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Differential Equations, Dynamical Systems, and an Introduction to Chaos

by: Hirsch; Smale; Devaney

Differential Equations, Dynamical Systems, and an Introduction to Chaos

by: Hirsch; Smale; Devaney

ISBN 13:
9780123497031
ISBN 10:
0123497035
Edition: 2nd
Format: Hardcover
Copyright: 10/22/2003
Publisher: Elsevier Science

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Summary

This best-selling, classic text by eminent mathematicians Hirsch and Smale has captured the beauty and relative accessibility of chaotic phenomena for three decades. This field has motivated scientists and engineers in many disciplines to understand how differential equations and dynamical systems phenomena appear in virtually every area of science: chemistry, electrical engineering, celestial mechanics, ecological systems, and beyond. Book jacket.

Preface

CHAPTER 1 First-Order Equations

(20)

1.1 The Simplest Example

(3)

1.2 The Logistic Population Model

(3)

1.3 Constant Harvesting and Bifurcations

(2)

1.4 Periodic Harvesting and Periodic Solutions

(3)

1.5 Computing the Poincaré Map

(3)

1.6 Exploration: A Two-Parameter Family

(6)

CHAPTER 2 Planar Linear Systems

(18)

2.1 Second-Order Differential Equations

(1)

2.2 Planar Systems

(2)

2.3 Preliminaries from Algebra

(3)

2.4 Planar Linear Systems

(1)

2.5 Eigenvalues and Eigenvectors

(3)

2.6 Solving Linear Systems

(3)

2.7 The Linearity Principle

(3)

CHAPTER 3 Phase Portraits for Planar Systems

(22)

3.1 Real Distinct Eigenvalues

(5)

3.2 Complex Eigenvalues

(3)

3.3 Repeated Eigenvalues

(2)

3.4 Changing Coordinates

(12)

CHAPTER 4 Classification of Planar Systems

(14)

4.1 The Trace-Determinant Plane

(3)

4.2 Dynamical Classification

(7)

4.3 Exploration: A 3D Parameter Space

(4)

CHAPTER 5 Higher Dimensional Linear Algebra

(32)

5.1 Preliminaries from Linear Algebra

(8)

5.2 Eigenvalues and Eigenvectors

(3)

5.3 Complex Eigenvalues

(3)

5.4 Bases and Subspaces

(6)

5.5 Repeated Eigenvalues

(6)

5.6 Genericity

101

(6)

CHAPTER 6 Higher Dimensional Linear Systems

107

(32)

6.1 Distinct Eigenvalues

107

(7)

6.2 Harmonic Oscillators

114

(5)

6.3 Repeated Eigenvalues

119

(4)

6.4 The Exponential of a Matrix

123

(7)

6.5 Nonautonomous Linear Systems

130

(9)

CHAPTER 7 Nonlinear Systems

139

(20)

7.1 Dynamical Systems

140

(2)

7.2 The Existence and Uniqueness Theorem

142

(5)

7.3 Continuous Dependence of Solutions

147

(2)

7.4 The Variational Equation

149

(4)

7.5 Exploration: Numerical Methods

153

(6)

CHAPTER 8 Equilibria in Nonlinear Systems

159

(30)

8.1 Some Illustrative Examples

159

(6)

8.2 Nonlinear Sinks and Sources

165

(3)

8.3 Saddles

168

(6)

8.4 Stability

174

(2)

8.5 Bifurcations

176

(6)

8.6 Exploration: Complex Vector Fields

182

(7)

CHAPTER 9 Global Nonlinear Techniques

189

(26)

9.1 Nullclines

189

(5)

9.2 Stability of Equilibria

194

(9)

9.3 Gradient Systems

203

(4)

9.4 Hamiltonian Systems

207

(3)

9.5 Exploration: The Pendulum with Constant Forcing

210

(5)

CHAPTER 10 Closed Orbits and Limit Sets

215

(20)

10.1 Limit Sets

215

(3)

10.2 Local Sections and Flow Boxes

218

(2)

10.3 The Poincaré Map

220

(2)

10.4 Monotone Sequences in Planar Dynamical Systems

222

(3)

10.5 The Poincaré-Bendixson Theorem

225

(2)

10.6 Applications of Poincaré-Bendixson

227

(3)

10.7 Exploration: Chemical Reactions That Oscillate

230

(5)

CHAPTER 11 Applications in Biology

235

(22)

11.1 Infectious Diseases

235

(4)

11.2 Predator/Prey Systems

239

(7)

11.3 Competitive Species

246

(6)

11.4 Exploration: Competition and Harvesting

252

(5)

CHAPTER 12 Applications in Circuit Theory

257

(20)

12.1 An RLC Circuit

257

(4)

12.2 The Lienard Equation

261

(1)

12.3 The van der Pol Equation

262

(8)

12.4 A Hopf Bifurcation

270

(2)

12.5 Exploration: Neurodynamics

272

(5)

CHAPTER 13 Applications in Mechanics

277

(26)

13.1 Newton's Second Law

277

(3)

13.2 Conservative Systems

280

(1)

13.3 Central Force Fields

281

(4)

13.4 The Newtonian Central Force System

285

(4)

13.5 Kepler's First Law

289

(3)

13.6 The Two-Body Problem

292

(1)

13.7 Blowing Up the Singularity

293

(4)

13.8 Exploration: Other Central Force Problems

297

(1)

13.9 Exploration: Classical Limits of Quantum Mechanical Systems

298

(5)

CHAPTER 14 The Lorenz System

303

(24)

14.1 Introduction to the Lorenz System

304

(2)

14.2 Elementary Properties of the Lorenz System

306

(4)

14.3 The Lorenz Attractor

310

(4)

14.4 A Model for the Lorenz Attractor

314

(5)

14.5 The Chaotic Attractor

319

(5)

14.6 Exploration: The Rössler Attractor

324

(3)

CHAPTER 15 Discrete Dynamital Systems

327

(32)

15.1 Introduction to Discrete Dynamital Systems

327

(5)

15.2 Bifurcations

332

(3)

15.3 The Discrete Logistic Model

335

(2)

15.4 Chaos

337

(5)

15.5 Symbolic Dynamics

342

(5)

15.6 The Shift Map

347

(2)

15.7 The Cantor Middle-Thirds Set

349

(3)

15.8 Exploration: Cubic Chaos

352

(1)

15.9 Exploration: The Orbit Diagram

353

(6)

CHAPTER 16 Homoclinic Phenomena

359

(24)

16.1 The Shil'nikov System

359

(7)

16.2 The Horseshoe Map

366

(6)

16.3 The Double Scroll Attractor

372

(3)

16.4 Homoclinic Bifurcations

375

(4)

16.5 Exploration: The Chua Circuit

379

(4)

CHAPTER 17 Existence and Uniqueness Revisited

383

(24)

17.1 The Existence and Uniqueness Theorem

383

(2)

17.2 Proof of Existence and Uniqueness

385

(7)

17.3 Continuous Dependence on Initial Conditions

392

(3)

17.4 Extending Solutions

395

(3)

17.5 Nonautonomous Systems

398

(2)

17.6 Differentiability of the Flow

400

(7)

Bibliography

407

(4)

Index

411

Amazon no longer offers textbook rentals. We do!

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Differential Equations, Dynamical Systems, and an Introduction to Chaos

9780123497031

0123497035

Summary

Author Biography

Table of Contents

Supplemental Materials

Write a Review

Amazon no longer offers textbook rentals. We do!

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Differential Equations, Dynamical Systems, and an Introduction to Chaos

9780123497031

0123497035

Summary

Author Biography

Table of Contents

Supplemental Materials

Write a Review

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