An Introduction To Chaotic Dynamical Systems

Robert Devaney

• Publisher: CRC Press
• Year: 2018
• ISBN:
  9780429981937
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.

Preface

The last twenty five years have seen an explosion of interest in the study of nonlinear dynamical systems. Scientists in all disciplines have come to realize the power and the beauty of the geometric and qualitative techniques developed during this period. More importantly, they have been able to apply these techniques to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Chaotic and random behavior of solutions of deterministic systems is now understood to be an inherent feature of many nonlinear systems, and the geometric theory developed over the past few decades handles this situation quite nicely.

Modern dynamical systems theory has a relatively short history. It begins with Poincaré (of course), who revolutionized the study of nonlinear differential equations by introducing the qualitative techniques of geometry and topology rather than strict analytic methods to discuss the global properties of solutions of these systems. To Poincaré, a global understanding of the gross behavior of all solutions of the system was more important than the local behavior of particular, analytically-precise solutions. Poincaré’s point of view was enthusiastically adopted and furthered by Birkhoff in the first part of the twentieth century. Birkhoff realized the importance of the study of mappings and emphasized discrete dynamics as a means of understanding the more difficult dynamics arising from differential equations.

The infusion of geometric and topological techniques during this period gradually led mathematicians away from the study of the dynamical systems themselves and to the study of the underlying geometric structures. Manifolds, the natural “state spaces” of dynamical systems, became objects of study in their own right. Fields such as differential topology and algebraic topology were born and eventually flourished. Rapid advances in these fields gave mathematicians new and varied techniques for attacking geometric problems. Meanwhile, the study of the dynamical systems themselves languished in relative disfavor, except in the Soviet Union, where mathematicians such as Liapounov, Pontryagin, Andronov and others, continued to study dynamics from various points of view.

All of this changed around 1960, due mainly to the influence of Moser and Smale in the United States, Peixoto in Brazil and Kolmogorov, Arnol’d and Sinai in the Soviet Union. Differential topological techniques enabled Smale, Peixoto and their followers to understand the chaotic behavior of a large class of dynamical systems known as hyperbolic or Axiom A systems. Geometry combined with hard analysis allowed Kolmogorov, Arnol’d and Moser to push through their celebrated KAM theory. Smooth ergodic theory, topological dynamics, Hamiltonian mechanics, and the qualitative theory of ordinary differential equations all developed as disciplines in their own right.

More recently, dynamical systems has benefited from an infusion of interest and techniques from a variety of fields. Physicists such as Feigenbaum have rekindled interest in low dimensional discrete dynamical systems. Breakthroughs in mathematical biology and economics have attracted a diverse group of scientists to the field. The discovery of stably chaotic systems such as the Lorenz system from meteorology have convinced scientists that there are many more stable types of dynamical behavior than just stable equilibrium points and limit cycles. And, by no means least of all, computer graphics has shown that the dynamics of simple systems can be at once beautiful and alluring.

All of these developments have made dynamical systems theory an attractive and important branch of mathematics of interest to scientists in many disciplines. Unfortunately, because of the background of many of the contemporary researchers in such advanced fields as differential topology, algebraic topology, and differential geometry, the available introductions to this subject presuppose a familiarity on the part of the student with several of these fields. It is our feeling that the elements of dynamical systems theory can be introduced without prerequisites such as the theory of differentiable manifolds, advanced analysis, etc. Dynamical systems on simple spaces like the real line or the plane exhibit all of the chaotic and interesting behavior that occur on more general manifolds. Without these unnecessary prerequisites, the basic ideas of the field should be accessible to junior and senior mathematics majors as well as to graduate students and scientists in other disciplines. This is the basic goal of this text.

The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in this century. While the field is still relatively young, there is no question that the field is becoming more and more important in a variety of scientific disciplines. We hope that this text serves to excite and to lure many others into this dynamic field.

A NOTE TO THE READER:

This is first of all a Mathematics text. Throughout, we emphasize the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory. The text begins at a relatively unsophisticated level and, by the end, has progressed so as to require not much more than the typical mathematics education of an engineer or a physicist. Fully three quarters of the text is accessible to students with only a solid advanced calculus and linear algebra background. Of course, a good dose of mathematical sophistication is useful throughout.

The first chapter, one-dimensional dynamics, is by far the longest. It is the author’s belief that virtually all of the important ideas and techniques of nonlinear dynamics can be introduced in the setting of the real line or the circle. This has the obvious advantage of minimizing the topological complications of the system and the algebraic machinery necessary to handle them. In particular, the only real prerequisite for this chapter is a good calculus course. (O.K., we do multiply a 2×2 matrix once or twice in §1.14 and we use the Implicit Function Theorem in two variables in §1.12, but these are exceptions.) With only these tools, we manage to introduce such important topics as structural stability, topological conjugacy, the shift map, homoclinic points, and bifurcation theory. To emphasize the point that chaotic dynamics occurs in the simplest of systems, we carry out most of our analysis in this section on a basic model, the quadratic mapping given by F_μ(x) = μx(1 − x). This map has the advantage of being perhaps the simplest nonlinear map yet one which illustrates virtually every concept we wish to introduce. A few topological ideas, such as the notion of a dense set or a Cantor set, are introduced in detail when needed.

The second chapter is devoted to higher dimensional dynamical systems. With many of the prerequisites already introduced in the first chapter, the discussion of such higher dimensional maps as Smale’s horseshoe, the hyperbolic toral automorphisms, and the solenoid become especially simple. This chapter assumes that the student is familiar with some multi-dimensional calculus as well as linear algebra, including the notion of eigenvalues and eigenvectors for 3 x 3 matrices. One of the major differences between one dimensional and higher dimensional dynamics, the possibility of both contraction and expansion at the same time, is treated at length in a section devoted to the proof of the Stable Manifold Theorem. We end the chapter with a lengthy set of exercises all centered on the important Hénon map of the plane. This section serves as a summary of many of the previous topics in the section as well as a good “final” project for the reader.

The last chapter should be regarded as a “special topics” chapter in that we presuppose a working knowledge of complex analysis. In this chapter we describe some of the fascinating and beautiful recent work on the dynamics of complex analytic maps and, in particular, the structure of the Julia set of polynomials. This gives a complementary view of the dynamics of maps such as the quadratic map, which receives so much attention in chapter one.

Each of the chapters is self-contained, assuming familiarity with the basic concepts of dynamics as outlined in the first chapter. Accordingly, we have numbered the Theorems, Figures, etc. consecutively within each subsection, without reference to the chapter number. As there is very little cross-referencing between chapters, this should cause no confusion.

There are many themes developed in this book. We have tried to present several different dynamical concepts in their most elementary formulation in chapter one and to return to these subjects for further refinement at later stages in the book. One such topic is bifurcation theory. We introduce the most elementary bifurcations, the saddle-node and the period-doubling bifurcations, early in chapter one. Later in the same chapter we treat the accumulation points of such bifurcations which occur when a homoclinic point develops. In chapter two, we return to bifurcation theory to discuss the Hopf bifurcation. Finally, in the last chapter, we explore several types of bifurcations that occur in analytic dynamics, including a discussion of the global aspects of the saddle-node bifurcation.

Another recurrent theme is symbolic dynamics. We think of symbolic dynamics as a tool whereby complicated dynamical systems are reduced to seemingly quite different systems which have the advantage that they can be analyzed quite easily. Symbolic dynamics appears quite early in chapter one when we first discuss the quadratic map. It is clear that the most elementary setting for the phenomena associated with the Smale horseshoe mapping occurs in one dimension and we fully exploit this idea. Later, symbolic dynamics is extended to the case of subshifts of finite type via another quadratic example. And finally the related concepts of Markov partitions and inverse limits are introduced in the second chapter.

Examples abound in the text. We often motivate new concepts by working through them in the setting of a specific dynamical system. In fact, we have often sacrificed generality in order to concentrate on a specific system or class of systems. Many of the results throughout the text are stated in a form that is nowhere near full generality. We feel that the general theory is best left to more advanced texts which presuppose more advanced Mathematics.

Much of what many researchers consider dynamical systems has been deliberately left out of this text. For example, we do not treat continuous systems or differential equations at all. There are several reasons for this. First, as is well known, computations with specific nonlinear ordinary differential equations are next to impossible. Secondly, the study of differential equations necessitates a much higher level of sophistication on the part of the student, certainly more than that necessary for chapter one of this text. We adopt instead the attitude that any dynamical phenomena that occurs in a continuous system also occurs in a discrete system, and so we might as well make life easy and study maps first. There are many texts currently available that treat continuous systems almost exclusively. We hope that this book presents an solid introduction to the topics treated in these more advanced texts.

Another topic that has been excluded is ergodic theory. It is our feeling that measure theory would take us too far afield in an elementary text. Of course, it can be argued that measure theory is no more advanced than the complex analysis necessary for chapter three. However, we feel that the topological approach adopted throughout this text is inherently easier to understand, at least for an undergraduate in Mathematics. There is no question, however, that ergodic theory would provide an ideal sequel to the material presented here, as would a course in nonlinear differential equations.

This text has benefited from the suggestions and comments of many people. I would like to thank Clark Robinson, Guido Sandri, Harvey Keynes, Phil Boyland, Paul Blanchard, Dick Hall, and Elwood Devaney for helpful comments on portions of the manuscript. Richard Millman, Chris Golé, and Steve Batterson read the entire text and made many useful suggestions regarding content and organization (and incorrect proofs). Finally, this text never would have been completed without the constant advice and encouragement of Phil Holmes. The book owes much to his experience and expertise.

The book was produced using T_EX at Boston University by Tom Orowan. Tom’s near-perfect typing and formatting of the text made the production of the book effortless and fun. Thanks are also due Chris Mayberry for his help designing the figures. And finally, it is a pleasure to thank Rick Mixter and his staff at Benjamin-Cummings for their enthusiastic support for the duration of this project.

Robert L. Devaney

Boston, Mass.

April, 1985

PREFACE TO THE SECOND EDITION:

The response to the first edition of this book has been most heartening. Accordingly, this edition maintains all of the features of the first edition. We have added new material on the orbit diagram and a new section on the Mandelbrot set. Apart from this, the only other major changes from the first edition include a revised treatment of elementary bifurcation theory and Sarkovskii’s Theorem.

This edition has benefited immensely from the suggestions of many mathematicians, including Susan Dabros, Odo Diekmann, David Doster, David Drasin, Bruce Elenbogen, Jenny Harrison, Henk Heijmans, Roger Kraft, Peter Landweber, Tyre Newton, John Milnor, Connie Overzet, Charles Pugh, Phil Rippon, Clark Robinson, Henk Roozen, Joe Silverman, and Mary Lou Zeeman. Scott Sutherland graciously assisted with many of the new figures. Elwood Devaney again digested the entire manuscript and returned many stylistic suggestions. And thanks are especially due to Gary Meisters and his class at the University of Nebraska for their enthusiastic response to this book, but I’m afraid that I’m not the coach they think I am!

Robert L. Devaney

Boston, Mass.

October, 1988

Chapter One

One-Dimensional Dynamics

The goal of this first chapter is to introduce many of the basic techniques from the theory of dynamical systems in a setting that is as simple as possible. Accordingly, all of the dynamical systems that we will encounter take place in one dimension, either on the real line or on the unit circle in the plane. For that reason, much of this chapter can be read with only a s...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• 0.1 Preface
• Part One: One-Dimensional Dynamics
• Part Two: Higher Dimensional Dynamics
• Part Three: Complex Analytic Dynamics
• Color Plates
• Index