On Poincaré’s legacyin dynamical systems

Henk Broer

Johann Bernoulli Institute for Mathematics and Computer Science

Rijksuniversiteit Groningen

Summary

i. Introduction

ii. Celestial mechanics

iii. Geometric ideas

iv. Multi- or quasi-periodicity

v. Invariant measures

vi. Concluding remarks

IntroductionClassical tool for problems in mathematical physics,

celestial mechanics, etc.:mainly calculus

New methods / toolsintroduced by Poincaré:geometry (in a wide sense) and groupsalso sense of probability

Legacy contains general theory of dynamical systems

J.H. Poincaré,Les Méthodes Nouvelles de la Mécanique CélesteI, II, III . Gauthier-Villars 1892,

93, 99. Blanchard 1987

J.H. Poincaré,Œuvres.Gauthier-Villars 1915–1956

Celestial mechanics ILa Meccanica é il paradiso delle

scienze matematiche, perché con quella si vieneal fruto matematico

Leonardo da Vinci

- Newton’s universal gravitation perturbation theory

- Laplace: is Solar system ‘stable’?

- Discovery Neptune 1821by ‘abberations’ in Uranus orbit(Adams, Leverrier, Bessel)

C.M. Linton,From Eudoxus to Einstein, A History of Mathematical Astronomy. Cambridge

University Press 2004

Celestial mechanics II- Three body problem:

Earth-Moon-Sun, Sun-Jupiter-asteroid

- Restricted Planar Three Body Problem stable?i.e. do perturbation series converge?

- Dirichlet and Weierstraß:Es werden die Nenner unendlich klein

- King Oscar II of Norway:60th birthday competition 1889

J.K. Moser,Stable and Random Motions in Dynamical Systems, with SpecialEmphasis on

Celestial Mechanics.Second Edition, Ann. Math. Studies, Vol.77, Princeton Univ. Press 2001

June Barrow-Green,Poincaré and the Three Body Problem.History of Mathematics, Volume11American Mathematical Society London Mathematical Society 1997

Complex Linearization- Given holomorphic germF : (C, 0) −→ (C, 0)

F (z) = λz + f(z)

with f(0) = f ′(0) = 0 andλ ∈ C parameter

- Problem: existence of bi-holomorphicdiffeomorphismΦ : (C, 0) −→ (C, 0), such that

Φ ◦ F = λ · Φ,

i.e., that linearizesF

- Hyperbolic case|λ| 6= 1 solved by Poincaré

J.H. Poincaré, Sur les propriétés des fonctions définies parles équations aux différences

partielles. In:ŒuvresI, Gauthier-Villars 1915-1956.

V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations.Grundlehren

der mathematischen Wissenschaften250, Springer 1983; 1988

Small divisors I- Formal solutionΦ(z) = z +

∑j≥2 φjz

j ⇒

φn =1

λ(1− λn)fn + known

- Formal series exists forλ 6= 0 andλ no root of 1

- Still small divisors . . .

Small divisors II- Convergence under Diophantine conditions:

there exist∃γ, τ > 0 such that∀p/q:

|λ− e2πip

q | ≥γ

qτ

in T1 meagre set of full measure

- Bruno condition

∑

n

log(qn+1)

qn< ∞

necessary and sufficient Fields medal Yoccoz

C.L. Siegel, Iteration of analytic functions.Ann. Math.42 (1942), 607-612

J.-C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques.Astérisque231(1995), 3-88

THE story- First solution: stability of the RTBP

Poincaré was awarded the prize

- Found mistake on query of Phragmén.Used prize money to buy all published volumesof Acta Math., nextpublished improved solution in same volume

J.H. Poincaré, Sur le problème des trois corps et les équations de la dynamique.Acta Math.13(1890), 1-270

The master himselfQue l’on cherche à se représenter la figure formée par

ces courbes. . . On sera frappé de la complexité decette figure, que je ne cherche même pas à tracer. Rien

n’est plus propre à nous donner une idée de lacomplication du problême des trois corps et, en

général, de tous les problèmes de dynamique où il n’ya pas d’intégrale uniforme et où les séries de Bohlin

sont divergentes

- Tangle= geometric obstruction for convergenceof perturbation series

- Transversal homo- and heteroclinic intersections

- Persistence⇒ divergence is significant

Tangle and the Smale Horseshoe

pq

D

Φn(D)

- Uses Poincaré map !

- Symbolic dynamics

- Positive topological entropy, weak form of chaos

S. Smale, Differentiable dynamical systems.Bull. Amer. Math. Soc.73 (1967), 747-817

Resonance of Jupiter satellites

W. de Sitter, Outlines of a new mathematical theory of Jupiter’s satellites.Koninklijke Akademie

van Wetenschappen te Amsterdam, Proceedings of the Sectionof Sciences20 (1918), 1289-99 and

1300-08;Ann. Sterrewacht Leidendeel12 (1918) 1, 1-53

W. de Sitter, Theory of Jupiter’s Satellites I, The intermediary orbit; II, The variations.

Koninklijke Akademie van Wetenschappen te Amsterdam. Proceedings of the Section of Sciences

21 (1919), 1156-63;22 (1920), 236-241

V. Lainey, L. Duriez and A. Vienne, New accurate ephemeridesfor the Galilean satellites of

Jupiter.A&A 420(2004), 1171-1183

[PS: relativistic correction in LDV due to De Sitter]

Geometric methods / tools I- Poincaré–Hopf index:

Consider isolated equilibriump in 2–dimensionalvector fieldXFollow X/||X|| in a small circle around this mapT1 → T

1

ThenIndexp(X) is its winding numberIf M is compact surface:

∑

X(p)=0

Indexp(X) = χ(M)

PS: Generalized to higher dimension by H. Hopf using Brouwer’s degree

Geometric methods / tools II- Poincaré–Bendixson Theorem:

For a vector field onR2 or S2

the only possibleω–limit sets are:• equilibria,• closed integral curves or• graphs of homo- / heteroclinic connections

Often used to prove existence of limit cyclePS: Mind Hilbert XVI !

Geometric methods / tools III- Poincaré–Birkhoff Theorem:

Given planar diffeomorphim that preserves areaConsider annulus between two invariant circleswith rotation numbers1 < 2Then, inside the annulus, for any rational

1 <p

q< 2

a periodic orbit exists with rotation numberp/q

- Complexity. . .

J.K. Moser, Lectures on Hamiltonian systems.Memoirs Amer. Math. Soc. 81 (1968), 1-60

Multi- or quasi-periodicity- Near integrability, persistence of quasi-periodicity,

each motion densely filling an invariant torus:⇒ stability of Solar system?

- Poincaré (conjecture): true for many initial states;Kolmogorov & al.: true for positive measure

A.N. Kolmogorov, On the persistence of conditionally periodic motions under a small change of

the Hamilton function.Dokl. Akad. Nauk SSSR98 (1954), 527–530 (in Russian); In: G. Casati

and J. Ford (eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems. Volta

Memorial Conference (1977),Lecture Notes in Phys.93, Springer 1979, 51–56; Reprinted in: Bai

Lin Hao (ed.),Chaos, World Scientific 1984, 81–86

Kolmogorov–Arnold–Moser

- Small divisors as before, overcome by Diophantineconditions

- Persistence of quasi-periodic motions / toriof positive measure also true when restrictingto an energy-hyper surface

C.L. Siegel and J.K. Moser,Lectures on Celestial Mechanics.Die Grundlehren der

mathematische Wissenschaften in Einzeldarstellungen, Band187, Springer 1955; 1971

J.K. Moser, On invariant curves of area-preserving mappings of an annulus,Nachr. Akad. Wiss.

Göttingen, Math.-Phys. Kl. II.1 (1962), 1-20

V.I. Arnold, Proof of a theorem by A.N. Kolmogorov on the persistence of conditionally periodic

motions under a small change of the Hamilton function,Russian Math. Surveys18 (5) (1963),

9-36

Another ‘toy’ example

Parametrically forced pendulum

x+ (a+ ε cos t) sin x = 0 (swing)

- Poincaré map swing in1 : 2 resonance

- periodicity, quasi-periodicity and chaos. . .

Invariant measures I- Liouville measure for Hamiltonian systems

Conditional Liouville measure on energyhypersurfaces

- Sinai–Bowen–Ruelle measureson certain invariant sets, like (strange) attractors

D. Ruelle and F. Takens, On the nature of turbulence.Comm. Math. Phys.20 (1971) 167-192;

23 (1971) 343-344

D. Ruelle,Thermodynamics Formalism.Addison-Wesley 1978

Invariant measures II- Poincaré recurrence:

Let Volume(D) be bounded andΦ : D −→ Dpreserve volumeThen∀x ∈ D and∀ nbhdU ∋ x ∃y ∈ U suchthatΦn(y) ∈ U for n > 0Indeed, consider

U,Φ(U),Φ2(U), . . . , all same volume

Volume(D) bounded⇒ ∃k > ℓ ≥ 0 such that

Φk(U) ∩ Φℓ(U) 6= ∅ ⇒ Φk−ℓ(U) ∩ U 6= ∅

Now taken = k − ℓ andy ∈ Φk−ℓ(U) ∩ U

Invariant measures III- Repeat argument withU ↔ Φn1(y) with n1 = k − ℓ infinite ‘return times’n1 < n2 < n3, . . .

- Return times many orders of magnitude larger thanestimated age of universe. . .

- Beginning of mathematical subject ‘ergodic theory’

J.H. Poincaré, Sur le problème des trois corps et les équations de la dynamique.Acta Math.13(1890), 1-270

V.I. Arnold, Mathematical Methods of Classical Mechanics.GTM 60, Springer 1978; 1989

R. Mañé,Ergodic Theory of Differentiable Dynamics, Springer 1983

S. Vandoren and G. ’t Hooft,Tijd in Machten van Tien, Natuurverschijnselen en hun Tijdschalen.

Van Veen Magazines 2011

Invariant measures IV- KAM theory⇒ Hamiltonian systems not ergodic

conflicts with Statistical Mechanics

in particular with Ergodic Hypothesis

e.g., see Kolmogorov’s closing addressof the IMC 1954 in the Concertgebouw

A.N. Kolmogorov,The general theory of dynamical systems and classical mechanics. In:

J.C.H. Gerretsen and J. de Groot (eds.), Proceedings of the International Congress of

Mathematicians1, Amsterdam 1954 North-Holland 1957, 315–333 (in Russian);Reprinted in:

International Mathematical Congress in Amsterdam, 1954 (Plenary Lectures), Fizmatgiz 1961,

187–208; Reprinted as Appendix in: R.H. Abraham and J.E. Marsden, Foundations of

Mechanics, Second Edition, Benjamin/Cummings (1978), 741–757.

Concluding remarks I

Stability of the Solar System?

- The KAM Theorem cannot establish stability for alarge set of initial states:the real non-integrability is too strong

- The inner Solar system is chaotic(positive Lyapunov exponent)‘problems’ in∼ 100 Miljon yrs . . .

V.I. Arnold, Small denominators and problems on the stability of motions in the classical and

celestial mechanics.Uspehy Math. Nauk18(6) (1963) 91-192.

J. Féjoz, Démonstration du «théorème d’Arnold» dur la stabilité du système planétaire (d’apres

M. Herman).Ergod. Th. & Dynam. Sys.24 (2004) 1-62

J. Laskar and M. Gastineau, Existence of collisional trajectories of Mercury, Mars and Venus with

the Earth.Nature Letters459 | 11 June 2009 | doi:10.1038/nature08096

Concluding remarks II

René Thom (1923-2002) Steve Smale (1930- )

- Fields medallists on Poincaré conjecturetransversality, genericity bifurcation theory⊂ singularity theory

R. Thom,Stabilité Structurelle et Morphogénèse, Benjamin 1972

V.I. Arnold (ed.), Dynamical Systems V. Bifurcation Theoryand Catastrophe Theory,

Encyclopædia of Mathematical Sciences, Vol. 5, Springer 1994

Thom himselfLa géométrie est une magie qui réussit.

J’aimerais énoncer une réciproque : toute magie,dans la mesure ou elle réussit, n’est-elle pas

nécessairement une géométrie ?

J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of

Vector Fields.Appl. Math. Sciences42, Springer 1983

F. Verhulst,Nonlinear Differential Equations and Dynamical Systems, Springer 1990; 1996

J. Palis and F. Takens,Hyperbolicity & Sensitive Chaotic Dynamics at Homoclinic Bifurcations,

Cambridge University Press 1993

H.W. Broer and F. Takens,Dynamical Systems and Chaos.Epsilon Uitgaven64, 2009; Appl.

Math. Sciences172, Springer 2011