (x,0) = (-'7r,0) (4.2.19)

for each I E [0, 27r) which is hyperbolic provided

0<-y<1. (4.2.20)

Henceforth, we will always assume that (4.2.20) is satisfied. In the full x1 - x2 - I

phase space we can view

M = (x1, .2i I) = (7r, 0, I) = (-7r, 0, I) , I E [0, 27r) (4.2.21)

as a periodic orbit. Two homoclinic trajectories which connect M to itself are given

by

(xlh(t),x2h(t)'I) _(±2 sin-1 [tanh 1 - -y sin I t], ±2 1 - 7 sin I sech 1 - y sin l t, I) (4.2.22)

where "+" refers to the homoclinic trajectory with x2 > 0 and "-" refers to thehomoclinic trajectory with x2 < 0. Thus M is connected to itself by a pair of twodimensional homoclinic orbits, denoted r±, which can be parametrized by

r:': _ {(x h(-to),x h(-to),I) E T1XER1xT1 I (to, 1) E 1R1xT1} . (4.2.23)

This system is therefore an example of System II with n = 1, m = 1, 1 = 0. Weremark that the unperturbed phase space of (4.2.17) is much the same as (4.2.3),which is sketched in Figure 4.2.2. The difference lies in the fact that x1 - x2coordinates of r± do not depend on the angular variable 00 in (4.2.3), but they do

depend on the angular variable I in (4.2.17).

We now consider the perturbed system (4.2.16). By Proposition 4.1.12, Mpersists (denoted by Me) as a periodic orbit of period . We want to determine


the behavior of the stable and unstable manifolds of ME. From 4.1b, v), the distance

between the manifolds is given by the scalar function

tEI DyH(M±(It5)yzh)(-to))II +0(E2) (4.2.24)

where

II DzH(xh(-to), xhl (-to)) I1 _[Dx1H(xlh(-to),x2h(_to))] 2 [Dx2H(xlh(-to),x2h(-to))]2. (4.2.25)

From (4.1.69), the Melnikov function is given by

fM±(I;6,,s2) = [-5(x2h(t))2+ Int(cosI)x2h(t)sinxlh(t)] dt. (4.2.26)

-00

Substituting (4.2.22) into (4.2.26) we obtain

M = M (I; 6, y, f2) =- M(I; 6, y, f2) _ 8b,11 - y sin I +4y12 cos I

1-rysinl(4.2.27)

Using (4.2.27) and Theorems 4.1.13 and 4.1.14 we obtain, after some algebra, an

equation whose graph in (6, y, f2) space is a surface above which transverse homo-

clinic orbits occur. This equation is given by

26

y= +O(E).1 + r-12

(4.2.28)

As in the previous example, we will present two graphs representing the shapes ofthe curves obtained from (4.2.28) when one of the parameters is viewed as fixed,

one in y - 6 space with n 0 0 fixed and the other in y - fl space with b # 0fixed. In each case, the curves in Figure 4.2.7 are such that quadratic homoclinic

tangencies occur on the curves, and transverse homoclinic orbits occur above the

curves. Note that our theory is not valid for y = 1, since in this case some of thefixed points in the unperturbed system are nonhyperbolic, and that would violate114.

The dynamical consequences of the transverse homoclinic orbits can be in-terpreted in terms of symbolic dynamics in a manner virtually identical to thatdescribed at the end of the previous example.

4.2. Examples 429

Y

I

(a)

I

Y

-E8

-------------

ER

(b)

Figure 4.2.7. a) Graph of (4.2.28), fl # 0 Fixed.b) Graph of (4.2.28), 6 # 0 Fixed.

4.2b. Slowly Varying Oscillators

We will now give two examples of System I that have the structure of a periodically

forced, single degree of freedom, nonlinear oscillator containing a parameter that

obeys a slow (0 (e)) first order ordinary differential equation.

The first example we shall consider arises from a class of third order nonau-tonomous systems proposed by Holmes and Moon [1983] to model certain feedback-

controlled mechanical devices. Imagine a mechanical device with multiple equilib-

rium positions in the absence of feedback. A controller is added to move the system

from one equilibrium position to another. A possible model for such systems with


first order feedback is

fi + 6z + k(x)x = -z + F(t)

Z + Ez = EG[x - xr(t)](4.2.29)

Equation (4.2.29) represents a mechanical oscillator with linear damping 6, non-linear spring constant k(x), and a linear feedback loop with a time constant 1/cand gain parameter G. F(t) represents an external force, and xr(t) represents thedesired position history of the device. We will present a specific example of (4.2.29),

which consists of the Duffing oscillator with a first order linear feedback loop. More

details on this example can be found in Wiggins and Holmes [1987].

The second example we will consider is that of a pendulum attached to arotating frame. This is an example of a class of systems which frequently arise inrotational dynamics, and it exhibits a very rich homoclinic structure. More details

of this example can be found in Shaw and Wiggins [1988].

i) The Duffing Oscillator with Weak Feedback Control

We consider the following system

xl = x2

y2=XI -xi-I-cSx2I = e(ryxl - aI + /3 cos 0)

9 =1

(x1,x2,I,0) E 1R1 xlR1 xIt1 xTl (4.2.30)

where a, /i, -y, and 6 are parameters, and a is small and fixed. The unperturbedsystem is given by

xl = x2

3±2=xl-xl-I(4.2.31)I =0

B =1and the xl - x2 component of (4.2.31) has the form of a 1-parameter family ofHamiltonian systems with Hamiltonian function given by

2 2 4

H(xl, X2; I) = 2 -

2+ 4 + Ixl . (4.2.32)

We now want to describe the geometrical structure of the unperturbed phase space.

4.2. Examples 431

Fixed Points. Fixed points of the x1 - x2 - I component of (4.2.31) are given by

(xl(I),0,I)

where xl(I) is a solution ofxi-x1+I=0.

(4.2.33)

(4.2.34)

For I E (-3

2 3 ,3

2 _3), (4.2.34) has three solutions, with the intermediate root

corresponding to a hyperbolic fixed point. For I >32 and I< - 32 there

exists only one solution of (4.2.34) corresponding to an elliptic fixed point and, for

I = f3

2 3 , (4.2.34) has two solutions corresponding to an elliptic fixed point and a

saddle-node fixed point. See Figure 4.2.8 for an illustration of the graph of (4.2.34).

We will only be interested in the hyperbolic fixed points.

Figure 4.2.8. Graph of (4.2.34).

Homoclinic Orbits. We denote the 1-manifold of hyperbolic fixed points of thexl - x2 - I component of (4.2.31) by

-Y(I) = (xl (I), 0, I) (4.2.35)

where x1 (I) is the intermediate solution of (4.2.34) for I E (-3

2 3, Each of

these fixed points is connected to itself by a pair of homoclinic orbits which satisfy


[2 - Z + 4 +Ixij - I -x121)2+ x141)4 +Ixl(I)1 =0. (4.2.36)

Thus, the phase space of (4.2.31) appears as in Figure 4.2.9.

I

Figure 4.2.9. Unperturbed Phase Space of (4.2.31).

So, in the context of the full xl - x2 - I - 0 phase space, (4.2.31) has a twodimensional normally hyperbolic invariant manifold with boundary

2 2(y(I), 0 ) , I E (--, - ) , 0 E [0, 2r) (4.2.37)

3,/3 3 V43

and M has three dimensional stable and unstable manifolds which coincide. There-

fore, (4.2.30) is an example of System I with n = m = I = 1.

We now turn our attention to the perturbed system (4.2.30). By Proposition4.1.5, we know that M persists an an invariant manifold Me, which we denote

Me=('Y(I)+0(e),0), 00 E[0,2,r). (4.2.38)

The procedure is to determine if Me contains any periodic orbits by using Proposi-

tion 4.1.6, and then to determine whether or not the stable and unstable manifolds

of these periodic orbits intersect by computing the appropriate Melnikov integral.

The Flow on Me. The perturbed vector field restricted to Me is given by

I = e [ryxl (I) - aI + cos 01 + 0 (e2)1 E - 2 ,

2(4.2.39)9-1 (

3 33,)

4.2. Examples 433

The averaged vector field is

277r

I = e f[xii) - al +,6cos01 dO27r (4.2.40)

0

=E['yxl(I)-aI] .

Fixed points of (4.2.40) must satisfy

I=axl(I) (4.2.41)

Using (4.2.41), along with the fact that xl(I) must satisfy (4.2.34), gives the fol-lowing expression for fixed points of (4.2.40):

I=0, fry 1-ry. (4.2.42)a a

So, for a < 1, (4.2.40) has three fixed points and, for > 1, (4.2.40) has one

fixed point (note: a pitchfork bifurcation is said to have occurred at « 1, see

Guckenheimer and Holmes [1983]). See figure 4.2.10.

We next want to calculate the nature of the stability of these fixed points ofthe averaged equations. This is given by the sign of

d,E(yxl(I) - aI) (4.2.43)

The fixed point is unstable (on ME) if (4.2.43) is positive and stable if (4.2.43) is

negative. A simple calculation shows that

fE(ry - a) for I = 0

[E(7xl (I) - a7) _ -2E(" - a) 7di for?=f- 1--.3«-2 a a(4.2.44)

Thus, I = 0 is stable for ry < a, unstable for ry > a, and I = f a 1 - are

unstable for < ry < a.By Proposition 4.1.6, these fixed points of the averaged equations (4.2.40)

correspond to periodic orbits of (4.2.39) of period 27r having the same stability type

as the fixed points of the averaged equations. If we consider a three dimensionalPoincare map formed from (4.2.30) by fixing 0 = 0 with the map taking points


(x1, x2, I) to their image under the perturbed flow after a time 27r (see Section 1.6),

then the periodic orbits on ME become fixed points of the Poincare map.

The Structure of the Poincare Map. Using our knowledge of the flow on ME as well

as the structure of the unperturbed phase space, we see that the Poincare map has

the following structure.

ry> a. Hyperbolic fixed point at (x1,x2iI) = (0,0,0) having a two dimensionalunstable manifold and a one dimensional stable manifold.

23 < ry < a. Three hyperbolic fixed points at (x1, x2, I) = (0,0,0) and

(f 1 - , 0, f 1 - ) , where (0, 0, 0) has a two dimensional stable manifold

and a one dimensional unstable manifold, and ( ± V 1- 0, f a 1 - have two

dimensional unstable manifolds and one dimensional stable manifolds.

ry < 2. Hyperbolic fixed point at (x1, x2,I) = (0,0,0) having a two dimensionalstable manifold and a one dimensional unstable manifold.

See Figure 4.2.10 for an illustration of the geometry of the Poincare map.

2a

3<ry<a


Calculation of the Melnikov Integrals. Next we calculate the Melnikov integrals.

These will give us sufficient conditions for the stable and unstable manifolds of the

hyperbolic fixed points of the Poincare map to intersect transversely.

4.2. Examples 435

From (4.1.47), the Melnikov integral is given by

eo

M1(90, a, Ij,'Y, 5) = f [(DzH, gz,\+ (DIH, g1)](zI (t), I, t + to) dt

-0O00

-(DIH(x(I), 0, I), f gI (xi (t), I, t + todt)-oo

0"f [_b(x2 (t)) 2 + 7 (xi(t)) 2 - alx1(t) + axf (t) cos(t + to)] dt

-0000

-xl ( I D [ xi (t) - aI +,3 cos(t + to)] dt-oo

(4.2.45)

where xI (t) = (xi(t), x4(t)) is a homoclinic trajectory of the unperturbed system

on the I=! level corresponding to the hyperbolic fixed point of the averagedvector field on ME. From (4.2.36) these homoclinic trajectories are found to be

r=0. x+(t) = (x (t)) = (+v secht,+f secht tanht)(t), X (4.2.46)2i

7=-a 1-7/a._ f 2cS + ab -2ad3ST

+ tx ( ) - 2bS-a' (2bS-a)2)3

(4.2.46)iST_ (2cS - ab 2ad- tx ( ) - \ 2bS+a' (2bS+a)2)

I=+ 1-rya.

x+(t) = r 2cS - ab 2ad3ST2bS - a ' (2bS - a) 2 )

(4.2.46) u

where

2cS + ab -2ad3ST2bS + a ' (2bS + a) 2

a=1/2y, b=,/1-2, c=1-2-Y, d=1/3-' -2V a a a

S = sech (dt), T = tanh(dt)


where the subscripts "u" and "1" refer to the "upper" homoclinic orbits on the I =

('y/a) 1 - y/a level and the "lower" homoclinic orbits on the 7=-(ry/a) 1-ryla

level.

Substituting (4.2.26), (4.2.26) u, and (4.2.26)1 into (4.2.45) and computing the

integrals gives

I = 0. M+(to, a, 0, -y, b) = 3 ba + 4,y ± y97rsech 2 cos to (4.2.47)±

I=-a 1-y/a.

M (to, a, 8, y, b) = -4b [d

+7b

3 fa 2+21[2d - 2fb(2 ±sin-11/gab)]

V ry

sinh(1 sin-1 ryd)+ 2N/2-7r#. sinh 'dd cos to

f sin-1 2ab)17

)JJ

I=+ 1

Mu (to, a, 6) _ -4b [3 + 7b ( f sin 12ab

fa 2 7

+2ry[2d-fb2 fsin-' 2ab)1

sinh(l sin-1

o'd)f 2 f7r/? cos tosinh d

(4.2.47)1

(4.2.47) u

where, on the I = ±a-%/l - -y/a levels, the superscript "+" refers to the largerhomoclinic loop.

We present graphs of (4.2.47)±, (4.2.47)±, and (4.2.47)1 in Figure 4.2.11 for

a = 1, ,0 = 1. In the region bounded by (4.2.47)u and (4.2.47)1 , the stable andunstable manifolds of the hyperbolic fixed point on the I = ±- 20 - ry/a corre-sponding to the small homoclinic orbit intersect transversely; in the region bounded

by (4.2.47) u and (4.2.47) j , the stable and unstable manifolds of the hyperbolic fixed

point on the I = f a 1 - y/a levels corresponding to the larger homoclinic orbit

intersect transversely; and in the region bounded by (4.2.47) + and (4.2.47) -, both

4.2. Examples 437

S

5.0

3.0

1.0

b-1.0

0.60-3.0

I

I

0.70

I

I0.80

I

0.90

Figure 4.2.11. Regions where Horseshoes Exist.

1.00

branches of the stable and unstable manifolds of the hyperbolic fixed point on the

I = 0 level intersect transversely.

In Figure 4.2.12, we illustrate the behavior of the stable and unstable manifolds

of the Poincare map for the four different parameter values indicated in Figure4.2.11.

Now let us interpret our results in the context of feedback control systems.There are two aspects which we want to consider: 1) modification of the region of

chaos in parameter space by the feedback loop, and 2) introduction of chaos by the

feedback loop. We emphasize that, in this example, chaos means Smale horseshoes.

We consider each aspect individually.

1. Modification of Chaos Via Feedback. Let us consider the situation of the gain,

-y, going to zero. In this case, the x1 - x2 component of the vector field decouples

from the I component. Thus, I can be solved for as an explicit function of time


(a)

(c)

Figure 4.2.12. Poincar6 Maps.

(b)

(d)

which is asymptotically periodic (I - eQ sin t + O (62) as t ---f oo), and the solution

can be substituted into the x1 - x2 components of the equation. The result is anequation for a periodically forced Dulling oscillator

xl = x2

x2 = x1 - x3 - 6[6x2 + 0 sin t] + 0 (e2) . (4.2.48)

Equation (4.2.48) has been studied in great detail by Holmes [1979] and Greenspan

and Holmes [1983], and the original Melnikov (1963] method gives a curve in Q - 6

space above which transverse homoclinic orbits to a hyperbolic periodic orbit exist.

4.2. Examples 439

This curve is given by46

r cosh(ir/2)3 f (4.2.49)

From (4.2.47) we see that a similar curve, above which there exist transverse ho-

moclinic orbits to a hyperbolic periodic orbit on the I = 0 level in the presence of

nonzero gain 7, is given by

Q=(4b 471 al I cosh .

22xThese curves are shown in Figure 4.2.13.

N

Figure 4.2.13. Graphs of (4.2.49) and (4.2.50).

(4.2.50)

Thus, we see from Figure 4.2.13 that the effect of the gain is to lower theboundary, and hence increase the area of the region in /6 - 6 space in which Smale

horseshoes are present in the dynamics of (4.2.50)-

2. Introduction of Chaos Via Feedback. The fact that the feedback loop has intro-duced chaos into the system is evident from Figure 4.2.11. The fixed points andhorseshoes on the I = f a 1 - 7/a levels are there solely as a result of thefeedback.


ii) The Whirling Pendulum

The whirling pendulum is shown in Figure 4.2.14. It consists of a rigid frame, that

freely rotates about a vertical axis and to which a planar pendulum is attached,the pivot being on the vertical axis. The behavior of this system is well knownif the frame rotation rate, 0, is held at a constant value, say fl. Below a critical11, the pendulum behaves essentially like a nonrotating pendulum; it has a stable

equilibrium at = 0 and an unstable one at 4 _ vr. Above the critical fl value,= 0 becomes unstable, and two new equilibria appear at _ = f cos-1(IC22)

.

As S2 -> oo, q --+ ±7r/2 as expected.

If one were to add small dissipation at the pendulum pivot and allow a small

periodic variation in B, i.e., set 0 = 1 + eIl cos(wt) (0 < c « 1), the system wouldbecome a forced planar oscillator, and the usual planar Melnikov [1963] analysiscould be used to predict the onset of chaotic motions. This type of perturbation is

a limiting case (see Shaw and Wiggins [1988]) of our more general system in which

B is allowed to vary in accordance with the equation which governs the behavior of

the angular momentum of the system.

The system considered here has "one and a half" degrees of freedom. Therotation of the frame is coupled to the motion of the pendulum via an angularmomentum relationship. The orientation of the frame, measured by the variable 0,

does not appear in the unperturbed equations of motion. In a Hamiltonian formu-

lation, one immediately obtains two constants of motion in the unperturbed case:

the energy and the conjugate momentum associated with 0; hence, this system iscompletely integrable. Upon the addition of small perturbations, the angular mo-

mentum and the energy will both vary slowly in time, and this variation affects the

occurrence of chaotic motions. These results should be of interest to experimen-talists, since often in rotating systems one can specify the applied torques but not

necessarily the rotation speed itself.

In dimensionless form (see Shaw and Wiggins [1988] for details), the equations

of motion are given by

=PO (4.2.51a)

ro = sin-1 + p8 cos 01(µ + sin2 0)2] eQO(po) (4.2.51b)

PoB =

µ+sin2o(4.2.51e)

4.2. Examples 441

T = cTo + eTl sin wt

Figure 4.2.14. The Whirling Pendulum.

18 = EQe(0,pe,0 (4.2.51d)

(O,pp O,pg) E Tl x ]R1 x T1 x 1Rl

where p = J/ml2, Q, = -copo, and QB = -c9p9/(p + sin2 ci) + To + T1 sin(wt).Physically, co > 0 is the damping constant representing viscous damping in thebearings of the frame, co > 0 is the damping constant associated with viscousdamping in the pendulum pivot, To represents a constant torque applied to theframe about the vertical axis, and T1 sin(wt) represents an oscillating torque applied

to the frame also about the vertical axis.


The form of these equations is quite interesting; (4.2.51a, b) are of the form of

a weakly damped oscillator with a particular form of parametric excitation. This

small excitation is applied through the po term in (4.2.51b) and is governed byits own differential equation, (4.2.51d). In particular, note that the 0 - po - pocomponents of (4.2.51) do not depend on 0. Thus, it suffices to analyze this three

dimensional nonautonomous subsystem, since its dynamics determine 0(t).

Thus, the system which we will analyze is

=PO

pO = sin[-1 + p2 cos o/(A+ sin10) 11 + 'Q 0 (PO)

pB =

=W

(4.2.52)

where we have rewritten the nonautonomous three dimensional subsystem (4.2.51a,

b, d) as a four dimensional autonomous subsystem by utilizing the time periodicity

of the perturbation and defining

O (t) = wt , mod 27r . (4.2.53)

We begin by describing the unperturbed system and the geometry of its phase

space. The unperturbed system is given by

=PO

P = sin [-1 + pe cos c/(µ + sin2 0) 2](4.2.54)

Po=0=w.

It is easily seen that the O-po components of (4.2.54) have the form of a 1-parameter

family of Hamiltonian systems with Hamiltonian function given by

2

H (0, po ; po) = 12 µ +

sin2

j+

2p

02 + (1 - cos O) . (4.2.55)

Fixed Points. Fixed points of the 0 - po - pa component of (4.2.54) are given by

Wpo), O, po) (4.2.56)

4.2. Examples 443

where 0(pg) is a solution of

sin 0 [-1 + pe cos o (g + sin2 ) 2 = 0. (4.2.57)

For p < pg, (4.2.57) has two solutions corresponding to 0 = 0 and 0=rr.These solutions exist for all pg, but a change of stability occurs at Ec = pg for

0 = 0. The solution 0 = 0 is a center for µ < pg and a saddle for p > pg. Atp = pg, a pitchfork bifurcation occurs at which two centers bifurcate from = 0

and approach f7r/2, respectively, as pg -> oc. The fixed points are shown in the0 - pg plane in Figure 4.2.15.

unstable

stable l\\

Figure 4.2.15. Solutions of (4.2.57).

Hornoclinic Orbits. Using the Hamiltonian function (4.2.55), it can be shown that

the saddle point

7P(pa) = (ir,O,pg) (4.2.58)

is connected to itself by a pair of homoclinic orbits for all values of pg, and thesaddle point

-YD(pg) = (O,O,pg) (4.2.59)

is connected to itself by a pair of homoclinic orbits for all pg > µ. The phase spaceof (4.2.54) appears as in Figure 4.2.16. We use the subscripts P in (4.2.58), since


the homoclinic orbits in that case are reminiscent of those in the simple pendulum,

and D in (4.2.59), since the homoclinic orbits in that case are reminiscent of those

in the Duffing oscillator (see Guckenheimer and Holmes [1983]).

Pe

0

Figure 4.2.16. Unperturbed Phase Space of (4.2.54).

Therefore, in the context of the full 0 - po - pp - 0 phase space, (4.2.54) hastwo two-dimensional normally hyperbolic invariant manifolds with boundary

MP = ('YP(po),00) , 00 E [0,2ir)

MD = (-ID W), bo) , p8 > A, 00 E [0,27x)

(4.2.60)

(4.2.61)

and Mp and MD have three dimensional stable and unstable manifolds whichcoincide. Thus, (4.2.51) is an example of System I with n = m = I = 1.

We now turn our attention to the perturbed system. By Proposition 4.1.5,we know that Mp and MD persist as invariant manifolds, which we denote asMp E

and MD E, respectively. We then use Proposition 4.1.6 to determine if thereare any periodic orbits on these manifolds and, if so, we compute the appropriate

Melnikov integral to determine whether or not the stable and unstable manifolds of

the periodic orbit intersect.

4.2. Examples 445

The Flow on MD,, and Mp,.

The perturbed vector field restricted to MD,, and MpE is identical and is given

by

pB = -e [ - = TD - T 1 sin'J + 0 (e2)

' =w.The averaged vector field is given by

P86Ceµe+Tof

This equation has a unique fixed point given by

µT0pg =

co

The fixed point is stable on MD,, and MpE, since

(4.2.62)

(4.2.63

(4.2.64)

-d(-E Lce +Tot ! -E <0. (4.2.65)

Now, by Proposition 4.1.6, these fixed points of the averaged equations corre-

spond to periodic orbits on MD,, and MpE, each having period 2w and the samestability type as the fixed points of the averaged equations.

In order to better understand the geometry of the phase space, we will consider

a three dimensional Poincare map constructed in the usual way of sampling the

variables (O, po, pg) at discrete times corresponding to the period of the external

forcing (i.e., at time intervals of 2-); cf., the previous example and Section 1.6. Inthis case the periodic orbits become fixed points of the Poincare map.

The Structure of the Poincare Map. Using our knowledge of the flow on MpE andXD,, as well as our knowledge of the structure of the unperturbed phase space, it is

easy to see that the Poincare map has one hyperbolic fixed point for pq < p which

has a two dimensional stable manifold and a one dimensional unstable manifold. For

pg > Et the Poincare map has two hyperbolic fixed points, each having two dimen-

sional stable manifolds and one dimensional unstable manifolds. See Figure 4.2.17

for an illustration of the geometry of the three dimensional Poincare map.

Calculation of the Melnikov Integrals. We now compute the Melnikov integrals for

the corresponding fixed point of the Poincare map so that we can determine theexistence of orbits homoclinic to the fixed points of the Poincare map.

446

P8

M P,E

Figure 4.2.17. Geometry of the Poincare Map, pe > µ.

From (4.1.47) the Melnikov vector is a scalar and is given by

MP,D (to; Fz, co, co, To, Ti, w)0f[(DpH)(Q) + (DpeH) (QB)] (OPB (t), p (t), p9, t + to) dt

-oo

Dp0H(YP,D(pB)) f QB(00(t),p (t),p9,t+to)dt-00

4. Global Perturbation Methods for Detecting Chaotic Dynamics

f [-cO I pPe (t) l 2+ 2O Z pp +To+TI sin w(t+to) I J dt\ 'k / IA+sin 0 (t) µ+sin B (t) /

00

PO J [ iCO +To+T1sinw(t+to)]dt (4.2.66)µ u+sngOpo(t)

-oo

where (Opv (t), pPO (t)) is the homoclinic trajectory of the unperturbed system on the

pB = p6 = level corresponding to the hyperbolic fixed point of the averaged

4.2. Examples 447

vector field on .MD e or Mp,. Unlike the previous example, we will not explicitly

compute the homoclinic orbits; however, with qualitative arguments we will be able

to go quite far.

The Melnikov integral is more conveniently written as

MP,D(to,cO,T1,w) _ -c0J1 -cgJ2+T1J3(w,to)

whereJCo

J1 = J (p (t))2dt-00

00

J2 ( pg - po)2dt= µ + sing ope (t))-00

00

1 (J3(w, to) pg - P) sinw(t+to)dt.=JJ µ + sing cbpe (t) Is

-00

(4.2.67)

(4.2.68)

(4.2.69)

(4.2.70)

We will consider µ, To, and co as fixed, and co, T1, and w as parameters.

Lemma 4.2.1. 1) J1 > 0

2)J2>000

3) J3 (w, to) = J3(-) sin wto where J3 (w) w f ON 0 (t) sin wt dt.-00

PROOF: 1) and 2) are obvious from (4.2.68) and (4.2.64), respectively. 3) First we00

consider the term -a f sinw(t+to)dt in (4.2.70). Expanding sinw(t+to) and-00

considering the improper integral as a limit of a sequence, we obtain

27tH 2nn00

WW

fsin fcos- w(t + to)dt =nliim [ g cos wto J sin wt dt - sin wt0 wt dt] = 0.

-00 27tHW W

(4.2.71)

We next treat the remaining part of (4.2.70). First, one notes thatPO/(µ + sin2op©(t)) can be taken to be an even function of t if we choose theunperturbed homoclinic trajectory such that OPO (0) = 0. Then, by expanding

sin(w(t + to)) = sin(wt) cos(wto) + cos(wt) sin(wto), the odd part of the integrand,

(pg/(µ + sin2 OPO (t)))sin wt cos(wto), can be eliminated, since it integrates to zero.


The remaining term, (pg/(µ + sin2 cPo (t)))cos(wt) sin(wt0), is still only condition-

ally convergent, see Section 4.1d. It is computed by considering the integral as alimit and integrating once by parts as follows:

J3(w, t0) = sin wt0 limn-+oo

pg sin wt

It + sin2 opo (t)

2,rnW 1

-2,rn w8P0 (t) sin wt dt

where BPo (t) is the solution of

W

BPo (t) = P8

A + sin2 OPe (t)

(4.2.72)

(4.2.73)

The result follows from (4.2.72).

Now for fixed cg, w, and To the integrals J1, J2, and 73- are constant, and theMelnikov function can be written in the form

M(to; c, ,TI) = -coJ1 - co J2 +T1J3(w) sinwt0 (4.2.74)

where Jl, J2, and J3(w) do depend on whether or not they are calculated along the

homoclinic orbits connecting the unperturbed fixed point on MD or .Mp. Thereare two distinct possibilities; either

(-J2/J1)D > (-J2/J1)P

or

(-J2/J1)P > (-J2/J1)D

(4.2.75)

(4.2.76)

where the subscript D represents the integrals computed along the unperturbedhomoclinic orbit connecting the fixed point on MD, and the subscript P represents

the integrals computed along the unperturbed homoclinic trajectory connecting the

fixed point on Mp. We will assume that (4.2.76) holds. In this case, we can appeal

to Theorems 4.1.9 and 4.1.10 and use (4.2.74) to plot regions in co -T1 space wheretransverse homoclinic orbits to hyperbolic fixed points of the Poincare map occur.

These are shown in Figure 4.2.18 as two wedge-shaped regions. Inside the wedge

labeled D, transverse homoclinic orbits to the hyperbolic periodic on MDSE exist,

and, inside the wedge labeled P, transverse homoclinic orbits to the hyperbolic fixed

4.2. Examples 449

Figure 4.2.18. Regions where Horseshoes Exist.

point on Mp, exist. In Figure 4.2.19, we show pictures of the homoclinic tanglesin the Poincare maps in the different regions.

Several interesting dynamic behaviors are possible. For instance, inside thewedge D, chaotic motions in which the pendulum erratically swings back and forth

past 0 = 0, but not over the top, are possible; this is very much like the chaosobserved in Duffing's equation, see Holmes [1979]. Inside the wedge P, chaoticmotions exist in which the pendulum undergoes arbitrary sequences of clockwiseand counterclockwise rotations about 0 = 0, see 4.2a, i. This example provides two

distinct types of chaotic motions that are commonly studied. In fact, in the region

where the interiors of both wedges intersect, both types of chaos are simultaneously

possible. The dynamics in that region have the potential for the system "hopping"

from one type of chaos to another; this cannot be proved using the present methods,

since the invariant manifolds for the two types of chaos remain bounded away from

one another. However, for pg = 0 (- ), using methods similar to the present ones, it

may be possible to predict when these manifolds mingle, thus proving the existence

of such "hopping."

The types of chaos that exist here involve arbitrary sequences of physicallydifferent events. For the pendulum type of chaos (P), there exist motions in which

the pendulum swings through approximately full 2-7r revolutions in arbitrary clock-

wise and counterclockwise sequences. For the Duffing type of chaos (D), there exist

motions which swing back and forth through 0 = 0 toward 0 > 0 and 0 < 0


Side View

Side View

Side View

Side View

(a) Along D +

(b) Inside D

(c) Along D

(d) Along P'

Figure 4.2.19. Poincare Maps.

Top View

Top View

Top View

Top View

4.2. Examples 451

Side View

(e) Inside P

Side View

(f) Along D-

Side View(g) Inside DAP

Figure 4.2.19 Continued.Poincare Maps.

Top View

Top View

in arbitrary orders. The proofs of these statements involve the use of symbol se-quences and symbolic dynamics and arguments similar to those given at the end of

Section 4.2a, i.

It must be pointed out that a different bifurcation diagram must be considered

if (-J2/J1)D > (J2/Jl)D. Essentially, the order of wedges P and D is switchedand the sequences of bifurcations are changed. The details of that case can easilybe worked out in the manner presented above.


4.2c. Perturbations of Completely Integrable, Two Degree of FreedomHamiltonian Systems

We now present two examples of perturbations of completely integrable, two degree

of freedom Hamiltonian systems.

The first is due to Holmes and Marsden [1982a], who first generalized Mel-nikov's method to such systems. Their technique utilized the "method of reduc-tion" for Hamiltonian systems (see Marsden and Weinstein [19741), which essentially

reduced the problem to the standard planar Melnikov theory (Melnikov [1963])when Hamiltonian perturbations were considered. Our techniques do not requirethe method of reduction, and the reader should compare our results with those of

Holmes and Marsden [1982a] (note: for an example where the method of reduction

is not applicable, see Holmes [1985]).

Our second example is due to Lerman and Umanski [1984]. It considers orbits

homoclinic to a hyperbolic fixed point in two strongly coupled nonlinear oscillators.

This is an example of an unperturbed system which contains a symmetry thatresults in the unperturbed homoclinic orbit being two dimensional.

i) A Coupled Pendulum and Harmonic Oscillator

We consider a linearly coupled simple pendulum and harmonic oscillator (see Holmes

and Marsden [1982a]). The equations of motion are given by

xl = x2

i2 = -sin xl + e(x1 - x3)

x3 = x4

y4 = -w 2x3 + e(x3 - xl)

(x1,x2,x3,x4)ET1xIRxIRx1R.. (4.2.77)

As a convenience, we will transform the harmonic oscillator to action-angle coordi-

nates via the transformation

xs = WF sin 8(4.2.78)

x4 = 2Iw cos 0 .

4.2. Examples 453

Using this transformation, (4.2.77) becomes

xl = x2

i2 =-sin x1 +e( Zw sin0-xi)

I=-e( 2Isin0-xl) Icos0

8=w+e( wlsin0-x1) s21w 0

(xi,x2,I,0) E T1xR.xR+ xT1 (4.2.79)

where R+ denotes the nonnegative real numbers. This system is Hamiltonian with

Hamiltonian function given by

He=H(xi,x2,I)+eH(xi,x2,I,0)= 2 -cosx1+Iw+2( wlsin0-x1)2.

(4.2.80)


(4.2.81)

It should be clear that the x1 - x2 component of (4.2.81) has a hyperbolic fixedpoint at (21, 22) = (7r, 0) = (-7r, 0) for all I E ]R+. This fixed point is connectedto itself by a pair of homoclinic trajectories

(x± (t), x± (t)) = (±2 sin- 1(tanh t), ±2sech t) . (4.2.82)1

Thus, in the full x1- x2 - I - 0 phase space, (4.2.81) has a two dimensional normally

hyperbolic invariant manifold with boundary

M = { (xl, x2, I, 0 I (x1, x2) = (7r, 0) , 16 ]R+, 0 E [0, 27r) } . (4.2.83)

M has three dimensional stable and unstable manifolds which coincide along the

three dimensional homoclinic orbit r parametrized by

r = { (f2 sin-1(tanh(-t0)), ±2sech (-to), I, Bo) E T1 x ]R x ]R+ x T1 I

(to, I, 00) E ]R X R+ X T 1 }. (4.2.84)


I

xI= -.1T x1= 7r

identify

Figure 4.2.20. Geometry of the Phase Space of (4.2.81).

T1

The geometry of the unperturbed phase space is shown in Figure 4.2.20. Thus,(4.2.77) has the form of System III with n = m = 1.

We now turn our attention to the perturbed system. By Proposition 4.1.16, M

persists as well as the collection of three dimensional energy manifolds given by the

level sets of (4.2.80). These energy manifolds intersect M in a periodic orbit which

can be parametrized by I (cf., Section 4.1c,v ). We now compute the Melnikovintegrals in order to determine if the two dimensional stable and unstable manifolds

of the periodic orbits intersect in the three dimensional energy surfaces.

By (4.1.92) and (4.1.85) the Melnikov vector is a scalar and is given by

MI}(to;w) _ -L00 D9H(x (t), xh(t),I,w(t+to))dt00

00_ -J .(V I sin(w(t+to)) -xlh(t)) wI cos(w(t+to))dt.

(4.2.85)

Evidently (4.2.85) converges at best conditionally. By Lemma 4.1.29 the improper

integral makes sense if we approach the limits ±oo along the sequences of times{T; }, {-T }, j = 1, 2,..., where for this problem we choose T = T -- iu) (2j+1).

4.2. Examples 455

Using (4.2.82), the integral becomes

MIA (t0; w) _ 27-, 7rW sin ,t0 . (4.1.86)

Thus, (4.2.86) has zeros at to j = 0, +1, ±2,. .., and Dt0MI± (TO; w) # 0for all I > 0. Thus, by Theorems 4.1.19 and 4.1.20 for each I > 0 the stableand unstable manifolds of the corresponding hyperbolic periodic orbit intersecttransversely yielding Smale horseshoes on the appropriate energy manifold. (Note:

the question of what happens to these horseshoes for I -p 0 is an open problem.)

ii) A Strongly Coupled Two Degree of Freedom System

The following example is due to Lerman and Umanski [1984]. They consider acompletely integrable two degree of freedom Hamiltonian system of the form

it A -w 0 0 x1

i2 = w A 0 0 x2- 2V aA

'i3 0 0 -A w x3

i4 0 0 -w -A x4

where A < 0 and w > 0. The two integrals are

(4.2.87)

H = K1 = A(xlx3 + x2x4) - w(x2x3 - x1x4) - -[(x1 + x2)2 + (x3 + x2)2]

K2 = 52x3 - 51x4 .(4.2.88)

The equation (4.2.87) has a hyperbolic fixed point at (xl, x2i x3, x4) = (0, 0, 0, 0)

where the eigenvalues of (4.2.87) linearized about the fixed point are easily seen to

be ±A ± iw. Thus, (0, 0, 0, 0) has a two dimensional stable manifold and a two

dimensional unstable manifold that coincide along the two dimensional homoclinic

manifold r, which can be parametrized by

r = l (xlh(t),x2h(t),z3h(t),x4h(t)) _2-1/4 cosh(-4,\t0)] -1/2 (eAto cos a, eat0 sin a, a-At0 cos a, e_At0 sin a) E I4

a = (-wt0 + a), (to, a) E 1R1 x T1 } .

(4.2.89)

We remark that the reason the stable and unstable manifolds coincide is that thesecond integral, K2, induces a rotational symmetry in the vector field (heuristically,


K2 can be thought of as "angular momentum"). The variable "a" in (4.2.89) arises

due to this rotational symmetry.

In this example, the origin plays the role of M in the general theory. So, for a

sufficiently small perturbation of (4.2.87), the fixed point persists (note: in particu-

lar, for our perturbation, the origin actually remains a hyperbolic fixed point of the

perturbed system). Whether or not any orbits homoclinic to the fixed point survive

can be determined by computing the appropriate Melnikov vector. There are twodistinct situations depending upon whether or not the perturbation is Hamiltonian.

We present two examples illustrating the different cases for this problem.

Non-Hamiltonian Perturbation of (4.2.87). Consider the following perturbation of

(4.2.87)

E9x(x;µ) = E

(lil + li2)x2

1x4A

-/42 3X

(4.2.90)

where µ = (µ1,µ2) E 1R.2 is a parameter.

So, in this case, the perturbed system is an example of System I with n = 2,m = I = 0. For this problem, the Melnikov vector has two components, which aregiven by

0Ml (a; µ) = f (DzH, 9y) (xlh(t), -20), X30), X40); µ1, µ2)dt (4.2.91a)

-oo

co

M2 (-;,U) = f (DaK2,9y)(xlh(t),x2h(t),x3h(t),x4h(t);µi,µ2)dt.-00

Using (4.2.89), the integrals are

(4.2.91b)

Ml (a; µ) = sechirw

sin 2a + µ2w + V27rA2w sechxw

cos 2a (4.2.92a)4 / 4a 4 / A 8A 4a

M2 (a;/) =-7rµ2

(4.2.92b)4faIt should be clear that M(a; µ) __ (Ml (a; µ), M2 (a; µ)) is zero at

µ2=T2=0, a=6 =0,2,7r, 32 (4.2.93)

4.2. Examples 457

and

;. 7rWl sech4a16A

7,'2

16x1-sech

4a

a = 0,7r

a= It 37r2' 2

(4.2.94)

Thus, by Theorem 4.1.9, for Fi2 near zero and for any jul 0 there exist four orbits

homoclinic to the origin in the perturbed system. We cannot immediately appeal

to the results of Section 3.2d, ii) to assert the existence of Smale horseshoe-likebehavior in the perturbed system. This is because the eigenvalues of the unper-turbed system do not satisfy the hypotheses of the example in Section 3.2d, ii) so,

necessarily, we would have to compute the 0 (e) correction to the eigenvalues due

to the perturbation and then recheck the hypotheses of the example.

Hamiltonian Perturbation of (4.2.87).

We consider a Hamiltonian perturbation of (4.2.67) where we perturb H byeH where

(4.2.95)

So in this case the perturbed system is an example of System III with n = 2,m = 0.

Since the perturbation is Hamiltonian and, therefore, the three dimensionalinvariant energy manifolds are preserved, the Melnikov vector is just a scalar and,by (4.1.92) and (4.1.85), it is given by

00

M2 (a) = J (DxK2, JDH) (zlh(t), X2h(t), z3h(t), X4h(t))dt . (4.2.96)

-00

Using (4.2.89), the integral can be computed and is found to be

It cosh 4aM2 (a) = sin 2a .

2A cosh 2a(4.2.97)

Thus, M2(a) = 0 at a = a = 0, 2,7r, 32 , and at these points DaM2 0. So,

by Theorem 4.1.19, the perturbed system has four orbits homoclinic to the origin

and, by Theorem 4.1.20, the two dimensional stable and unstable manifolds of the

origin intersect transversely along these orbits in the three dimensional energy sur-

face. Thus, by Devaney's theorem (Theorem 3.2.22), the perturbed system contains


horseshoes near these orbits (note: the reader can verify directly by computationthat Ml = 0 for this problem for all a).

4.2d. Perturbation of a Completely Integrable Three Degree of FreedomSystem : Arnold Diffusion

We now give an example of a three degree of freedom Hamiltonian system due to

Holmes and Marsden [1982b] which exhibits the phenomenon of Arnold diffusion,

see Section 4.ld,v. Our methods differ slightly from those of Holmes and Marsden

in that they utilize the method of reduction to first reduce the order of the system.

Our methods are more direct in that we avoid this preliminary transformation ofthe system.

We consider the following system

xl = x2

i2=- sin x1+e[ 2Ilsin01+ 212sin02-2x1]

I1=-E[ 211sin01-xl] 2I1cos01

I2 = -E 2I2 sin 82 - xl] 2I2 cos 02

sin 8181 = DI1G1(I1)+E[ 2Ilsin01 - xlI 211

sin B282 = DI2G2(12) + E [ 2I2 sin 82 - x2l

2I2

(x1 x2, Il I2+8182) G

Tlx]RxLR+xlR+xTl xTl.

(4.2.98)

This system is Hamiltonian with Hamiltonian function given by

HE = H(xl, x2, I1I2) + EH(xl x2, Il 12, 81, 82)

= 2 - cos x1 + G1(Il) + G2(12) + 2 [( 2Il sin 81 - xl) 2 (4.2.99)

+ (.\/2I2 sin 82 - x2) 2]

where G1(11) and G2(12) are arbitrary C2 functions which satisfy the nondegener-

acy requirement for the KAM Theorem 4.1.17 given by

D2 2DI2G2(I2) 54 0. (4.2.100)

4.2. Examples 459


xl = x2

z2 = -sin x1

I1=0(4.2.101)

12 = 0

01 = DI1G1(Il)

02 = DI2G2(I2)

We can think of the Il - 01 and 12 - 02 components of (4.2.101) as being nonlinear

oscillators expressed in action angle coordinates, with DI1G1(Il) and DI2G2(I2)

being the frequencies of the oscillators. The xl - x2 component has a hyperbolicfixed point at (1x2) = (ir, 0) _ (- 7r, 0) which is connected to itself by a pair ofhomoclinic trajectories given by (x1h(t),x2h(t)) = (t2 sin- 1 (tanh t), ±2 secht).Thus, in the full x1 - x2 - Il - I2 - 01- 02 phase space, (4.2.101) has a 4 dimensional

normally hyperbolic invariant manifold given by

M = { (x1,x2,I1,I2,01,02) E T1xIRxIR+xlR+xT1xT1 I (X1, X2) = (ir,0) = (-ir,0)}(4.2.102)

which has the structure of a two parameter family of 2-tori. M has a 5 dimensional

stable manifold and a 5 dimensional unstable manifold that coincide along the 5

dimensional homoclinic orbits r±, which can be parametrized by

07= [(f2 sin-' (tanh(-to)), ±2sech (-t0), I1, 12,010,020)

E T1 x ]R x ]R+ x T1 x T1 I to E IR}. (4.2.103)

Note that on M, for each I = (11,12), there corresponds a two torus T(I) havingthree dimensional stable and unstable manifolds coinciding along the three dimen-

sional homoclinic orbits rI , where rI is obtained from (4.2.103) by fixing the Icomponent. Thus, (4.2.98) is an example of System III with n = 1, m = 2.

We now turn our attention to the perturbed system where the perturbationcorresponds to linear coupling of the nonlinear oscillators with the pendulum. By

Proposition 4.1.16, M persists (denoted by ME) and intersects each 5 dimensional

invariant energy surface given by the level sets of (4.2.99) in a three dimensionalset of which, by the KAM Theorem 4.1.17, "most" of a one parameter family


of invariant two tori persist (note: (4.2.100) is equivalent to the nondegeneracy

hypothesis of the KAM theorem for our system). We then compute the Melnikov

vector in order to determine if the stable and unstable manifolds of the KAM tori

intersect transversely. If so, we can conclude that (4.2.98) exhibits Arnold diffusion.

From (4.1.92) and (4.1.85), the Melnikov vector has two components given by

00

M2 (810,820,II,I2) = f Dgl I(x h(t),x h(t),I1,I2,w1t+B10,W2t+020)dt-00

(4.2.104a)00

N I3 (810, 820, I1, I2) = J ( t ) ,

(4.2.1046)

Using (4.2.103) (cf., equations (4.2.85) and (4.2.86)) the Melnikov integrals can be

computed and are found to be

t ZM x-1(810, 820) _ + 21isech za sin 0io , i = 1,2 (4.2.105)

a

where fti - DIzGi(Ii) with

If 87r2 I1I2sech

rSZlsech f112 cosdet[D(910,920)M (010,020)] = 111n2 2 2

010 cos 020 -

(4.2.106)

Thus, M1±(910,026) has zeros at (010, 020) = (810, 820) = (kwr, kir),

k = 0, ±1, ±2,..., and D(e10,e20)MI } (010, 020) has rank 2 at these points. So,

by Theorems 4.1.19 and 4.1.20, the stable and unstable manifolds of the KAM tori

intersect transversely, and hence Arnold diffusion occurs in (4.2.98).

4.2e. Quasiperiodically Forced Single Degree of Freedom Systems

We now consider two examples of single degree of freedom systems subjected to

quasiperiodic excitation. In these examples the Melnikov vector will detect orbits

homoclinic to normally hyperbolic invariant tori with the resulting chaotic dynamics

being characterized by Theorem 3.4.1.

The first example is the quasiperiodically forced Duffing oscillator studied by

Wiggins [1987]. In this example, the existence of transverse homoclinic tori isestablished, and the effect of the number of forcing frequencies on the chaotic region

in parameter space is examined. Additionally, a relationship of these theoretical

4.2. Examples 461

results with experimental work of Moon and Holmes [1985] on the quasiperiodically

forced beam is discussed.

The second example is the parametrically excited pendulum whose base isvertically oscillated with a combination O (E) amplitude 0(1) frequency and 0(1)

amplitude 0(e) frequency excitation.

i) The Duffing Oscillator: Forced at 0(c) Amplitude with N 0(1) Fre-quencies

We first consider the quasiperiodically forced Duffing oscillator forced with two

frequencies

xl = x2

i2 = 2x1(1 - x1) + CI f cos 01 + f cos 02 - 6x2]

01 =w1

02=w2

(x1,x2,01,02) E Rx1RxT1 xT1

(4.2.107)

where f and 6 are positive, and w1 and w2 are positive real numbers. We can reduce

the study of (4.2.107) to the study of an associated three dimensional Poincare map

obtained by defining a three dimensional cross-section to the four dimensional phase

space by fixing the phase of one of the angular variables and allowing the remaining

three variables that start on the cross-section to evolve in time under the action of

the flow generated by (4.2.107) until they return to the cross-section, see Section 1.6.

The return occurs after one period of the angular variable whose phase was fixed

in order to define the cross-section. To be more precise, the cross-section, E020, is

given by

E020 ={(xi,x2,01,02) ElRx]RxT1xT1102 (4.2.108)

where, for definiteness, we fix the phase of 02. The Poincare map is then defined to

be

PP: E020 , E020

l l(x1(0),x2(0),01(0) = 0i0) -' (x1 (w2)'x2 02/01 (\w'/ = 2 z +010)(4.2.109)


For e = 0, the x1 - x2 component of (4.2.107) is a completely integrableHamiltonian system with Hamiltonian function given by

2 2 4

H(x1,x2) = 2 4 + 8 .(4.2.110)

It also has a hyperbolic fixed point at (x1, x2) = (0,0) which is connected toitself by a symmetric pair of homoclinic trajectories given by (x1(t),x2(t)) =(+/sech _, +sech f tank k). Thus, in the full x1 - x2 - 01 - 02 phase space,

the unperturbed system has a two dimensional normally hyperbolic invariant torusgiven by

M = { (x1,x2,91,02) E ]Rx]RxT1xT1 I X1 = x2 = 0, 01,02 E [0,27x)} (4.2.111)

with trajectories on the torus given by (x1(t) , x2 (t), 01(t), 02 (t)) = (0, 0, w1t +010, w2t + 020). The torus has a symmetric pair of coincident three dimensionalstable and unstable manifolds with trajectories in the respective branches given by

(xih(t),x2h(t),01(t),02(t))=(f sech,+sech tanh ,w1t+010,w2t+920)

(4.2.112)

Thus, (4.2.107) is an example of System I with n = 1, m = 0, 1 = 2.

Utilizing this information, we can obtain a complete picture of the global in-

tegrable dynamics of the unperturbed Poincare map, PO. In particular, PO has aone dimensional normally hyperbolic invariant torus, TO = M n E920, that hasa symmetric pair of two dimensional stable and unstable manifolds, W3(TO) and

Wu(TO), that are coincident, see Figure 4.2.21.

By Proposition 4.1.5, for e 0 and small, the perturbed Poincare map, PE,still possesses an invariant one dimensional normally hyperbolic invariant torusTE = M. n E920 having two dimensional stable and unstable manifolds, W8(TE)

and Wu(TE), which may now intersect transversely yielding transverse homoclinic

orbits to TE. Intersections of W8(TE) and Wu(TE) can be determined from theMelnikov vector. From (4.1.47), the Melnikov vector has one component and isgiven by

I0P0

M±(010,020;f,6,w1,w2) =J

(DzH,9z)(x h(t),x h(t),w1t+010,w2t+020)dt.-00

(4.2.113)

4.2. Examples 463

Figure 4.2.21. Homoclinic Geometry of the Phase Space of P0, Cut Away Half View.

Using (4.2.110) and from (4.2.107), gx = (0, f cos o1 + f cos 02 - 6x2), (4.2.113)

becomes

M±(01o, 020; f, 6, w1, w2)00

J [-6 (x2h (t)) 2 + f x2h (t) cos(wlt + 610) + f x2h(t) cos(w2t + 020)] dt

-00

- -23 6 f 27r f wl sech x sin 010 + 27r f w2sech sin 020

It should be clear that (4.2.114) has zeros provided

f . 37r(wlsech f + w2sech 2) .

(4.2.114)

(4.2.115)

We have the following theorem.

Theorem 4.2.2. For all f,6,wl,w2 such that (4.2.115) is satisfied, W3(TE) andWu(TE) intersect transversely in a transverse homoclinic torus.

PROOF: From (4.2.114), Mf = 0 implies

_ fv/z6f37r [wisech sin 010 + w2sech' sin 020]

V2 f (4.2.116)


So, if (4.2.115) is satisfied, then (4.2.116) has solutions for some (010,020). Also,

we have

h±2 8 (4 2 117a)T01 p

7r f wlsec= 10 ,r- /cos . .

fh±2 7 s 0 2 117b)(4

a02 O7r f wlsec- 20 ,co ..

so, if (4.2.115) is satisfied, (4.2.117a) and (4.2.117b) cannot each be zero simultane-

ously. Then, by Theorem 4.1.9, we can conclude that W5(T) and W'u (TE) intersect

at some (010, 020) = (010, 020) and, by Theorem 4.1.10, we can conclude that the

intersection is transversal.

Transverse Homoclinic Tori

x2

Figure 4.2.22. Transverse Homoclinic Torus for PE, Cut Away Half View.

This establishes that Ws(TE) and W"(TE) intersect transversely at some point

(01, 02) = (#1, #2); we now need to argue that (4.2.115) is actually a sufficientcondition for W8(T6) and W 3 (TE) to intersect transversely in a 1-torus. The ar-gument goes as follows: since, by (4.2.115), aB

ofand ae

ofcannot both vanish

simultaneously, suppose for definiteness, a f (810, #20) yl 0. Then, by the globalimplicit function theorem (Chow and Hale [1982]), there exists a function of 020,

4.2. Examples 465

say ht (020), whose graph is a zero of tf -, i.e., M± (ht(020), 020) for all 020 such

that am-h (h7'(020),020) 34 0. At points where aM- = 0 then, since 8M± is nota01o aelo ae2o

also zero, there exists a function g±(010) such that aeo

(010, 9±(010)) 54 0. Then,

20 are never both zero, graph h± (020)U graph g±(010) forms asinceaslo

fand

x8f

differentiable circle which is a zero of M.

Thus, Theorem 3.4.1 applies to this system and implies the existence of chaotic

dynamics for parameter values satisfying (4.2.24).

We illustrate the geometry of the homoclinic orbits of PE in Figures 4.2.22and 4.2.23. In Figure 4.2.22 we show a transverse homoclinic torus for PE. Using

an argument similar to that given for concluding the existence of the homoclinictangle for orbits homoclinic to fixed points of maps (see Section 3.4 or Abrahamand Shaw [1984]), we can conclude that a homoclinic torus tangle results as shown

in Figure 4.2.23.

The homoclinic torus tangle appears to form the backbone of the strange at-tractor experimentally observed by Moon and Holmes [1985] for this system. They

studied the structure of the strange attractor by utilizing a technique due to Lorenz[1984], which involves constructing a double Poincare section or Lorenz cross-section

by fixing the phase of one of the angular variables and a small window about a fixed

phase of the remaining angular variable. The map of this "section of a section" into

itself revealed a fractal nature of the strange attractor similar to that found in theusual Duffing-Holmes strange attractor (Holmes [1979]) which was not apparent in

the three dimensional Poincare map. Our results give much insight into the nature

of this phenomenon. In Figure 4.2.23, it is clear that the intersection of W3(TE)and Wu(TE) with the double Poincare section yields a geometric structure that is

quite similar to the homoclinic tangle which occurs in the periodically forced buff-

ing equation and that is responsible for the fractal structure of the Duffing-Holmesstrange attractor.

Next we want to consider the effect on the region where transverse homoclinic


Double Poincare Section

x2

xl

Figure 4.2.23. Homoclinic Torus Tangle for PE and the Double Poincare Section,

Cut Away Half View.

4.2. Examples 467

tori exist caused by adding additional forcing functions to (4.2.107), i.e., we consider

xl = x21x2= 2x1(1-xl)+e[fcosoj+ fcos02+ +fcosOn - by]

B1 = wl (4.2.118)

Ef

Es

Figure 4.2.24. Regions of Chaos in f - b Space as a Function of the Number

of Forcing Frequencies.

We reduce the study of (4.2.118) to the study of an associated n-1 dimensional

Poincare map having an n - 2 dimensional normally hyperbolic invariant manifold

with n - 1 dimensional stable and unstable manifolds. Intersection of the stableand unstable manifolds is determined by calculating the Melnikov function. In

Figure 4.2.24, the lines f = m1b, f = m2b, and f = mnb represent lines abovewhich transverse homoclinic tori occur for the Duffing oscillator forced at 1, 2, and n

frequencies, respectively; ml, m2, and Mn are obtained from the Melnikov function

and are given by

ml37rwlsech

2


m2 =

and

Mn =

3:r (wlsech f + w2sech f )

37r (wlsech f + w2sech + . + wnsech )Vf2- ,, 52

Thus, we see that the effect of increasing the number of forcing frequencies is toincrease the area in parameter space where chaotic behavior can occur, and hence

to increase the likelihood of chaotic dynamics.

ii) The Pendulum: Parametrically Forced at 0(e) Amplitude with 0(1)Frequency and 0 (1) Amplitude with 0 (e) Frequency

We consider the same system as in Section 4.2a but with the combined forcingfunctions of Examples 4.2a, i and 4.2a, ii. More specifically, we have

it + E6x1 + (1 - EI' sin Gt - -y sin cwt) sin xl = 0. (4.2.119)

Writing (4.2.119) as a system gives

xl = x2

i2 = -(1 - y sin I) sin xi + c[r sin B sin xl - 6x2]

I=cw (x1ix2,I,0) E T1 x]RxT1 xT1 .

(4.2.120)


±1 = x2

i2 = -(1 - y sin I) sin xl

I=08=0

(4.2.121)

and the xi - x2 component of (4.2.121) is Hamiltonian with Hamiltonian function

given by2

H(x1,x2) =2

- (1 - ysinl) cosxi . (4.2.122)

4.2. Examples 469

The unperturbed system has a hyperbolic fixed point at (x1, -t2) = (-,0) = Hr, 0)

for each I E (0, 2xr], 0 E (0, 2xr] provided 0 < ry < 1. Each of these fixed points is

connected to itself by a pair of homoclinic trajectories given by

(.'r 1h(t), x2h(t)) =(t2 sin 1 [tanh 1 - 7 sin I t 2 1 - ry sin I sech 1 - 'Y sin I t)

(4.2.123)

Thus, the unperturbed system has a normally hyperbolic invariant two torus whose

stable and unstable manifolds coincide; see Figure 4.2.25 for an illustration of the

unperturbed phase space of (4.2.121). Thus, (4.2.120) is an example of System II

with n=m=l=1.

x

T1

X. =-7r

Figure 4.2.25. The Phase Space of (4.2.121).

The Melnikov function will give us information concerning the behavior of the

stable and unstable manifolds of the torus in the perturbed system. From (4.1.89),

the Melnikov function is given by

00

JL

6(x2h(t)) 2 + YWt cos Ix2h(t) sin xlh(t)(4 2 124). .-00

+ Fx2h(t) sin x1h(t) sin f1(t + to)} dt.


Using (4.2.31), we obtain

M+(to, I; 6,'Y, w,r,12) = M (t0, I; 6,'Y, w, r, n) = M(t0, I; b, 1, w, r, D)

_ -86 1 - i sin I +4ryw cos I + 2?rSt2I'

cos 12t0 .1-1sinl(4.2.125)

?rflsi l2 1-ry sin I

Using arguments similar to those given for the quasiperiodically forced Duffing os-

cillator, (4.2.125) can be used to prove the existence of transverse homoclinic toriin (4.2.120). However, the term sinh 7M renders the necessary computa-

2 1-7sinItions analytically intractable, and the details will probably need to be carried out

numerically. We leave this to the interested reader.

4.3. Final Remarks

1) Heteroclinic Orbits

In this chapter, we developed techniques for measuring the distance between the sta-

ble and unstable manifolds of an invariant torus in three general classes of ordinary

differential equations, i.e., we developed techniques for determining the existenceof orbits homoclinic to tori. Analogous techniques for detecting heteroclinic orbits

in the same classes of systems are developed in precisely the same manner. Theonly difference is that the unperturbed system is assumed to have two normallyhyperbolic invariant manifolds, say M1 and M2, which are connected to each other

by a manifold of heteroclinic orbits. The geometry of the splitting of the manifolds,

the measure of the splitting of the manifolds, and the Melnikov vector are the same

as in the homoclinic case for orbits heteroclinic to tori of the same dimension. The

case of orbits heteroclinic to different types of invariant sets (e.g., a periodic orbit

and a two torus) has not been worked out as yet.

ii) Additional Applications of Melnikov's Method

The following are references to additional applications of Melnikov's Method which

are grouped according to the different fields.

Fluid Mechanics. Holmes [1985], Knobloch and Weiss [1986], Slemrod and Marsden

[1985], Suresh [1985], Ziglin [1980], Rom-Kedar, Leonard, and Wiggins [1987].


The Josephson Junction. Holmes [1981., Salam and Sastry [1985], Hockett andHolmes [1986].

Power System Dynamics. Kopell and Washburn [1982], Salam, Marsden, andVaraiya [1984].

Condensed Matter Physics. Coullet and Elphick [1987], Koch [1986].

Rigid Body Dynamics. Holmes and Marsden [1981], [1983], Koiller [1984], Shaw

and Wiggins [1988].

Yang-Mills Field Theory. Nikolaevskii and Shchur [1983].

Power Spectra of Strange Attractors. Brunsden and Holmes [1987]_

iii) Exponentially Small Melnikov Functions

The methods of averaging and normal forms are often useful techniques for trans-

forming analytically intractable problems into "almost" tractable problems, seeGuckenheimer and Holmes [1983] or Sanders and Verhulst [1985]. In particular,these techniques can often be utilized to transform systems into "near integrable"systems which appear to be ideal candidates for Melnikov type analyses. However,

formidable mathematical difficulties lie lurking in the background. We will briefly

sketch the main problem.

Suppose we have a planar, time periodic ordinary differential equation in the

standard form for application of the method of averaging, i.e.,

x = Ef(x,t) + E2g(x,t;E) (4.2.126)

with 0 < c << 1, f and g Cr, r > 2, and bounded on bounded subsets of 1R2 for each

t E [0, T) where T is the period, and l og(x,t; E) exists uniformly. Application ofthe averaging transformation to (4.2.126) yields

± = CI(X) + E2g(x, t; E) (4.2.127)

Twhere f (x) _ c f f (x, t)dt (the exact form of g" is not important to us but it

0can be found in Guckenheimer and Holmes [1983] or Sanders and Verhulst [1985]).

Rescaling time, t -r t/E, transforms (4.2.127) into

tf (X) + Eg(x, E, E) . (4.2.128)


Now suppose that, for c = 0, (4.2.128) is Hamiltonian with Hamiltonian function

H(x) and, in particular, it has a homoclinic orbit qo(t) connecting a hyperbolicfixed point po to itself. Then the Melnikov function for (4.2.128) is given by

0o

M(to) = J (DaH(g0(t)),g(go(t), t Et0;0))dt . (4.2.129)

-00

From (4.2.129) the problem should be apparent; that is, the Melnikov functiondepends explicitly on c. Moreover, the relatively rapid oscillation (period eT)

in general results in the Melnikov function being exponentially small in e. Thus,

without a careful consideration of the errors in the formula for the expansion of the

distance between the manifolds in powers of c, we cannot claim that the 0(e) term

(i.e. the Melnikov function) dominates the higher order terms for sufficiently small

c. In particular, Theorems 4.1.9, 4.1.13, and 4.1.19 are not valid. These problems

were first pointed out by Sanders [1982]. Let us illustrate with a specific calculation.

Consider the simple pendulum

x + sin x = e sin wt . (4.2.130)

The Melnikov function for (4.2.130) is given by

M(to) = 27rsech(2) coswt0. (4.2.131)

The splitting distance of the separatrices is proportional to

dsplit -- e maax IM(to) I + 0 (e2) . (4.2.132)t

Next consider the rapidly forced pendulum

t2+sinx=Esin-.E

(4.2.133)

Using (4.2.131), for (4.2.133) we find

max M(to)j .:: 27re 2E . (4.2.134)to

Thus the Melnikov function is smaller than any power of E.

A theoretical breakthrough on problems of this sort has recently been made by

Holmes, Marsden and Scheurle [1987 a,b], and there are two results dealing with

these problems which we now mention.


Upper Estimate Consider

A + sin .r = b sin(t/E) .

For any n > 0 there is a 60 > 0 and a constant C = C(rt,6p) such that, for all Eand 6 satisfying 0 < E < 1 and 0 < 6 < 60, we have

ITsplitting distance < Cb exp rr -

2 E

Lower Estimate And Sharp Upper Estimate Consider

x + sin x = EP6 sin(t/E) .

If p > 8, then there is a 80 > 0 and (absolute) constants Cl and C2 such that, forall c, 6 satisfying 0 < c < 1 and 0 < 6 < bp, we have

C2EPbe-7f/2E < splitting distance < Clc1be-xf2e.

These estimates are special cases of estimates for a planar system

Ti = g(u,E) +EP6hI U, E, tJ'E

uE]R.2

where one assumes:

1) g and h are entire in u and q2) h is of Sobolev class Hl (for the splitting distance results) and T-periodic in

the variable 0 = t/E;

3) it = g(u, c) has a homoclinic orbit it (c, t) which is analytic on a strip in thecomplex t-plane, with width r.

One needs to make additional assumptions on the fundamental solution of the

first variation equation

v = Dµg(u, E) v

which can be checked to hold in the pendulum example. There are analogues of the

upper and lower estimates above for this general situation, with 7r/2 replaced by a

positive constant r; we refer to Holmes, Marsden, and Scheurle [1987a] for details.

The proofs depend on detailed estimates of the terms in an iterative process in the

complex strip that are used to define the invariant manifolds. It is important toextend these iterates to the complex strip in the proper way; for example, sin(,.)


becomes very large for complex t, and naively extended iteration procedures for the

stable and unstable manifolds will lead to unbounded sequences of functions.

The Holmes-Marsden-Scheurle techniques have immediate applications to the

structure of the resonance bands in KAM theory and to the unfolding of degenerate

singularities of vector fields. Specific examples can be found in Holmes, Marsden,

and Scheurle [1987b].

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INDEX

absolute convergence 408, 411, 413

adapted metric 146admissible string of length k 103-106almost periodic functions 16Arnold diffusion 334-335, 338,

394-396,458-460asymptotic behavior 20-22asymptotic phase 410atlas 34-35attracting set 21autonomous systems 8

averaged equations 357-358averaging 471

Baire space 60basin of attraction, see domain of

attractionbasic frequencies 16bifurcation 62-67,199-206,213-

216, 219, 223, 227, 240, 248-251,258, 299-300, 332-333, 423, 433,443, 451global 66, 226local 65-66period-doubling 248, 332pitchfork 433, 443saddle-node 248

bifurcation point 63bifurcation set 62bifurcation values 66

buckled beam 173-178bump function 354-355bursting 313

Ck 23Ck-close 58Ck-conjugate 23-24Cke-close 58Ck equivalent 24Ck generic 60Ck topology 59locally Ck conjugate 23

Cr 2Cr (Rn, Ru) 58Cr curve 40Cr diffeomorphism 34Cr manifold of dimension m 34Cr manifold of dimension mwith boundary 49Cr perturbation theorem 50

Cantor set 93,99-100,121,386of p-dimensional surfaces 158of tori 320, 322, 333

cardinality 99chaos 75-76, 93-94; see also Smale

horseshoecriteria for, hyperbolic case 108-150criteria for, nonhyperbolic case 150-170

chart 34-35circle 12codimension 63-65

490

commute 23completely integrable Hamilton.ian

systems, see under Hamiltoniansystems

condensed matter physics 471conditional convergence 412, 448cone field, see sector bundlesconjugacies 22-26conjugate 22 -26

topologically 23continuation 5coordinate chart 35-36coordinate transformations 22coupled pendulum, see under

pendulumcritical point, see fixed pointcross-section 68cylinder 12, 213, 221

deformations 65dense orbit 88Di f f r (IR n, lRa) 58diffeomorphism 11discrete space 97discrete time system 14domain of attraction 21double Poincare section 465Duffing-Holmes strange attractor

465buffing oscillator 173, 334-335,

430-439,461-468

energy manifold 394, 455energy surfaces 391,393c - neighborhood 99equilibrium point, see fixed pointexponential dichotomies 33exponentially small Melnikov

functions 471-474

feedback control 429-439

first variational equation 3, 404,408, 411

fixed point 15,24:3-251

of saddle-focus type 276-286

Index

of saddle type 208-226fluid mechanics 470-471full shift on N symbols 101

generalized Lyapunov-type numbers,see Lyapunov-type numbers

general position 57genericity 60-62global bifurcation, see under

bifurcationglobal cross-section 71global perturbation 334global perturbation methods 335-

396System I 336-337, 339-369

- Melnikov vector for 396-397

- perturbed phase space of 352-369

- unperturbed phase space of340-

352System II 336-337; 370-380

- Melnikov vector for 397- perturbed phase space of 373-

380- unperturbed phase space of

370-373System III 336-337, 380-396

- Melnikov vector for 397-398- perturbed phase space of 384-

396- unperturbed phase space of

381-384

Hamiltonian systems 275-298,380-396completely integrable 341, 371,381,394,452-458

harmonic oscillator 452-455heteroclinic 171, 181, 300heteroclinic cycle 300-301heteroclinic orbit 181-182, 470

to hyperbolic fixed points 300-313homeomorphism 11homoclinic 173-181

Index

homoclinic coordinates 343, 350-352,372-373,383-384

homoclinic explosion 226-227homoclinic manifold 341, 346-347,

350, 371homoclinic motion 173-176homoclinic orbit 181-182

double pulse 251-253subsidiary 251-253to hyperbolic fixed points 182-

300to invariant tori 313-332to periodic orbits 313-332

homoclinic torus 319,321,463-464homoclinic torus tangle 465horizontal slab 108-109, 112-114,

116-118, 153--159, 425full intersection of 114, 154horizontal boundary of 112, 154vertical boundary of 112, 154width of 116, 156-157

horizontal slice 111-112, 151-153horseshoe, see Smale horseshoehyperbolic fixed point 29, 341-345hyperbolic invariant manifold 347,

357, 371, 373, 382hyperbolic invariant set 145-149hyperbolic invariant tori 318-332hyperbolic set 145-149

nonuniformly hyperbolic 146uniformly hyperbolic 146

improper integral 408, 447, 454integral curve 5invariant manifold 26-56, 352

inflowing 47-48, 355locally 355normally hyperbolic 347, 357,

371, 373, 382overflowing 47-48,355

perturbed 30-32

stability of 33, 48-56unperturbed 27-30,32

invariant set 20, 79-85, 121-126,145-149

491

hyperbolic 145-149negative 20positive 20

invariant splitting 321invariant tori 313-332

normally hyperbolic 318-332iterate 14

Josephson junction 471

KAM theorem 337, 386, 394-395,401, 458, 474

KAM tori 386-387, 393, 395, 398knot theory 333

knot-type periodic orbit 227Kupka-Smale theorem 60

A-lemma 323toral A-lemma 324

linearization 18-19Liouville's theorem 275local bifurcation, see under bifurcationlocal cross-section 417Lorenz cross-section 465Lyapunov-type numbers 33, 49-54,

347, 349, 357

manifoldatlas on 34-35Cr manifold of dimension m 34

-with boundary 46chart on 34-35coordinate chart on 35-36differentiable 33-34, 46

- with boundary 46global stable 29-30global unstable 29-30invariant, see under invariant

manifoldlocal stable 29-30local unstable 29splitting of 359-369, 375-380,

387-393stable, see under stable manifoldunstable, see under unstable

492

manifoldwith boundary 46

map 14asymptotic behavior of 20period k point of 15

maximal interval of existence 5

Melnikov vector 366,379, 392,396-418,446time-dependent 402-403

metric on EN 97-98Mobius strip 214-215, 221motion, see trajectory

negative invariant set 20negatively invariant subbundle 48Newhouse sinks 333nonautonomous systems 9noncontinuable 5

nonresonance 358-359nonresonant motions 382-385nonwandering point 20nonwandering set 20normal forms 471normally hyperbolic 55, 320-321normally hyperbolic invariant

manifold 347, 357, 371, 373normally hyperbolic invariant tori

318-332

orbit 6-7, 346; see also individualentries

ordinary differential equations 1

asymptotic behavior of 16autonomous systems of 8continuation of solutions 4

dependence on initial conditionsand parameters 3

existence and uniqueness of

solutions 2

maps of solutions 14nonautonomous systems of 8-10noncontinuable solutions of 5periodic solutions of 15quasiperiodic solutions of 16

Index

special solutions of 15stability 16-20

orientation-preserving 74

pendulum 12-13, 171-172, 418-428,468-470coupled 452-458whirling 440-451

perfect set 99period-doubling bifurcation, see under

bifurcationperiodic orbit 15

of knot-type 227of period k 15

periodic motions 15period k point 15persistence 354-355phase curve, see trajectoryphase flows 11phase space 1, 12-15

fixed point in 15with structure of circle 12with structure of cylinder 12with structure of sphere 13with structure of torus 12

phase transitions 181-182pitchfork bifurcation, see under

bifurcationPoincare map 67-74,183-184,188-

198,200-201,208-211, 228-232,

241-243, 254-255,261-265,268-

272,278-281,288-298, 302-305,

308-309, 417, 434, 445, 450point vortices 178-180Poisson bracket 341, 409positively invariant subbundle 349power spectra 471power system dynamics 471

quasiperiodic excitation 460quasiperiodic function 16quasiperiodic motions 15-16quasiperiodic orbit 16

residual set 60

Index

resonance bands 474resonance phenomena 373rest point, see fixed pointrigid body dynamics 176-178, 471

saddle-node bifurcation, see underbifurcation

sector 128

stable sector at pp 163stable sector at z0 128unstable sector at pp 163unstable sector at z0 128

sector bundles 128-129,161-165sensitive dependence on initial

conditions 94separatrix 173

shift map 86-87, 100-107Silnikov phenomena 227, 251-252,

258Silnikov-type strange attractor 258

singular point, see fixed pointslab, see horizontal slab and vertical

slabslice, see horizontal slice and vertical

sliceslowly varying oscillators 429-451Smale-Birkhoff homoclinic theorem

332, 421, 423Smale horseshoe 76-94, 176, 221-

223, 231-240, 242, 257-260, 266-267, 272-274, 286, 298, 306, 312,437, 449

space of bi-infinite sequences of twosymbols 87

sphere 13

splitting distance 472splitting of manifolds 359-369,375-

380, 387-393stability 16-20, 216-219

asympotic 17linearization method for 18-19Lyapunov 17

stable manifold 26,29-31,147,347,355-356

state space, see phase space

493

stationary point, see fixed pointstrange attractor 226-227

Duffing-Holmes 465Silnikov-type 258

structural stability 58-62subshift of finite type 101-108subshift of infinite type 107symbolic dynamics 86-90, 94-107

tangent bundle 45, 321tangent space 40-44

tangent vector 41

sectors of 128toral A-lemma 324torus 12, 161, 318-332, 347, 352,

373trajectory 5, 346transition chain 395transition matrix 102-103transversality 56-57, 367-369, 380,388-389,393transverse homoclinic point 314transverse homoclinic torus 319, 321,

463-464traveling wave solutions 180-181

unfoldings 65universal 65

unstable 17unstable manifold 26, 29-31, 147,

347, 355-356

vector field 2

autonomous 8cross-section to 68nonautonomous 8

versal deformations 65vertical slab 108-109, 114-118,

155-159horizontal boundary of 115, 156vertical boundary of 115, 156width of 116, 156-157

vertical slice 111-112, 152-153

whirling pendulum, see under

494Index

pendulumWhitney-sum 48

Yang-Mills field theory 471

Applied Mathematical Sciences

cont. from page ii

55. Yosida: Operational Calculus: A Theory of Hyperfunctions.56. Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications.57. Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations.58. Dwover/Hussaini/Voigr (eds.): Theoretical Approaches to Turbulence.59. Sunders/Verhulsi: Averaging Methods in Nonlinear Dynamical Systems.60. Ghil/Childress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate

Dynamics.61. Sattinger/Weaver: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics.62. LaSa/le: The Stability and Control of Discrete Processes.63. Grasman: Asymptotic Methods of Relaxation Oscillations and Applications.64. Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems.65. Rend/Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra.66. H/oviu'ek/Has/inger/Nei is/Lovisek: Solution of Variational Inequalities in Mechanics.67. Cereignani: The Boltzmann Equation and Its Applications.68. Temarn: Infinite Dimensional Dynamical System in Mechanics and Physics.69. Golubitskv/Stewart/Schaeffer: Singularities and Groups in Bifurcation Theory, Vol. IL70. Cot..stontin/Foios/NicolaenkolTemam: Integral Manifolds and Inertial Manifolds for Dissipative Partial

Differential Equations.71. Catlin: Estimation. Control, and the Discrete Kalman Filter.72. Lochak/Meunier: Multiphase Averaging for Classical Systems.73. Wiggins: Global Bifurcations and Chaos.74. Mawhin/Wi/lem: Critical Point Theory and Hamiltonian Systems.75. AbrahamlMarsden/Ratiu: Manifolds, Tensor Analysis, and Applications. 2nd ed.76. Lagerstrom: Matched Asymptotic Expansion: Ideas and Techniques.

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