Differential Equations, Dynamical Systems, and An Introduction to Chaos
2nd Edition

Morris W. Hirsch
University of California, Berkeley

Stephen Smale
University of California, Berkeley

Robert L. Devaney
Boston University


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 Poincaré 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 Poincaré Map 220
10.4 Monotone Sequences in Planar Dynamical Systems 222
10.5 The Poincaré-Bendixson Theorem 225
10.6 Applications of Poincaré-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 R?ssler 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

Bibliography 407
Index 411