The Easy Lectureson Nonlinear Dynamical Systemswith emphasis on physicochemical phenomena

Antonis Karantonis

April 2002

2

Preface

This collection of notes on Nonlinear Dynamical Systems emerged through a halfsemester seminar in the Department of Chemistry, Saitama University.

The main scope of the lectures (as well as the present notes) was to presentthe basic concepts of the theory of nonlinear dynamical systems with emphasison related physical phenomena. Since, the theory of dynamical systems is mainlya mathematical theory, most of the material is concentrated on mathematical def-initions, theorems and techniques but in a very applied manner. At the end ofeach chapter I tried to introduce some examples from the physical world, in orderto present some simplified applications. Additionally, a whole chapter is dedi-cated to numerical techniques suitable for studying nonlinear dynamical systems,in order to give the students a chance to try their own examples or problems.

Obviously, this collection of notes do not give the complete picture of thetheory of dynamical systems and its applications. The interested reader shouldconsult the bibliography for a more complete and accurate information.

Finally, readers should be careful to notice that this book is an uncorrectedmanuscript (even though, at this point I should thank Masao Gohdo for findingmany mistakes in the text). There might be a large number of errors throughoutthe text, but I hope the basic material still conserves some value.

1Typeset with LATEX

Contents

1 Introduction and Definitions 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Some additional definitions . . . . . . . . . . . . . . . . . . . . . 71.4 Types of dynamical response . . . . . . . . . . . . . . . . . . . . 71.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Linear Systems 152.1 Concepts from linear algebra . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . 16

2.2 Solution of linear ODEs . . . . . . . . . . . . . . . . . . . . . . . 192.3 Plane autonomous systems . . . . . . . . . . . . . . . . . . . . . 202.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Linear Stability Analysis 293.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Linearized stability . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Elementary Bifurcations 374.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Center manifold theory . . . . . . . . . . . . . . . . . . . . . . . 384.3 Static bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.3.1 The saddle-node bifurcation . . . . . . . . . . . . . . . . 404.3.2 The transcritical bifurcation . . . . . . . . . . . . . . . . 434.3.3 The pitchfork bifurcation . . . . . . . . . . . . . . . . . . 44

4.4 Normal forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.5 The Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . 49

4.5.1 The normal form for a pair of pure imaginary eigenvalues 49

3

4 CONTENTS

4.5.2 The normal form in various coordinate systems . . . . . . 524.5.3 Simplified analysis of the Hopf bifurcation . . . . . . . . 52

4.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5 Numerical Methods and Tools 675.1 Aim of this chapter . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Steady states: The Newton-Raphson method . . . . . . . . . . . . 68

5.2.1 Gauss elimination . . . . . . . . . . . . . . . . . . . . . 695.2.2 LU decomposition . . . . . . . . . . . . . . . . . . . . . 72

5.3 Use of libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Integration of ODEs: Runge-Kutta method . . . . . . . . . . . . . 745.5 The AUTO 97 package . . . . . . . . . . . . . . . . . . . . . . . . 755.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6 Chaos through examples 876.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.2 The logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . 886.3 More fun with maps . . . . . . . . . . . . . . . . . . . . . . . . . 916.4 The Rossler equations . . . . . . . . . . . . . . . . . . . . . . . . 93

Chapter 1

Introduction and Definitions

1.1 Introduction

In the physical world with the term dynamical system we define any physical phe-nomenon evolves in time. Since a physical system can be described by differentphysical variables we can say that a dynamical system is a system in which some(or all) variables evolve in time. Since most of the phenomena we observe in na-ture are progressing in time it seems clear that the study of dynamical systems is ofgreat interest. In the present series of lectures we will mainly concentrate on dis-sipative dynamical systems, i.e. systems that exchange energy and/or mass withtheir environment (most of physicochemical systems belong to this category). Inother words, we will study systems which are kept away from the thermodynamicequilibrium.

Dynamical systems are studied systematically by mathematicians, physicistsand lest commonly by chemists, for more than 150 years. So, why we usuallyhear that “the theory of dynamical systems is modern”? Probably this is becauseof the “discovery” of the strange or chaotic attractor. We will come back to thisissue later but for the moment we can say chaotic response is a dynamical behaviorwhich is very sensitive to the initial conditions (the system evolves very differentlyif we just change slightly the initial state of the system), is unpredictable in thelong term but it is given by completely deterministic laws.

5

6 CHAPTER 1. INTRODUCTION AND DEFINITIONS

1.2 Background

Throughout this text we will deal mainly with physical systems which can bedescribed by systems of equations of the form,

dx1

dt= f1(x1, x2, ..., xn; µ1, µ2, ..., µp),

dx2

dt= f2(x1, x2, ..., xn; µ1, µ2, ..., µp),

...

dxn

dt= fn(x1, x2, ..., xn; µ1, µ2, ..., µp).

(1.1)

Obviously, Eq.(1.1), can be written in a more compact form,

x = f(x; µ), (1.2)

with x ∈�

n, t ∈�

1 and µ ∈�

p. In Eq.(1.2), the over-dot means “ ddt

” andbold symbols are used for vectors. We call x the dynamical variables, µ theparameters of the system and t the time. In the physical world, x are the variablesof the system that we observe experimentally, either directly or via a responsefunction and µ are the parameters of the system that we keep constant. Obviously,t is also a variable but since we cannot affect its course it is considered as asindependent variable. If we take as an example the evolution of a homogeneouschemical reaction, then, x is the concentration of the reactants or the productsand µ might be the temperature, pressure or volume. Equation (1.2) is oftencalled ordinary differential equation, vector field or simply dynamical system. Inthe case when the right hand side of Eq.(1.2) is a nonlinear function, we call itnonlinear dynamical system. The dimension of the dynamical system is definedas the number of variables which are required to describe the system. In Eq.(1.2)the dimension is n. Equation (1.2) together with some initial conditions is oftencalled initial value problem. Finally, since the right hand side of Eq.(1.2) doesn’tdepend explicitly on time, the dynamical system is called autonomous.

Dynamical systems of the form of Eq.(1.2) can describe evolutionary pro-cesses with specific properties which have special interest to physical chemistry.This properties are [Arnold, 1973]:

• Determinacy: The entire future course and the entire past are uniquely de-termined by its state at the present instant of time. (The system is said to bedeterministic.)

• Finite-dimensionality: The number of variables required to describe the sys-tem is finite.

1.3. SOME ADDITIONAL DEFINITIONS 7

-20 -10 0 10 20 30x

-20

-15

-10

-5

0

5

10

y

(a)

fixed point

0 10 20 30 40 50t

-0.0005

0

0.0005

0.001

x

(b)

Figure 1.1: (a) Trajectories to a stable fixed point and (b) projection of the integralcurve.

• Differentiability: The change of state with time is described by differen-tiable functions.

Some examples of physical systems which are not described by equations ofthe form considered here is the motion of a quantum particle (it is not determinis-tic), the motion of fluids (it is not finite-dimensional), the motion of shock waves(it is not differentiable). We will try to deal with infinite dimensional dynamicalsystems in the last chapter of these notes.

1.3 Some additional definitions

Some additional definitions are also useful. Hence, the space defined by the dy-namical variables of the system is called phase space. Obviously, the dimensionof this space is n. The solution of Eq.(1.2) under a specified initial conditionx0 ≡ x(t = t0) is written x(t, t0,x0) and is called the trajectory or phase curvethrough the point x0 at t = t0. The graph of x(t, t0,x0) versus t is called an inte-gral curve. Finally, the set of points in phase space that lie on a trajectory passingthrough x0 is called and orbit through x0.

1.4 Types of dynamical response

In the following chapters we will give several examples of different types of dy-namical response. Here we will present just some definitions and geometric rep-resentations.

8 CHAPTER 1. INTRODUCTION AND DEFINITIONS

−0.1

0

0.1

−1

0

1

−1

0

1

PSfrag replacements

x1

x2

x3

0.07829

10

20

30

40

48.33

−20

−10

0

10

2027.65

−17.62 −10 0 10 19.81

PSfrag replacements

x1

x2

x3

Figure 1.2: (a) Trajectories on a torus, and (b) the Lorenz chaotic attractor

• Fixed points. A fixed point of the dynamical system Eq.(1.2) is a pointx ∈

�n such that,

f(x) = 0. (1.3)

From Eq.(1.3) we see that the fixed point is a solution which does not changein time. Fixed points are also called equilibrium points or steady states.There are different types of fixed points, but we will come back to this later.In Fig.1.1 we present an example of a fixed point in

�2 and the integral

curve.

• Limit cycles. Concerning the limit cycle behavior we will only give a veryrough definition. Consider Eq.(1.2) where x ∈

�2. A periodic solution of

this dynamical system is called limit cycle if another solution of the systemapproaches this periodic solution for t → ±∞. A graphic interpretation isgiven in Fig.1.5.

(A limit cycle can be defined also as an isolated periodic orbit.)

• Tori. Once again, we will deal with trajectories on a torus later. At thispoint, we can say that the response on a torus can be non-periodic, and morespecifically, quasiperiodic. Quasiperiodic response is defined as a solutionconsisting of at least two incommensurate frequencies, i.e.,

θ

φ= α 6=

p

q, (1.4)

where q ∈�∗ and q ∈ � ∗.

An example of quasiperiodicity on a torus is shown in Fig.1.2(a)

• Chaos. Roughly speaking, the chaotic response is a non-periodic response,sensitive to the initial conditions. A more accurate definition will be givenlater. Here we just present the famous Lorenz chaotic attractor (Fig.1.2(b)).

1.5. STABILITY 9

0

(a)

(t)x

(t)yε

δ

tt

(t)

x(t)

(b)

tt0

β

y

Figure 1.3: (a) Lyapunov stability and (b) asymptotic stability.

1.5 Stability

Consider the autonomous dynamical system Eq.(1.2). Let x(t) be a solution ofthe system. Then, the solution x(t) is stable if solutions starting close to x(t) ata given time remain close to x(t) for all later times. It is asymptotically stable ifnearby solutions converge to x(t) as t → ∞. In a formal way these definitionscan be written [Wiggins, 1990]:

Definition 1.1 (Lyapunov stability) The solution x(t) is said to be stable (Lya-punov stable) if, given ε, there exists δ = δ(ε) such that, for any other solutiony(t) satisfying |x(t0)− y(t0)| < δ, then |x(t)− y(t)| < ε for t > t0.

A geometrical interpretation of Lyapunov stability is shown in Fig.1.3a.

Definition 1.2 (Asymptotic stability) The solution x(t) is said to be asymptoti-cally stable if it is Lyapunov stable and if there exists a constant β > 0 such that,if |x(t0)− y(t0)| < β then limt→∞ |x(t)− y(t)| = 0.

A geometrical interpretation of asymptotic stability is shown in Fig.1.3b.

1.6 Examples

Example 1.1 Consider the linear dynamical system,

x1 + kx1 = 0 (1.5)

with x1(t = 0) = 1 and x1(t = 0) = 0. Let k = 1.

1. Write Eq.(1.5) as a system of ODEs.

2. What is the dimension of the system?

10 CHAPTER 1. INTRODUCTION AND DEFINITIONS

x1

x2(a)

x1

x2(b)

PSfrag replacements

x 1

x2

t-1

-1 00

0

1

1

10 20

(c)

Figure 1.4: (a) An orbit, (b) a trajectory through (1,0) and (c) an integral curve ofEqs.(1.6-1.7).

3. Write Eq.(1.6-1.7) in a vector-matrix form.

4. Draw the orbit of the system (Hint: divide Eq.(1.6) by Eq.(1.7) and inte-grate).

5. Draw a trajectory of the system.

6. Draw the integral curve of the system (Hint: use complex coordinates).

(Note: Equation (1.5) describes small oscillations of a plane pendulum.)

Solution:

1. Equation (1.5) can be written as a system of ODEs by letting,

x1 = x2.

Under this change of variables, Eq.(1.5) is written,

x1 = x2, (1.6)

x2 = −x1, (1.7)

1.6. EXAMPLES 11

with x1(t = 0) = 1 and x2(t = 0) = 0.

2. Since the system is described by two dynamical variables, the dimension ofthe system is n = 2.

3. The system can be written in a vector matrix form by letting,

x =

(

x1

x2

)

, (1.8)

A =

(

0 1−1 0

)

, (1.9)

that is,x = Ax. (1.10)

4. The orbit of the system can be drawn by dividing Eq.(1.6) with Eq.(1.7) andintegrating,

dx1

dx2

= −x2

x1

. (1.11)

This equation can be readily solved by separation of variables,

x2

1(t) = −x2

2(t) + c, (1.12)

where c is the constant of integration. But, due to the initial condition, c =1,thus,

x2

1(t) + x2

2(t) = 1. (1.13)

We observe from Eq.(1.13) that the orbit is a circle of unit radius with itscenter at the origin (Fig.1.4a).

The direction of rotation can be drawn by noticing that the tangent on theorbit determined by the fraction −x2

x1

, thus it is clockwise.

5. A trajectory is plotted in Fig.1.4b.

6. In order to plot the integral curve of the system, we must find the solution.Even though Eqs.(1.6-1.7) can be solved easily, here we will introduce away which will prove useful in future examples.

Let us follow the hint and introduce complex variables, i.e.,

z = x1 + ix2,

z = x1 − ix2,(1.14)

12 CHAPTER 1. INTRODUCTION AND DEFINITIONS

where the over-bar represents the complex conjugate. Equation (1.14) iswritten in vector-matrix form,

z = Sx, (1.15)

where S =(

1 1−i i

)

. The inverse relation1 can be also defined,

x = S−1z, (1.16)

where S−1 = 1

2

(

1 i1 −i

)

and S−1S = I. Turning now to Eq.(1.10), we canwrite,

x = AS−1Sx. (1.17)

Left multiplication by S gives,

Sx = SAS−1Sx. (1.18)

Or, by using Eqs.(1.15) and (1.16),

z = Jz, (1.19)

where J =(

−i 0

0 i

)

. Since in Eq.(1.19) the first line is the complex conjugateof the second, we can study only the first (or only the second) equation inthis system, i.e.,

z = −iz, (1.20)

with solution z(t) = e−it. Since the second equation is just the complexconjugate, its solution is z = eit. Using DeMoivre’s formula2 and Eq.(1.16)we have,

x1(t) = cos t, (1.21)

x2(t) = − sin t. (1.22)

As expected, the solution is a periodic function with period and amplitudeunity. An integral curve is shown in Fig.1.4c.

Example 1.2 Consider the system [Nemytskii and Stepanov, 1989],

x = −y +x

√

x2 + y2(1− (x2 + y2)), (1.23)

y = x +y

√

x2 + y2(1− (x2 + y2)). (1.24)

1. Find the solution of the system (Hint: use polar coordinates).

1.6. EXAMPLES 13

-1 0 1x

-1

0

1

y

inner trajectory

outer trajectory

limit cycle

Figure 1.5: A limit cycle and trajectories approaching it.

2. Draw the solution in the phase space and the trajectories starting fromwithin and outside the periodic solution.

Solution:

1. We start by using polar coordinates,

x = r cos θ,

y = r sin θ.(1.25)

Under this transformation Eqs.(1.23) and (1.24) are written,

x = −y +x

r(1− r2), (1.26)

y = x +y

r(1− r2). (1.27)

Multiplying Eq.(1.26) by x and Eq.(1.27) by y and adding we obtain,3

r = 1− r2, (1.28)

where r ≥ 0. Similarly, multiplying Eq.(1.26) by y and Eq.(1.27) by x andsubtracting we obtain,4

θ = 1. (1.29)1The inverse of a n× n matrix S is S

−1 = 1

detSadjS where adjS is the adjoint, provided that

the determinant detS 6= 0.2DeMoivre’s formula: eiα = cosα + i sin α.3Here we use the identity: xx + yy = rr4Here we use the identity: xy + yx = r2θ.

14 CHAPTER 1. INTRODUCTION AND DEFINITIONS

At this point we must observe that Eq.(1.28) has a fixed point,

1− r2 = 0 ⇒ r = ±1. (1.30)

Since, r represents the radius, the negative sign does not have any meaning,thus the only fixed point is r = 1. Also we must note that fixed pointsfor Eq.(1.28) represent periodic orbits for the full system, Eqs.(1.28) and(1.29). Thus, the system has a periodic orbit with radius unity.

Solving Eq.(1.28) we get,

r =

{

Ae2t−1

Ae2t+1for 0 < r < 1

Ae2t+1

Ae2t−1

for r > 1(1.31)

where A = |1+r0

1−r0

|. Obviously, in both cases r → 1 as t → +∞.

2. Since r → 1 as t → +∞ when r is within or outside the periodic orbit, thelimit cycle is stable. A graphic representation of the solution as well as thetrajectories towards the limit cycle are shown in Fig. 1.5.