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Nonlinear Dynamics and Chaos : With Applications to Physics, Biology, Chemistry, and Engineering

by
ISBN13:

9780738204536

ISBN10:
0738204536
Format:
Paperback
Pub. Date:
1/18/2001
Publisher(s):
PERSEUS BOOKS
List Price: $60.00

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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


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