Chaos and Dynamical Systems

by David P. Feldman

Overview

Chaos and Dynamical Systems presents an accessible, clear introduction to dynamical systems and chaos theory, important and exciting areas that have shaped many scientific fields. While the rules governing dynamical systems are well-specified and simple, the behavior of many dynamical systems is remarkably complex. Of particular note, simple deterministic dynamical systems produce output that appears random and for which long-term prediction is impossible. Using little math beyond basic algebra, David Feldman gives readers a grounded, concrete, and concise overview.

In initial chapters, Feldman introduces iterated functions and differential equations. He then surveys the key concepts and results to emerge from dynamical systems: chaos and the butterfly effect, deterministic randomness, bifurcations, universality, phase space, and strange attractors. Throughout, Feldman examines possible scientific implications of these phenomena for the study of complex systems, highlighting the relationships between simplicity and complexity, order and disorder.

Filling the gap between popular accounts of dynamical systems and chaos and textbooks aimed at physicists and mathematicians, Chaos and Dynamical Systems will be highly useful not only to students at the undergraduate and advanced levels, but also to researchers in the natural, social, and biological sciences.

Product Details

ISBN-13:	9780691189390
Publisher:	Princeton University Press
Publication date:	08/06/2019
Series:	Primers in Complex Systems , #7
Sold by:	Barnes & Noble
Format:	eBook
Pages:	264
File size:	19 MB
Note:	This product may take a few minutes to download.

About the Author

David P. Feldman is professor of physics and mathematics at the College of the Atlantic. He is the author of Chaos and Fractals: An Elementary Introduction.

Table of Contents

Preface ix

1 Introducing Iterated Functions 1

1.1 Iterated Functions 1

1.2 Thinking Globally 5

1.3 Stability: Attractors and Repellors 7

1.4 Another Example 9

1.5 One More Example 10

1.6 Determinism 14

1.7 Summary 15

2 Introducing Differential Equations 17

2.1 Newton's Law of Cooling 17

2.2 Exact Solutions 21

2.3 Calculus Puzzles 22

2.4 Qualitative Solutions 24

2.5 Numerical Solutions 27

2.6 Putting It All Together 32

2.7 More about Numerical Solutions 35

2.8 Notes on Terminology and Notation 36

2.9 Existence and Uniqueness of Solutions 39

2.10 Determinism and Differential Equations 40

2.11 Iterated Functions vs. Differential Equations 42

2.12 Further Reading 44

3 Interlude: Mathematical Models and the Newtonian Worldview 45

3.1 Why Isn't This the End of the Book? 45

3.2 Newton's Mechanistic World 46

3.3 Laplacian Determinism and the Aspirations of Science 47

3.4 Styles of Mathematical Models 50

3.5 Levels of Models 54

3.6 Pluralistic View of Mathematical Models 59

3.7 Further Reading 61

4 Chaos I: The Butterfly Effect 62

4.1 The Logistic Equation 62

4.2 Periodic Behavior 67

4.3 Aperiodic Behavior 70

4.4 The Butterfly Effect 74

4.5 The Butterfly Effect Defined 80

4.6 Chaos Defined 83

4.7 Lyapunov Exponents 85

5 Chaos II: Deterministic Randomness 90

5.1 Symbolic Dynamics 91

5.2 As Random as a Coin Toss 93

5.3 Deterministic Sources of Randomness 95

5.4 Implications of the Butterfly Effect 99

5.5 Further Reading 104

6 Bifurcations: Sudden Transitions 106

6.1 Logistic Differential Equation 106

6.2 Logistic Equation with Harvest 109

6.3 Bifurcations and Bifurcation Diagrams 113

6.4 General Remarks on Bifurcations 119

6.5 Catastrophes and Tipping Points 120

6.6 Hysteresis 123

6.7 Further Reading 128

7 Universality in Chaos 129

7.1 Logistic Equation Bifurcation Diagram 129

7.2 Exploring the Bifurcation Diagram 137

7.3 Some Words about Emergence 141

7.4 The Period-Doubling Route to Chaos 143

7.5 Universality in Maps 145

7.6 Universality in Physics 149

7.7 Renormalization 151

7.8 Phase Transitions, Critical Phenomena, and Power Laws 159

7.9 Conclusion: Lessons and Limits to Universality 165

7.10 Further Reading 169

8 Higher-Dimensional Systems and Phase Space 170

8.1 A Quick Review of One-Dimensional Differential Equations 170

8.2 Lotka-Volterra Differential Equations 172

8.3 The Phase Plane 176

8.4 Phase Planes in General 181

8.5 The Rössler Equations and Phase Space 183

8.6 Further Reading 188

9 Strange Attractors 189

9.1 Chaos in Three Dimensions 190

9.2 The Rössler Attractor 195

9.3 Strange Attractors 201

9.4 Back to ID: The Lorenz Map 203

9.5 Stretching and Folding 207

9.6 Poincaré Maps 210

9.7 Delay Coordinates and Phase Space Reconstruction 212

9.8 Determinism vs. Noise 219

9.9 Further Reading 221

10 Conclusion 223

10.1 Summary 223

10.2 Complex Systems 225

10.3 Emergence(?) 227

10.4 But Not Everything Is Simple 229

10.5 Further Reading 230

10.6 Farewell 230

Bibliography 231

Index 243

What People are Saying About This

From the Publisher

“This book is a readable tour and deep dive into chaotic dynamics and related concepts from the field of dynamical systems theory. Appropriate for use in a sequence at the undergraduate level, this book will also appeal to graduate students, postdocs, and faculty in the biological and social sciences and engineering.”—James P. Crutchfield, University of California, Davis

“With clear and explanatory prose and simple yet precisely constructed examples, this book conveys a deep understanding of chaos, bifurcations, and other core concepts of dynamical systems to a much larger audience than was previously possible. Feldman achieves this all without relying on a deep knowledge of math. An impressive balancing act, this is certainly a significant contribution to the field.”—Van Savage, University of California, Los Angeles

“Chaos and Dynamical Systems is a great introduction to nonlinear dynamics, bifurcations, and chaos. It is easy to follow and understand, yet also provides a generous amount of mathematical detail, which will satisfy technically oriented minds too. This book’s core take-home message, that simple mathematical systems can produce complex dynamics, has implications for many real-world complex systems.”—Hiroki Sayama, Binghamton University, State University of New York

Editorial Reviews

"Feldman succeeds in introducing the reader to the world of dynamic systems and the, almost mythical, chaos that they can produce."—-Adhemar Bultheel, European Mathematical Society

"[A] gentle and loving introduction to dynamical systems. . . . Chaos and Dynamical Systems is a book for everyone from the layman to the expert."—-David S. Mazel, MAA Reviews

From the Publisher