Lecture notes 4 Nonlinear Dynamics YFX1520

Lecture 4 2-D homogeneous linear systems classifica-tion of fixed points in 2-D systems Lyapunov stabilitybasin of attraction

Contents

1 Linear homogeneous 2-D systems 211 Introduction 212 2-D phase portrait plotting 213 Why bother with linear homogeneous 2-D systems 314 Examples 4

141 Harmonic oscillator 4142 Damped harmonic oscillator 4

2 Theory of 2-D linear systems 4

3 Classification of fixed points in 2-D linear systems 531 CASE 1 Saddle node (also called saddle point or saddle) 532 CASE 2a Node 633 CASE 2b Spiral 734 CASE 3 Center 835 CASE 4a Degenerate node (of the first type) 936 CASE 4b Degenerate node (of the second type Star) 1037 CASE 5a Non-isolated fixed point a line of fixed points 1138 CASE 5b Non-isolated fixed point a plane of fixed points 1239 Summary overview 12310 A note on degenerate nodes 14

4 Basin of attraction 14

Handout Classification of fixed points of linear homogeneous 2-D systems

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Lecture notes 4 Nonlinear Dynamics YFX1520

1 Linear homogeneous 2-D systems

11 Introduction

We continue our discussion of second-order systems in the form

x = 983187f(983187x) (1)

where 983187f is an arbitrary function 983187x = (x y)T and 983187x isin R2 The component form of Eq (1) is983083x = f(x y)

y = g(x y)(2)

where f and g are functions The fixed point in the case (1) is defined as follows

x = 0 rArr 983187f(983187xlowast) = 9831870 (3)

and for the component form (2)983083x = 0

y = 0rArr

983083f(xlowast ylowast) = 0

g(xlowast ylowast) = 0(4)

In this lecture we will discuss second-order linear homogeneous systems with constant coefficients Thefollowing slide show an example of a homogeneous differential equation and compares it to its nonhomo-geneous counterpart

Slide 3

Second-order dicrarrerential equations

Homogeneous dicrarrerential equation

ax+ bx+ cx = 0 (x = y

ay = by cx(1)

Nonhomogeneous dicrarrerential equation

ax+ bx+ cx = f(x) (x = y

ay = by cx+ f(x)(2)

Non-autonomous dicrarrerential equation

ax+ bx+ cx = f(t) (x = y

ay = by cx+ f(t)(3)

In all of the above cases a b and c are the constant coecients and

f is an arbitrary (linear) function

In a more general setting a b and c may depend on time tDKartofelev YFX1520 3 15

Also included here is the non-autonomous system mentioned in Lecture 1

12 2-D phase portrait plotting

As introduced in the previous lecture phase portrait of second-order Sys (2) is constructed by plottingvectorfield x = (x y)T = (f(x y) g(x y))T against the independent variables x and y in xy-plane as shownin Fig 1 and on Slide 4 A trajectory of vector field represents a single solution of Sys (2) correspondingto an initial conditionmdasha single point in the plane (x0 y0) where x0 = x(0) y0 = y(0)

Essence of the course (Henri Poincareacute) Construction of phase portrait allows us to find all qualita-tively different trajectories (solutions) of a system without solving the system itself explicitly (analytically)

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Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 1 (Left) Phase portrait of a 2-D or a second-order problem A trajectory is shown with thecontinuous line where t1 lt t2 lt t3 lt t4 (Right) Idea behind the construction of a 2-D phase portrait(entire vector field for the shown ranges of x and y) using computer The vector field vectors are shown forpoints (x y) placed in a uniform grid that is shown with the dashed lines

Slide 42-D phase portrait1

(x = f(x y)

y = g(x y)(4)

1See Mathematica nb file uploaded to course webpageDKartofelev YFX1520 4 15

An example of phase portrait plotting procedure and different visualisation styles of phase portraits areshown in the following numerical file

Numerics cdf1 nb1Vector field plotting Integrated solution and phase portrait of damped harmonic oscillator

13 Why bother with linear homogeneous 2-D systems

In nonlinear systems it is harder to determine the stability of fixed points due to a possibility of morecomplex dynamics present in the phase portraits For this reason in future lectures we will be linearisingnonlinear systems and then studying the stability and dynamics of the corresponding linearised systemswith the aim to gain insight into the original nonlinear systems The conditions under which this approachis allowed and feasible will be discussed in the lectures to come This means that we need to familiariseourselves with linear second-order systems and their dynamics

The general form of linear homogeneous Sys (2) is the following983083x = ax+ by

y = cx+ dy(5)

where a b c d isin R are the constant coefficients By defining 983187x = (x y)T we rewrite Sys (5) in the matrixform

x = A983187x equiv983061a bc d

983062983187x (6)

where matrix A is the system or coefficient matrix of Sys (5) Dynamics of linear systems is fully determinedby the eigenvalues λ and eigenvectors 983187v of matrix A

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Lecture notes 4 Nonlinear Dynamics YFX1520

1 Linear homogeneous 2-D systems

11 Introduction

We continue our discussion of second-order systems in the form

x = 983187f(983187x) (1)

where 983187f is an arbitrary function 983187x = (x y)T and 983187x isin R2 The component form of Eq (1) is983083x = f(x y)

y = g(x y)(2)

where f and g are functions The fixed point in the case (1) is defined as follows

x = 0 rArr 983187f(983187xlowast) = 9831870 (3)

and for the component form (2)983083x = 0

y = 0rArr

983083f(xlowast ylowast) = 0

g(xlowast ylowast) = 0(4)

In this lecture we will discuss second-order linear homogeneous systems with constant coefficients Thefollowing slide show an example of a homogeneous differential equation and compares it to its nonhomo-geneous counterpart

Slide 3

Second-order dicrarrerential equations

Homogeneous dicrarrerential equation

ax+ bx+ cx = 0 (x = y

ay = by cx(1)

Nonhomogeneous dicrarrerential equation

ax+ bx+ cx = f(x) (x = y

ay = by cx+ f(x)(2)

Non-autonomous dicrarrerential equation

ax+ bx+ cx = f(t) (x = y

ay = by cx+ f(t)(3)

In all of the above cases a b and c are the constant coecients and

f is an arbitrary (linear) function

In a more general setting a b and c may depend on time tDKartofelev YFX1520 3 15

Also included here is the non-autonomous system mentioned in Lecture 1

12 2-D phase portrait plotting

As introduced in the previous lecture phase portrait of second-order Sys (2) is constructed by plottingvectorfield x = (x y)T = (f(x y) g(x y))T against the independent variables x and y in xy-plane as shownin Fig 1 and on Slide 4 A trajectory of vector field represents a single solution of Sys (2) correspondingto an initial conditionmdasha single point in the plane (x0 y0) where x0 = x(0) y0 = y(0)

Essence of the course (Henri Poincareacute) Construction of phase portrait allows us to find all qualita-tively different trajectories (solutions) of a system without solving the system itself explicitly (analytically)

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Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 1 (Left) Phase portrait of a 2-D or a second-order problem A trajectory is shown with thecontinuous line where t1 lt t2 lt t3 lt t4 (Right) Idea behind the construction of a 2-D phase portrait(entire vector field for the shown ranges of x and y) using computer The vector field vectors are shown forpoints (x y) placed in a uniform grid that is shown with the dashed lines

Slide 42-D phase portrait1

(x = f(x y)

y = g(x y)(4)

1See Mathematica nb file uploaded to course webpageDKartofelev YFX1520 4 15

An example of phase portrait plotting procedure and different visualisation styles of phase portraits areshown in the following numerical file

Numerics cdf1 nb1Vector field plotting Integrated solution and phase portrait of damped harmonic oscillator

13 Why bother with linear homogeneous 2-D systems

In nonlinear systems it is harder to determine the stability of fixed points due to a possibility of morecomplex dynamics present in the phase portraits For this reason in future lectures we will be linearisingnonlinear systems and then studying the stability and dynamics of the corresponding linearised systemswith the aim to gain insight into the original nonlinear systems The conditions under which this approachis allowed and feasible will be discussed in the lectures to come This means that we need to familiariseourselves with linear second-order systems and their dynamics

The general form of linear homogeneous Sys (2) is the following983083x = ax+ by

y = cx+ dy(5)

where a b c d isin R are the constant coefficients By defining 983187x = (x y)T we rewrite Sys (5) in the matrixform

x = A983187x equiv983061a bc d

983062983187x (6)

where matrix A is the system or coefficient matrix of Sys (5) Dynamics of linear systems is fully determinedby the eigenvalues λ and eigenvectors 983187v of matrix A

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Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 1 (Left) Phase portrait of a 2-D or a second-order problem A trajectory is shown with thecontinuous line where t1 lt t2 lt t3 lt t4 (Right) Idea behind the construction of a 2-D phase portrait(entire vector field for the shown ranges of x and y) using computer The vector field vectors are shown forpoints (x y) placed in a uniform grid that is shown with the dashed lines

Slide 42-D phase portrait1

(x = f(x y)

y = g(x y)(4)

1See Mathematica nb file uploaded to course webpageDKartofelev YFX1520 4 15

An example of phase portrait plotting procedure and different visualisation styles of phase portraits areshown in the following numerical file

Numerics cdf1 nb1Vector field plotting Integrated solution and phase portrait of damped harmonic oscillator

13 Why bother with linear homogeneous 2-D systems

In nonlinear systems it is harder to determine the stability of fixed points due to a possibility of morecomplex dynamics present in the phase portraits For this reason in future lectures we will be linearisingnonlinear systems and then studying the stability and dynamics of the corresponding linearised systemswith the aim to gain insight into the original nonlinear systems The conditions under which this approachis allowed and feasible will be discussed in the lectures to come This means that we need to familiariseourselves with linear second-order systems and their dynamics

The general form of linear homogeneous Sys (2) is the following983083x = ax+ by

y = cx+ dy(5)

where a b c d isin R are the constant coefficients By defining 983187x = (x y)T we rewrite Sys (5) in the matrixform

x = A983187x equiv983061a bc d

983062983187x (6)

where matrix A is the system or coefficient matrix of Sys (5) Dynamics of linear systems is fully determinedby the eigenvalues λ and eigenvectors 983187v of matrix A

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Lecture notes 4 Nonlinear Dynamics YFX1520

14 Examples

141 Harmonic oscillator

Harmonic oscillator is described by a linear system in the following form

x+ x = 0 rArr983083x = y

y = minusx(7)

where x is the displacement The matrix form of this system for 983187x = (x y)T and appropriately selectedsystem matrix A is

x =

9830610 1minus1 0

983062983187x (8)

The solution and phase portrait of harmonic oscillation (7) are shown in the following numerical file

Numerics cdf1 nb1Vector field plotting Integrated solution and phase portrait of damped harmonic oscillator

142 Damped harmonic oscillator

Damped harmonic oscillator is described by a linear system in the following form

x+ x + x983167983166983165983168Damping

= 0 rArr983083x = y

y = minusxminus y(9)

where x is the displacement and the last term on the right-hand side of the equation is the damping orfriction term The matrix form of this system for 983187x = (x y)T and appropriately selected system matrix Ais

x =

9830610 1minus1 minus1

983062983187x (10)

The solution and phase portrait of damped harmonic oscillation (9) are shown in the following numericalfile

Numerics cdf1 nb1Vector field plotting Integrated solution and phase portrait of damped harmonic oscillator

2 Theory of 2-D linear systems

As stated above fixed point dynamics of a linear 2-D system is determined by the eigenanalysis of itssystem matrix A The following is a short reminder of your linear algebra courses

Slides 6ndash9

2-D linear systems

Letrsquos consider 2-D linear system given in the form

~x = A~x (7)

where

A =

a bc d

(8)

a b c d 2 R and ~x = (x y)T The fixed point ~x = ~0 and the

phase portrait near ~x = ~0 is fully determined by the eigenvalues and

eigenvectors of system matrix A

What are the eigenvalues and vectors of a system Following is a

short reminder of your linear algebra courses

DKartofelev YFX1520 6 15

2-D linear systems

We seek a straight line solution (assumption) in the following form

~x(t) = ~vet (9)

where ~v is the eigenvector and is the eigenvalue

Eqs (7) and (9) yield

~x = ~vet A~x = A(~vet) (10)

~vet = A~vet | divide et (11)

~v = A~v (12)

a useful relationship between the eigenvalues and eigenvectors of a

system

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Lecture notes 4 Nonlinear Dynamics YFX1520

2-D linear systems

The (straight line) solution exists if one can find eigenvalues and

eigenvectors ~vEigenvalue is given by

det(A I) = 0 )a bc d

= 2 + = 0 (13)

where the boxed part is called characteristic equation of a system and

where

= a+ d is the trace of matrix A (14)

and

= ad bc is the determinant of matrix A (15)

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2-D linear systems

2 + = 0

Algebraic form of two eigenvalues

12 = plusmn

p 2 4

2 (16)

Additional nice propertied of and are the following

= 1 + 2 (17)

= 12 (18)

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Dynamics of linear 2-D systems is determined by system matrix determinant ∆ and trace τ

One can use computer to perform eigenanalysis of square matrices The following numerical file containsan example of eigenanalysis

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systems

3 Classification of fixed points in 2-D linear systems

All possible behaviours found in linear homogeneous second-order systems are presented here The classi-fication of fixed points is presented in terms of system matrix determinant ∆ and trace τ

31 CASE 1 Saddle node (also called saddle point or saddle)

Criterion for determining the fixed point∆ lt 0 (11)

This criterion holds for real and distinct (not equal) eigenvalues and linearly independent (not on the sameline) eigenvectors Since ∆ = λ1λ2 we must have λ1 ≶ 0 and λ2 ≷ 0 The determination of saddle nodefixed point does not depend on trace τ (minusinfin lt τ lt infin) Figures 2 and 3 show the dynamics of a saddleStability The fixed point 983187xlowast = 9831870 = (0 0)T is always unstable although the phase portrait has one stableeigendirection (other one is unstable)

Figure 2 Phase portrait of saddle node fixed point where eigenvalues λ1 gt 0 and λ2 lt 0

General solution of the system983187x(t) = C1e

λ1t983187v1 + C2eλ2t983187v2 (12)

where Ci are the integration constants and 983187vi are the eigenvectors See Figs 2 and 3

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Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 2

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Figure 3 Phase portrait and eigenvectors Eigenvectors are shown with the red arrows (Left) Saddlenode (Right) Saddle node different flow directions

32 CASE 2a Node

Criterion for determining the fixed point

∆ gt 0 and τ2 minus 4∆ gt 0 (13)

This criterion holds for real and distinct eigenvalues having the same sign λ1λ2 ≷ 0 Eigenvectors arelinearly independent The criterion τ2minus4∆ gt 0 simply means that we are located outside the region definedby τ2 minus 4∆ = 0 shown in the overview plot in Fig 17 and Slide 10 Figures 4 and 5 show the dynamics ofa node

Figure 4 Phase portrait of stable node where λ1 lt λ2 lt 0 As t rarr infin trajectories approach the fixed pointtangentially to the slower eigendirection

Stability The fixed point 983187xlowast = 9831870 may be either stable (attracting sink) or unstable (repelling source) If

τ lt 0 (14)

then we have a stable node and from (13) it also follows that λ1λ2 lt 0 If

τ gt 0 (15)

then we have an unstable node and from (13) it follows that λ1λ2 gt 0General solution of the system

983187x(t) = C1eλ1t983187v1 + C2e

λ2t983187v2 (16)

where Ci are the integration constants and 983187vi are the eigenvectors See Figs 4 and 5

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Lecture notes 4 Nonlinear Dynamics YFX1520

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 4

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y(t)

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Figure 5 Phase portrait and eigenvectors Eigenvectors are shown with the red arrows (Left) Stablenode (Right) Unstable node

33 CASE 2b Spiral

Criterion for determining the fixed point

∆ gt 0 and τ2 minus 4∆ lt 0 (17)

This criterion holds for complex and distinct eigenvalues The criterion τ2 minus 4∆ lt 0 simply means that weare located inside the region defined by τ2 minus 4∆ = 0 shown in the overview plot in Fig 17 and Slide 10Figures 6 and 7 show the dynamics of a spiral

Figure 6 Phase portrait of stable spiral in the case micro lt 0

Physical interpretation of complex eigenvalues λ = micro + iω is the following real part micro is the decay rate(micro lt 0) of the spiral and imaginary part ω is the rotation rateStability The fixed point 983187xlowast = 9831870 may be either stable (attracting sink) or unstable (repelling source) If

τ lt 0 (18)

witch also implies that micro lt 0 then we have a stable spiral If

τ gt 0 (19)

witch also implies that micro gt 0 then we have an unstable spiralGeneral solution of the system A linear combination of

emicro1t cosω1t (20)

andemicro2t sinω2t (21)

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Lecture notes 4 Nonlinear Dynamics YFX1520

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 6

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y(t)

Figure 7 Phase portrait (Left) Stable spiral (Right) Unstable spiral

Which way is vectorfield rotating This question is not an issue when one uses computer to constructthe phase portrait But letrsquos say you need to sketch the portrait by hand System matrix A does notexplicitly give you the direction In order to determine the direction simply calculate one vector and thedirection of the entire flow becomes clear

Figure 8 Determination of the rotation direction Flow vector calculated for coordinate (x y)T = (1 0)T is shown with the bold arrow Phase portrait features Lyapunov stable fixed point at its origin

Example We consider a simple case that was introduced abovemdashharmonic oscillator defined by Eq (7)983083x = y

y = minusx

Letrsquos say we are interested in a vector located at (x y)T = (1 0)T (shown with the red bullet in Fig 8)The resulting vector is 983083

x = y = 0

y = minusx = minus1rArr x =

983061xy

983062=

9830610minus1

983062 (22)

This vector is shown on the phase portrait in Fig 8 From here it is clear that the field is rotating ina clockwise direction 2-D systems feature a new type of stability called the Lyapunov stability A fixedpoint is said to be Lyapunov stable if a solution starts near a fixed point and it stays near it for t rarr infin(note a more rigorous definition exists) The fixed point of harmonic oscillator shown in Fig 8 is Lyapunovstable

34 CASE 3 Center

Criterion for determining the fixed point

∆ gt 0 and τ = 0 (23)

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Lecture notes 4 Nonlinear Dynamics YFX1520

This criterion holds for pure imaginary eigenvalues λ = plusmniω (complex conjugate pair) Figures 9 and 10show the dynamics of a center

Figure 9 Phase portrait of Lyapunov stable center

Stability The fixed point 983187xlowast = 9831870 is Lyapunov stable This is true because the real part associated withthe decay rate of the rotation of the pure imaginary eigenvalues micro = 0General solution of the system A linear combination of

cosω1t (24)

andsinω2t (25)

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 9

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y(t)

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Figure 10 Phase portrait (Left) Center featuring clockwise rotation (Right) Center featuring counter-clockwise rotation

35 CASE 4a Degenerate node (of the first type)

Criterion for determining the fixed point

∆ gt 0 and τ2 minus 4∆ = 0 (26)

This criterion holds for real and repeated (equal) eigenvalues and for one uniquely determined eigenvector(the other can be anything) The criterion τ2minus4∆ = 0 simply means that we are located on the line definedby τ2 minus 4∆ = 0 shown in the overview plot in Fig 17 and Slide 10 Figures 11 and 12 show the dynamicsof this fixed pointStability The fixed point 983187xlowast = 9831870 can be either stable or unstable If

τ lt 0 (27)

which also implies λ1 = λ2 lt 0 then the fixed point is stable If

τ gt 0 (28)

which implies λ1 = λ2 gt 0 then the fixed point is unstableGeneral solution Omitted from this document

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Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 11 Phase portrait of stable degenerate node

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 11

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Figure 12 Phase portrait and non-unique eigenvector (unique one is coincidentally |983187v| = 0) Eigenvec-tor is shown with the red arrow (Left) Stable degenerate node (Right) Unstable degenerate node

36 CASE 4b Degenerate node (of the second type Star)

Criterion for determining the fixed point

∆ gt 0 and τ2 minus 4∆ = 0 (29)

This criterion holds for real and repeated (equal) eigenvalues and for no uniquely determined eigenvectors(both can be anything every direction is an eigendirection) The criterion τ2 minus 4∆ = 0 simply means thatwe are located on the line defined by τ2 minus 4∆ = 0 shown in the overview plot in Fig 17 and Slide 10Figures 13 and 14 show the dynamics of this fixed point

Figure 13 Phase portrait of stable degenerate node of the second type aka stable star

Stability The fixed point 983187xlowast = 9831870 can be either stable or unstable If

τ lt 0 (30)

which also implies λ1 = λ2 lt 0 then the fixed point is stable If

τ gt 0 (31)

which implies λ1 = λ2 gt 0 then the fixed point is unstableGeneral solution Omitted from this document

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Lecture notes 4 Nonlinear Dynamics YFX1520

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 13

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Figure 14 Phase portrait and non-unique eigenvectors Eigenvectors are shown with the red arrows(Left) Stable star (Right) Unstable star

37 CASE 5a Non-isolated fixed point a line of fixed points

Criterion for determining the fixed point

∆ = 0 and τ ∕= 0 (32)

This criterion holds for real eigenvalues On the overview plot shown in Fig 17 and Slide 10 we are locatedon the vertical τ -axis Figures 15 and 16 show the dynamics of non-isolated fixed point

Figure 15 Phase portrait of non-isolated fixed points (Left) Lyapunov stable non-isolated fixed point(Right) Unstable non-isolated fixed point

Stability The fixed points (line) can be either Lyapunov stable or unstable If

τ lt 0 (33)

then the fixed point is Lyapunov stable Ifτ gt 0 (34)

then the fixed point is unstableGeneral solution of the system

983187x(t) = C1eλ1t983187v1 + C2e

λ2t983187v2 (35)

where Ci are the integration constants and 983187vi are the eigenvectors See Figs 15 and 16

Numerics cdf2 nb2Calculation of matrix determinant trace eigenvalues and eigenvectors using computer Classificationof fixed points in 2-D linear homogeneous systemsAn example of quantitatively accurate phase portrait of the dynamics shown in Fig 15

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Figure 16 Phase portrait and eigenvectors Eigenvectors are shown with the red arrows (Left)Lyapunov stable non-isolated fixed point (Right) Unstable non-isolated fixed point

38 CASE 5b Non-isolated fixed point a plane of fixed points

Criterion for determining the fixed point

∆ = 0 and τ = 0 (36)

All points on the plane are fixed points Nothing happens and nothing can happen On the overview plotshown in Fig 17 and on Slide 10 criterion (36) is located at the origin

39 Summary overview

A nice and concise way of summarising the classification discussed in this lecture is shown on Slide 10 andin Fig 17

Slide 10

Classification of fixed points in 2-D linear systems

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Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 17 Classification of fixed points of linear homogeneous 2-D systems where trace τ = λ1 + λ2 anddeterminant ∆ = λ1λ2 are determined by 2times 2 system matrix A

The presented classification can also be summarised by the following flowchart

1003317 if ∆ lt 0Isolated fixed pointCASE 1 Saddle node

1003317 if ∆ = 0Non-isolated fixed points

bull if τ lt 0CASE 5a Line of Lyapunov stable fixed points

bull if τ = 0CASE 5b Plane of fixed points

bull if τ gt 0CASE 5a Line of unstable fixed points

1003317 if ∆ gt 0Isolated fixed point

bull if τ lt minusradic4∆

CASE 2a Stable nodebull if τ = minus

radic4∆

if there is one uniquely determined eigenvector (the other is non-unique)

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Lecture notes 4 Nonlinear Dynamics YFX1520

CASE 4a Stable degenerate node if there are no uniquely determined eigenvectors (both are non-unique)

CASE 4b Stable starbull if minus

radic4∆ lt τ lt 0

CASE 2b Stable spiralbull if τ = 0

CASE 3 Centerbull if 0 lt τ lt

radic4∆

CASE 2b Unstable spiralbull if τ =

radic4∆

if there is one uniquely determined eigenvector (the other is non-unique)CASE 4a Unstable degenerate node

if there are no uniquely determined eigenvectors (both are non-unique)CASE 4b Unstable star

bull ifradic4∆ lt τ

CASE 2a Unstable node

310 A note on degenerate nodes

Figure 18 Comparison of critically damped system solution time-series shown on the left to damped systemsolution time-series shown on the right

The degenerate nodes rarely occur in engineering applications They represent critically damped or over-damped systems These second-order systems achieve equilibrium without being allowed to oscillate Figure18 compares a possible solution of a critically damped oscillator to a solution of damped oscillator

4 Basin of attraction

The definition of the basin of attraction is given on Slide 13 The notion of basin of attraction will beusedexpanded in future lectures

Slide 13Basin of attraction

Basin of attraction of a fixed point (or an attractor) is the region

of the phase space over which integration (or iteration) is defined

such that any point (any initial condition) in that region will

eventually be integrated into an attracting region (an attractor) or to

a particular stable fixed point

In the case of linear systems with a stable fixed point every point

in the phase space is in the basin of attraction

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Notions of iteration and attractor shown within the brackets will be explained in future lectures

D Kartofelev 1415 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

Revision questions

1 How to plot 2-D phase portrait of a system2 What are 2-D homogeneous linear systems3 What are non-homogeneous systems4 Classification of fixed points in 2-D systems5 Sketch a saddle node fixed point6 Sketch a stable node fixed point7 Sketch an unstable node fixed point8 Sketch a stable spiral (fixed point)9 Sketch an unstable spiral (fixed point)

10 Sketch a center (fixed point)11 Sketch a stable non-isolated fixed point12 Sketch an unstable non-isolated fixed point13 2-D homogeneous nonlinear systems14 What does it mean that a fixed point is Lyapunov stable15 Give an example of Lyapunov stable fixed point

D Kartofelev 1515 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

Figure 17 Classification of fixed points of linear homogeneous 2-D systems where trace τ = λ1 + λ2 anddeterminant ∆ = λ1λ2 are determined by 2times 2 system matrix A

The presented classification can also be summarised by the following flowchart

1003317 if ∆ lt 0Isolated fixed pointCASE 1 Saddle node

1003317 if ∆ = 0Non-isolated fixed points

bull if τ lt 0CASE 5a Line of Lyapunov stable fixed points

bull if τ = 0CASE 5b Plane of fixed points

bull if τ gt 0CASE 5a Line of unstable fixed points

1003317 if ∆ gt 0Isolated fixed point

bull if τ lt minusradic4∆

CASE 2a Stable nodebull if τ = minus

radic4∆

if there is one uniquely determined eigenvector (the other is non-unique)

D Kartofelev 1315 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

CASE 4a Stable degenerate node if there are no uniquely determined eigenvectors (both are non-unique)

CASE 4b Stable starbull if minus

radic4∆ lt τ lt 0

CASE 2b Stable spiralbull if τ = 0

CASE 3 Centerbull if 0 lt τ lt

radic4∆

CASE 2b Unstable spiralbull if τ =

radic4∆

if there is one uniquely determined eigenvector (the other is non-unique)CASE 4a Unstable degenerate node

if there are no uniquely determined eigenvectors (both are non-unique)CASE 4b Unstable star

bull ifradic4∆ lt τ

CASE 2a Unstable node

310 A note on degenerate nodes

Figure 18 Comparison of critically damped system solution time-series shown on the left to damped systemsolution time-series shown on the right

The degenerate nodes rarely occur in engineering applications They represent critically damped or over-damped systems These second-order systems achieve equilibrium without being allowed to oscillate Figure18 compares a possible solution of a critically damped oscillator to a solution of damped oscillator

4 Basin of attraction

The definition of the basin of attraction is given on Slide 13 The notion of basin of attraction will beusedexpanded in future lectures

Slide 13Basin of attraction

Basin of attraction of a fixed point (or an attractor) is the region

of the phase space over which integration (or iteration) is defined

such that any point (any initial condition) in that region will

eventually be integrated into an attracting region (an attractor) or to

a particular stable fixed point

In the case of linear systems with a stable fixed point every point

in the phase space is in the basin of attraction

DKartofelev YFX1520 13 15

Notions of iteration and attractor shown within the brackets will be explained in future lectures

D Kartofelev 1415 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

Revision questions

1 How to plot 2-D phase portrait of a system2 What are 2-D homogeneous linear systems3 What are non-homogeneous systems4 Classification of fixed points in 2-D systems5 Sketch a saddle node fixed point6 Sketch a stable node fixed point7 Sketch an unstable node fixed point8 Sketch a stable spiral (fixed point)9 Sketch an unstable spiral (fixed point)

10 Sketch a center (fixed point)11 Sketch a stable non-isolated fixed point12 Sketch an unstable non-isolated fixed point13 2-D homogeneous nonlinear systems14 What does it mean that a fixed point is Lyapunov stable15 Give an example of Lyapunov stable fixed point

D Kartofelev 1515 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

CASE 4a Stable degenerate node if there are no uniquely determined eigenvectors (both are non-unique)

CASE 4b Stable starbull if minus

radic4∆ lt τ lt 0

CASE 2b Stable spiralbull if τ = 0

CASE 3 Centerbull if 0 lt τ lt

radic4∆

CASE 2b Unstable spiralbull if τ =

radic4∆

if there is one uniquely determined eigenvector (the other is non-unique)CASE 4a Unstable degenerate node

if there are no uniquely determined eigenvectors (both are non-unique)CASE 4b Unstable star

bull ifradic4∆ lt τ

CASE 2a Unstable node

310 A note on degenerate nodes

Figure 18 Comparison of critically damped system solution time-series shown on the left to damped systemsolution time-series shown on the right

The degenerate nodes rarely occur in engineering applications They represent critically damped or over-damped systems These second-order systems achieve equilibrium without being allowed to oscillate Figure18 compares a possible solution of a critically damped oscillator to a solution of damped oscillator

4 Basin of attraction

The definition of the basin of attraction is given on Slide 13 The notion of basin of attraction will beusedexpanded in future lectures

Slide 13Basin of attraction

Basin of attraction of a fixed point (or an attractor) is the region

of the phase space over which integration (or iteration) is defined

such that any point (any initial condition) in that region will

eventually be integrated into an attracting region (an attractor) or to

a particular stable fixed point

In the case of linear systems with a stable fixed point every point

in the phase space is in the basin of attraction

DKartofelev YFX1520 13 15

Notions of iteration and attractor shown within the brackets will be explained in future lectures

D Kartofelev 1415 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

Revision questions

1 How to plot 2-D phase portrait of a system2 What are 2-D homogeneous linear systems3 What are non-homogeneous systems4 Classification of fixed points in 2-D systems5 Sketch a saddle node fixed point6 Sketch a stable node fixed point7 Sketch an unstable node fixed point8 Sketch a stable spiral (fixed point)9 Sketch an unstable spiral (fixed point)

10 Sketch a center (fixed point)11 Sketch a stable non-isolated fixed point12 Sketch an unstable non-isolated fixed point13 2-D homogeneous nonlinear systems14 What does it mean that a fixed point is Lyapunov stable15 Give an example of Lyapunov stable fixed point

D Kartofelev 1515 1003211 As of March 2 2020

Lecture notes 4 Nonlinear Dynamics YFX1520

Revision questions

1 How to plot 2-D phase portrait of a system2 What are 2-D homogeneous linear systems3 What are non-homogeneous systems4 Classification of fixed points in 2-D systems5 Sketch a saddle node fixed point6 Sketch a stable node fixed point7 Sketch an unstable node fixed point8 Sketch a stable spiral (fixed point)9 Sketch an unstable spiral (fixed point)

10 Sketch a center (fixed point)11 Sketch a stable non-isolated fixed point12 Sketch an unstable non-isolated fixed point13 2-D homogeneous nonlinear systems14 What does it mean that a fixed point is Lyapunov stable15 Give an example of Lyapunov stable fixed point

D Kartofelev 1515 1003211 As of March 2 2020