Differential equations, dynamical systems, and an introduction to chaos / Morris W. Hirsch, Stephen Smale, Robert L. Devaney.

Format:

Book

Author/Creator:

Hirsch, Morris W., 1933-

Contributor:

Smale, Stephen, 1930-

Devaney, Robert L., 1948-

Hirsch, Morris W., 1933-

Alumni and Friends Memorial Book Fund.

Series:

Pure and applied mathematics (Academic Press) ; 60.

Pure and applied mathematics ; v. 60, 2nd ed.

Language:

English

Subjects (All):

Differential equations.

Algebras, Linear.

Chaotic behavior in systems.

Physical Description:

xiv, 417 pages : illustrations ; 24 cm.

Edition:

Second edition.

Place of Publication:

Amsterdam ; Boston : Elsevier Academic Press, [2004]

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.

Contents:

Chapter 1 First-Order Equations 1

1.1 The Simplest Example 1

1.2 The Logistic Population Model 4

1.3 Constant Harvesting and Bifurcations 7

1.4 Periodic Harvesting and Periodic Solutions 9

1.5 Computing the Poincare Map 12

1.6 Exploration: A Two-Parameter Family 15

Chapter 2 Planar Linear Systems 21

2.1 Second-Order Differential Equations 23

2.2 Planar Systems 24

2.3 Preliminaries from Algebra 26

2.4 Planar Linear Systems 29

2.5 Eigenvalues and Eigenvectors 30

2.6 Solving Linear Systems 33

2.7 The Linearity Principle 36

Chapter 3 Phase Portraits for Planar Systems 39

3.1 Real Distinct Eigenvalues 39

3.2 Complex Eigenvalues 44

3.3 Repeated Eigenvalues 47

3.4 Changing Coordinates 49

Chapter 4 Classification of Planar Systems 61

4.1 The Trace-Determinant Plane 61

4.2 Dynamical Classification 64

4.3 Exploration: A 3D Parameter Space 71

Chapter 5 Higher Dimensional Linear Algebra 75

5.1 Preliminaries from Linear Algebra 75

5.2 Eigenvalues and Eigenvectors 83

5.3 Complex Eigenvalues 86

5.4 Bases and Subspaces 89

5.5 Repeated Eigenvalues 95

5.6 Genericity 101

Chapter 6 Higher Dimensional Linear Systems 107

6.1 Distinct Eigenvalues 107

6.2 Harmonic Oscillators 114

6.3 Repeated Eigenvalues 119

6.4 The Exponential of a Matrix 123

6.5 Nonautonomous Linear Systems 130

Chapter 7 Nonlinear Systems 139

7.1 Dynamical Systems 140

7.2 The Existence and Uniqueness Theorem 142

7.3 Continuous Dependence of Solutions 147

7.4 The Variational Equation 149

7.5 Exploration: Numerical Methods 153

Chapter 8 Equilibria in Nonlinear Systems 159

8.1 Some Illustrative Examples 159

8.2 Nonlinear Sinks and Sources 165

8.3 Saddles 168

8.4 Stability 174

8.5 Bifurcations 176

8.6 Exploration: Complex Vector Fields 182

Chapter 9 Global Nonlinear Techniques 189

9.1 Nullclines 189

9.2 Stability of Equilibria 194

9.3 Gradient Systems 203

9.4 Hamiltonian Systems 207

9.5 Exploration: The Pendulum with Constant Forcing 210

Chapter 10 Closed Orbits and Limit Sets 215

10.1 Limit Sets 215

10.2 Local Sections and Flow Boxes 218

10.3 The Poincare Map 220

10.4 Monotone Sequences in Planar Dynamical Systems 222

10.5 The Poincare-Bendixson Theorem 225

10.6 Applications of Poincare-Bendixson 227

10.7 Exploration: Chemical Reactions That Oscillate 230

Chapter 11 Applications in Biology 235

11.1 Infectious Diseases 235

11.2 Predator/Prey Systems 239

11.3 Competitive Species 246

11.4 Exploration: Competition and Harvesting 252

Chapter 12 Applications in Circuit Theory 257

12.1 An RLC Circuit 257

12.2 The Lienard Equation 261

12.3 The van der Pol Equation 262

12.4 A Hopf Bifurcation 270

12.5 Exploration: Neurodynamics 272

Chapter 13 Applications in Mechanics 277

13.1 Newton's Second Law 277

13.2 Conservative Systems 280

13.3 Central Force Fields 281

13.4 The Newtonian Central Force System 285

13.5 Kepler's First Law 289

13.6 The Two-Body Problem 292

13.7 Blowing Up the Singularity 293

13.8 Exploration: Other Central Force Problems 297

13.9 Exploration: Classical Limits of Quantum Mechanical Systems 298

Chapter 14 The Lorenz System 303

14.1 Introduction to the Lorenz System 304

14.2 Elementary Properties of the Lorenz System 306

14.3 The Lorenz Attractor 310

14.4 A Model for the Lorenz Attractor 314

14.5 The Chaotic Attractor 319

14.6 Exploration: The Rossler Attractor 324

Chapter 15 Discrete Dynamical Systems 327

15.1 Introduction to Discrete Dynamical Systems 327

15.2 Bifurcations 332

15.3 The Discrete Logistic Model 335

15.4 Chaos 337

15.5 Symbolic Dynamics 342

15.6 The Shift Map 347

15.7 The Cantor Middle-Thirds Set 349

15.8 Exploration: Cubic Chaos 352

15.9 Exploration: The Orbit Diagram 353

Chapter 16 Homoclinic Phenomena 359

16.1 The Shil'nikov System 359

16.2 The Horseshoe Map 366

16.3 The Double Scroll Attractor 372

16.4 Homoclinic Bifurcations 375

16.5 Exploration: The Chua Circuit 379

Chapter 17 Existence and Uniqueness Revisited 383

17.1 The Existence and Uniqueness Theorem 383

17.2 Proof of Existence and Uniqueness 385

17.3 Continuous Dependence on Initial Conditions 392

17.4 Extending Solutions 395

17.5 Nonautonomous Systems 398

17.6 Differentiability of the Flow 400.

Notes:

Rev. ed. of: Differential equations, dynamical systems, and linear algebra / Morris W. Hirsch and Stephen Smale. 1974.

Includes bibliographical references (pages 407-409) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

Related Work:

Hirsch, Morris W., 1933- Differential equations, dynamical systems, and linear algebra.

ISBN:

0123497035

OCLC:

52631423

Online:

Publisher description