Introduction to Chaos and Coherence

Jan Froyland

• Publisher: Routledge
• Year: 2022
• ISBN:
  9781351437172
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This book provides an introduction to the theory of chaotic systems and demonstrates how chaos and coherence are interwoven in some of the models exhibiting deterministic chaos. It is based on the lecture notes for a short course in dynamical systems theory given at the University of Oslo.

1

INTRODUCTION

Many of the fundamental properties of classical dynamical systems were discovered in the last century by people like Poincaré and Birkhoff. The advent of the new quantum physics distracted the attention of physicists and chemists from these problems for half a century. The revival of the field is closely related to the availability of modern computers to perform the necessary numerical experiments. In particular, Feigenbaum’s discovery of universality gave the field a big push forward. The implications are that many totally unrelated systems have in common features both qualitatively and, even more amazingly, also numerically.

Any system that develops in time in a non-trivial manner may be considered a dynamical system, but to make it somewhat clearer what dynamical systems theory is about it is perhaps appropriate to call it nonlinear dynamical systems theory since all the interesting effects that we shall encounter are closely related to the nonlinearity of the systems under consideration. In particular, we shall consider systems that possess elements of so-called deterministic chaos and how chaos is reached from non-chaotic states as controlling parameters are varied.

Dynamical systems are normally regulated by parameters. When the parameters change, so do the properties of the system. In particular, the stability of a system may be investigated by considering the results of small disturbances. If the disturbances die with time the system is stable, and if the disturbances grow the system is unstable. At some points in the space of parameters some of these properties may change discontinuously as a function of one or more of the parameters. When linear systems lose stability it is usually obvious why. It may be because the controlling parameters have changed exponential decay into exponential growth or because some boundary condition is ‘violated’. However, many nonlinear dynamical systems lose stability for no obvious reason, in which case more or less dramatic changes of dynamical patterns take place. This kind of phenomenon—unknown to the linear theory—is known as a bifurcation. Investigating bifurcations in some detail is one of the major subjects of this book.

Before we proceed it may help to make a rough classification of various dynamical systems and their basic characteristics. The classification is given in table 1.1. The concept phase space is somewhat loosely taken to mean the entire space spanned by the minimum number of dynamical, i.e. time-dependent, variables of the problem. We shall mainly be concerned with low-dimensional maps and differential equations, but we shall also have a brief look at the simplest properties of cellular automata. The systems under consideration are mostly deterministic, i.e. they are systems where the next time step is always exactly predictable. Exceptions to this are the autoregressive models where the updating in time has a stochastic component.

Table 1.1 Basic properties of classes of dynamical systems

Class	Phase space dimension	Variable types	Time
Partial diff. equations	infinite	continuous	continuous
Ordinary diff. equations	finite	continuous	continuous
Iterative maps	finite	continuous	discrete
Cellular automata	finite	discrete	discrete

We shall not deal with other interesting aspects of nonlinear dynamics, such as solitons and related topics.

2

FRACTALS

Exercises in traditional classical mechanics usually consist of considering a set of integrable equations of motion. This means that the equations of motion may be completely separated into as many independent equations as there are degrees of freedom. The equations are then said to be an integrable set of equations even though it may not be possible to express the solutions in terms of elementary functions. In ideal—fortunate—cases the integrations may be performed and the results—the solutions—be presented as functions of time and initial conditions. Unfortunately, this lucky state of affairs is in some sense abnormal. In realistic models of systems one will frequently find that the equations of motion are not completely integrable, meaning that there are fewer independent separation constants than the number of degrees of freedom. One might make an analogy with the real numbers where the irrational numbers outnumber the rational ones. The non-integrability usually comes about because of nonlinearities or because of the presence of external forces. If the system is non-integrable the orbit that the system traces out in phase space will normally be chaotic. The most important (necessary but not sufficient) property of a chaotic orbit is that it never returns to a point previously visited and that it is also not approaching a periodic orbit.

Although it may not be possible—even in principle—to express the motion of a system in phase space as a function of time and initial conditions, the desire to find some quantity or object that does not change with time is very strong. If the motion is periodic one may give coordinates of points on the orbit—the phase space trajectory—but only rarely is this of much help, and if the orbit is chaotic this is quite useless unless the orbit is subjected to some kind of analysis. Even if the motion is chaotic the orbit may nevertheless be bounded in all directions in phase space and attracted to geometrical objects called strange attractors with strange and unfamiliar properties. In particular, a strange attractor does not change with time.

Strange attractors are members of a family of mathematically defined objects named fractals. The term fractal was invented by Mandelbrot who gives the following explanation of the word:

Fractal comes from the Latin adjective fractus, which has the same roots as fraction and fragment and means ‘irregular or fragmented’. It is related to frangere which means ‘to break’.

Typically it is not sensible to assign an integer dimension to a fractal. However, there are several alternative ways to arrive at a non-integer dimension of a given fractal. This we shall return to later. The notion of fractals is useful in many circumstances both in describing phenomena in nature (clouds, mountains, rivers, trees, coast lines, etc) and to give a frame of reference for strange mathematical concepts previously a...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction
• 2 Fractals
• 3 The logistic map
• 4 The circle map
• 5 Higher dimensional maps
• 6 Dissipative maps in higher dimensions
• 7 Conservative maps
• 8 Cellular automata
• 9 Ordinary differential equations
• 10 The Lorenz model
• 11 Time series analysis
• Appendices
• Further Reading
• Index