Nonlinear Dynamics and Chaos

by Strogatz, Steven H.

• ISBN13: 9780738204536
• ISBN10: 0738204536
• Format: Paperback
• Copyright: 2001-01-18
• Publisher: Westview Pr

Summary

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 the mathematical theory.Richly illustrated, and with many exercises and worked examples, this book is ideal for an introductory course at the junior/senior or first-year graduate level. It is also ideal for the scientist who has not had formal instruction in nonlinear dynamics, but who now desires to begin informal study. The prerequisites are multivariable calculus and introductory physics.

Author Biography

Steven H. Strogatz is professor of applied mathematics at Cornell University. He received his Ph.D. from Harvard University in 1986. Professor Strogatz has been honored with several awards including MIT's highest teaching prize, the E.M. Baker Award for Excellence in Undergraduate Teaching, as well as a Presidential Young Investigator Award from the National Science Foundation. His research on a wide variety of nonlinear systems--from synchronized fireflies to small-world networks--has been featured in the pages of Scientific American, Nature, Discover, Business Week, and The New York Times .

Table of Contents

Preface	ix
Overview	1	(14)
Chaos, Fractals, and Dynamics	1	(1)
Capsule History of Dynamics	2	(2)
The Importance of Being Nonlinear	4	(5)
A Dynamical View of the World	9	(6)
Part I. One-Dimensional Flows
Flows on the Line	15	(29)
Introduction	15	(1)
A Geometric Way of Thinking	16	(2)
Fixed Points and Stability	18	(3)
Population Growth	21	(3)
Linear Stability Analysis	24	(2)
Existence and Uniqueness	26	(2)
Impossibility of Oscillations	28	(2)
Potentials	30	(2)
Solving Equations on the Computer	32	(12)
Exercises	36	(8)
Bifurcations	44	(49)
Introduction	44	(1)
Saddle-Node Bifurcation	45	(5)
Transcritical Bifurcation	50	(3)
Laser Threshold	53	(2)
Pitchfork Bifurcation	55	(6)
Overdamped Bead on a Rotating Hoop	61	(8)
Imperfect Bifurcations and Catastrophes	69	(4)
Insect Outbreak	73	(20)
Exercises	79	(14)
Flows on the Circle	93	(30)
Introduction	93	(1)
Examples and Definitions	93	(2)
Uniform Oscillator	95	(1)
Nonuniform Oscillator	96	(5)
Overdamped Pendulum	101	(2)
Fireflies	103	(3)
Superconducting Josephson Junctions	106	(17)
Exercises	113	(10)
Part II. Two-Dimensional Flows
Linear Systems	123	(22)
Introduction	123	(1)
Definitions and Examples	123	(6)
Classification of Linear Systems	129	(9)
Love Affairs	138	(7)
Exercises	140	(5)
Phase Plane	145	(51)
Introduction	145	(1)
Phase Portraits	145	(3)
Existence, Uniqueness, and Topological Consequences	148	(2)
Fixed Points and Linearization	150	(5)
Rabbits versus Sheep	155	(4)
Conservative Systems	159	(4)
Reversible Systems	163	(5)
Pendulum	168	(6)
Index Theory	174	(22)
Exercises	181	(15)
Limit Cycles	196	(45)
Introduction	196	(1)
Examples	197	(2)
Ruling Out Closed Orbits	199	(4)
Poincare-Bendixson Theorem	203	(7)
Lienard Systems	210	(1)
Relaxation Oscillators	211	(4)
Weakly Nonlinear Oscillators	215	(26)
Exercises	227	(14)
Bifurcations Revisited	241	(60)
Introduction	241	(1)
Saddle-Node, Transcritical, and Pitchfork Bifurcations	241	(7)
Hopf Bifurcations	248	(6)
Oscillating Chemical Reactions	254	(6)
Global Bifurcations of Cycles	260	(5)
Hysteresis in the Driven Pendulum and Josephson Junction	265	(8)
Coupled Oscillators and Quasiperiodicity	273	(5)
Poincare Maps	278	(23)
Exercises	284	(17)
Part III. Chaos
Lorenz Equations	301	(47)
Introduction	301	(1)
A Chaotic Waterwheel	302	(9)
Simple Properties of the Lorenz Equations	311	(6)
Chaos on a Strange Attractor	317	(9)
Lorenz Map	326	(4)
Exploring Parameter Space	330	(5)
Using Chaos to Send Secret Messages	335	(13)
Exercises	341	(7)
One-Dimensional Maps	348	(50)
Introduction	348	(1)
Fixed Points and Cobwebs	349	(4)
Logistic Map: Numerics	353	(4)
Logistic Map: Analysis	357	(4)
Periodic Windows	361	(5)
Liapunov Exponent	366	(3)
Universality and Experiments	369	(10)
Renormalization	379	(19)
Exercises	388	(10)
Fractals	398	(25)
Introduction	398	(1)
Countable and Uncountable Sets	399	(2)
Cantor Set	401	(3)
Dimension of Self-Similar Fractals	404	(5)
Box Dimension	409	(2)
Pointwise and Correlation Dimensions	411	(12)
Exercises	416	(7)
Strange Attractors	423	(32)
Introduction	423	(1)
The Simplest Examples	423	(6)
Henon Map	429	(5)
Rossler System	434	(3)
Chemical Chaos and Attractor Reconstruction	437	(4)
Forced Double-Well Oscillator	441	(14)
Exercises	448	(7)
Answers to Selected Exercises	455	(10)
References	465	(10)
Author Index	475	(3)
Subject Index	478