Differential equations, dynamical systems, and an introduction to chaos.

Format:

Book

Author/Creator:

Hirsch, Morris W., 1933-

Contributor:

Smale, Stephen, 1930-

Devaney, Robert L., 1948-

Class of 1953 Fund.

Language:

English

Subjects (All):

Differential equations.

Algebras, Linear.

Chaotic behavior in systems.

Physical Description:

xiv, 418 pages : illustrations ; 24 cm

Edition:

Third edition / Morris W. Hirsch, Stephen Smale, Robert L. Devaney.

Place of Publication:

Waltham, MA : Academic Press, [2013]

Summary:

Differential Equations, Dynamical Systems, arid an Introduction to Chaos, now in its third edition, covers the dynamical aspects of ordinary differential equations. It explores the relations between dynamical systems and certain fields outside pure mathematics, and continues to be the standard textbook for advanced undergraduate and graduate courses in this area.

Written for students with a background in calculus and elementary linear algebra, the text is rigorous yet accessible and contains examples and explorations to reinforce learning.

Key Features

Developed by three of the world's leading mathematicians, including Steven Smale, a recipient of the Field's Medal for his work in dynamical systems

Includes discrete dynamics, various introductions to chaos, background material on linear algebra, and bifurcation theory throughout

New to Third Edition

New Explorations including: introducing chaos via numerical methods, regarding the SIRS model and zombies, and concerning the flight of gliders

Major revisions of the Lorenz section proofs and the stable. curve theorem

Simplifications and new updates throughout the text Book jacket.

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 10

1.5 Computing the Poincare Map 11

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 73

5.1 Preliminaries from Linear Algebra 73

5.2 Eigenvalues and Eigenvectors 82

5.3 Complex Eigenvalues 85

5.4 Bases and Subspaces 88

5.5 Repeated Eigenvalues 93

5.6 Genericity 100

Chapter 6 Higher-Dimensional Linear Systems 107

6.1 Distinct Eigenvalues 107

6.2 Harmonic Oscillators 114

6.3 Repeated Eigenvalues 120

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

7.6 Exploration: Numerical Methods and Chaos 156

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 175

8.6 Exploration: Complex Vector Fields 182

Chapter 9 Global Nonlinear Techniques 187

9.1 Nullclines 187

9.2 Stability of Equilibria 192

9.3 Gradient Systems 202

9.4 Hamiltonian Systems 206

9.5 Exploration: The Pendulum with Constant Forcing 209

Chapter 10 Closed Orbits and Limit Sets 213

10.1 Limit Sets 213

10.2 Local Sections and Flow Boxes 216

10.3 The Poincare Map 218

10.4 Monotone Sequences in Planar Dynamical Systems 220

10.5 The Poincaré-Bendixson Theorem 222

10.6 Applications of Poincare-Bendixson 225

10.7 Exploration: Chemical Reactions that Oscillate 228

Chapter 11 Applications in Biology 233

11.1 Infectious Diseases 233

11.2 Predator-Prey Systems 237

11.3 Competitive Species 244

11.4 Exploration: Competition and Harvesting 250

11.5 Exploration: Adding Zombies to the SIR Model 251

Chapter 12 Applications in Circuit Theory 257

12.1 An RLC Circuit 257

12.2 The Liénard Equation 261

12.3 The van der Pol Equation 263

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 282

13.4 The Newtonian Central Force System 285

13.5 Kepler's First Law 290

13.6 The Two-Body Problem 293

13.7 Blowing Up the Singularity 294

13.8 Exploration: Other Central Force Problems 298

13.9 Exploration: Classical Limits of Quantum Mechanical Systems 299

13.10 Exploration: Motion of a Glider 301

Chapter 14 The Lorenz System 305

14.1 Introduction 306

14.2 Elementary Properties of the Lorenz System 308

14.3 The Lorenz Attractor 312

14.4 A Model for the Lorenz Attractor 316

14.5 The Chaotic Attractor 321

14.6 Exploration: The Rössler Attractor 326

Chapter 15 Discrete Dynamical Systems 329

15.1 Introduction 329

15.2 Bifurcations 334

15.3 The Discrete Logistic Model 337

15.4 Chaos 340

15.5 Symbolic Dynamics 344

15.6 The Shift Map 349

15.7 The Cantor Middle-Thirds Set 351

15.8 Exploration: Cubic Chaos 354

15.9 Exploration: The Orbit Diagram 355

Chapter 16 Homoclinic Phenomena 361

16.1 The Shilnikov System 361

16.2 The Horseshoe Map 368

16.3 The Double Scroll Attractor 375

16.4 Homoclinic Bifurcations 377

16.5 Exploration: The Chua Circuit 381

Chapter 17 Existence and Uniqueness Revisited 385

17.1 The Existence and Uniqueness Theorem 385

17.2 Proof of Existence and Uniqueness 387

17.3 Continuous Dependence on Initial Conditions 394

17.4 Extending Solutions 397

17.5 Nonautonomous Systems 401

17.6 Differentiability of the Flow 404.

Notes:

Includes bibliographical references and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Class of 1953 Fund.

ISBN:

0123820103

9780123820105

OCLC:

773025052

Publisher Number:

99948287818