Synopses & Reviews
This textbook is aimed at newcomers to nonlinear dynamics and chaos, especially students taking a first course in the subject. The presentation stresses analytical methods, concrete examples, and geometric intuition. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
A unique feature of the book is its emphasis on applications. These include mechanical vibrations, lasers, biological rhythms, superconducting circuits, insect outbreaks, chemical oscillators, genetic control systems, chaotic waterwheels, and even a technique for using chaos to send secret messages. In each case, the scientific background is explained at an elementary level and closely integrated with mathematical theory.
In the twenty years since the first edition of this book appeared, the ideas and techniques of nonlinear dynamics and chaos have found application to such exciting new fields as systems biology, evolutionary game theory, and sociophysics. This second edition includes new exercises on these cutting-edge developments, on topics as varied as the curiosities of visual perception and the tumultuous love dynamics in Gone With the Wind.
Review
The new edition has a friendly yet clear technical style . . . One of the books biggest strengths is that it explains core concepts through practical examples drawn from various fields and from real-world systems . . . the authors excellent use of geometric and graphical techniques greatly clarifies what can be amazingly complex behavior.”
Physics TodayNonlinear Dynamics and Chaos is an excellent book that effectively demonstrates the power and beauty of the theory of dynamical systems. Its readers will want to learn more.” Mathematical Association of America
Praise for the prior edition:
"Exceptionally well-written. Time after time, Strogatz gives explanations of concepts that are among the most lucid I have ever read...One of the best introductions to nonlinear dynamics currently available."
SIAM Review
"The examples impressed me with their subtlety and incisiveness. Important, delicate distinctions and exceptions are highlighted and accessible."
Physics Today
"More than any undergraduate book that I have seen in recent years, this book can lure students into the mathematical sciences, make them want to change their major, and spark in them some real intellectual curiosity."
UMAP Journal
Synopsis
This textbook introduces chaos and nonlinear systems to students taking a first course in the subject. Richly illustrated, the second edition includes updated examples, illustrations and applications to science and engineering. The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.
Synopsis
An accessible introduction to chaos and nonlinear systems, with numerous examples, illustrations, and applications to science and engineering.
About the Author
Steven Strogatz is the Schurman Professor of Applied Mathematics at Cornell University. His honors include MITs highest teaching prize, a lifetime achievement award for the communication of mathematics to the general public, and membership in the American Academy of Arts and Sciences. His research on a wide variety of nonlinear systemsfrom synchronized fireflies to small-world networkshas been featured in the pages of
Scientific American, Nature, Discover, Business Week, and
The New York Times.
Table of Contents
Preface1. Overview
1.0 Chaos, Fractals, and Dynamics
1.1 Capsule History of Dynamics
1.2 The Importance of Being Nonlinear
1.3 A Dynamical View of the World
PART I. ONE-DIMENSIONAL FLOWS
2. Flows on the Line
2.0 Introduction
2.1 A Geometric Way of Thinking
2.2 Fixed Points and Stability
2.3 Population Growth
2.4 Linear Stability Analysis
2.5 Existence and Uniqueness
2.6 Impossibility of Oscillations
2.7 Potentials
2.8 Solving Equations on the Computer
Exercises
3. Bifurcations
3.0 Introduction
3.1 Saddle-Node Bifurcation
3.2 Transcritical Bifurcation
3.3 Laser Threshold
3.4 Pitchfork Bifurcation
3.5 Overdamped Bead on a Rotating Hoop
3.6 Imperfect Bifurcations and Catastrophes
3.7 Insect Outbreak
Exercises
4. Flows on the Circle
4.0 Introduction
4.1 Examples and Definitions
4.2 Uniform Oscillator
4.3 Nonuniform Oscillator
4.4 Overdamped Pendulum
4.5 Fireflies
4.6 Superconducting Josephson Junctions
Exercises
PART II. TWO-DIMENSIONAL FLOWS
5. Linear Systems
5.0 Introduction
5.1 Definitions and Examples
5.2 Classification of Linear Systems
5.3 Love Affairs
Exercises
6. Phase Plane
6.0 Introduction
6.1 Phase Portraits
6.2 Existence, Uniqueness, and Topological Consequences
6.3 Fixed Points and Linearization
6.4 Rabbits versus Sheep
6.5 Conservative Systems
6.6 Reversible Systems
6.7 Pendulum
6.8 Index Theory
Exercises
7. Limit Cycles
7.0 Introduction
7.1 Examples
7.2 Ruling Out Closed Orbits
7.3 Poincaré-Bendixson Theorem
7.4 Liénard Systems
7.5 Relaxation Oscillators
7.6 Weakly Nonlinear Oscillators
Exercises
8. Bifurcations Revisited
8.0 Introduction
8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations
8.2 Hopf Bifurcations
8.3 Oscillating Chemical Reactions
8.4 Global Bifurcations of Cycles
8.5 Hysteresis in the Driven Pendulum and Josephson Junction
8.6 Coupled Oscillators and Quasiperiodicity
8.7 Poincaré Maps
Exercises
PART III. CHAOS
9. Lorenz Equations
9.0 Introduction
9.1 A Chaotic Waterwheel
9.2 Simple Properties of the Lorenz Equations
9.3 Chaos on a Strange Attractor
9.4 Lorenz Map
9.5 Exploring Parameter Space
9.6 Using Chaos to Send Secret Messages
Exercises
10. One-Dimensional Maps
10.0 Introduction
10.1 Fixed Points and Cobwebs
10.2 Logistic Map: Numerics
10.3 Logistic Map: Analysis
10.4 Periodic Windows
10.5 Liapunov Exponent
10.6 Universality and Experiments
10.7 Renormalization
Exercises
11. Fractals
11.0 Introduction
11.1 Countable and Uncountable Sets
11.2 Cantor Set
11.3 Dimension of Self-Similar Fractals
11.4 Box Dimension
11.5 Pointwise and Correlation Dimensions
Exercises
12. Strange Attractors
12.0 Introductions
12.1 The Simplest Examples
12.2 Hénon Map
12.3 Rössler System
12.4 Chemical Chaos and Attractor Reconstruction
12.5 Forced Double-Well Oscillator
Exercises
Answers to Selected Exercises
References
Author Index
Subject Index