#### Preface

#### Part 1. Classical theory

### Chapter 1. Introduction 3

1.1. Newton's equations 3

1.2. Classi cation of di erential equations 6

1.3. First order autonomous equations 8

1.4. Finding eplicit solutions 13

1.5. Qualitative analysis of rst-order equations 19

1.6. Qualitative analysis of rst-order periodic equations 26

#### Chapter 2. Initial value problems 31

2.1. Fied point theorems 31

2.2. The basic eistence and uniqueness result 33

2.3. Some etensions 36

2.4. Dependence on the initial condition 39

2.5. Etensibility of solutions 44

2.6. Euler's method and the Peano theorem 47

#### Chapter 3. Linear equations 51

3.1. The matri eponential 51

3.2. Linear autonomous rst-order systems 56

3.3. Linear autonomous equations of order n 62

3.4. General linear rst-order systems 69

3.5. Periodic linear systems 75

3.6. Appendi: Jordan canonical form 80

#### Chapter 4. Di erential equations in the comple domain 87

4.1. The basic eistence and uniqueness result 87

4.2. The Frobenius method for second-order equations 90

4.3. Linear systems with singularities 101

4.4. The Frobenius method 105

#### Chapter 5. Boundary value problems 111

5.1. Introduction 111

5.2. Compact symmetric operators 114

5.3. Regular Sturm-Liouville problems 120

5.4. Oscillation theory 127

5.5. Periodic operators 133

Part 2. Dynamical systems

#### Chapter 6. Dynamical systems 145

6.1. Dynamical systems 145

6.2. Thef ow of an autonomous equation 146

6.3. Orbits and invariant sets 149

6.4. The Poincare map 154

6.5. Stability of ed points 155

6.6. Stability via Liapunov's method 156

6.7. Newton's equation in one dimension 158

#### Chapter 7. Local behavior near ed points 163

7.1. Stability of linear systems 163

7.2. Stable and unstable manifolds 165

7.3. The Hartman-Grobman theorem 172

7.4. Appendi: Integral equations 177

#### Chapter 8. Planar dynamical systems 185

8.1. The Poincare{Bendison theorem 185

8.2. Eamples from ecology 189

8.3. Eamples from electrical engineering 194

#### Chapter 9. Higher dimensional dynamical systems 199

9.1. Attracting sets 199

9.2. The Lorenz equation 203

9.3. Hamiltonian mechanics 208

9.4. Completely integrable Hamiltonian systems 212

9.5. The Kepler problem 216

9.6. The KAM theorem 218

#### Part 3. Chaos

#### Chapter 10. Discrete dynamical systems 225

10.1. The logistic equation 225

10.2. Fied and periodic points 228

10.3. Linear di erence equations 230

10.4. Local behavior near ed points 232

#### Chapter 11. Discrete dynamical systems in one dimension 235

11.1. Period doubling 235

11.2. Sarkovskii's theorem 238

11.3. On the de nition of chaos 239

11.4. Cantor sets and the tent map 242

11.5. Symbolic dynamics 245

11.6. Strange attractors/repellors and fractal sets 250

11.7. Homoclinic orbits as source for chaos 254

#### Chapter 12. Periodic solutions 259

12.1. Stability of periodic solutions 259

12.2. The Poincare map 260

12.3. Stable and unstable manifolds 262

12.4. Melnikov's method for autonomous perturbations 265

12.5. Melnikov's method for nonautonomous perturbations 270

#### Chapter 13. Chaos in higher dimensional systems 273

13.1. The Smale horseshoe 273

13.2. The Smale-Birkho homoclinic theorem 275

13.3. Melnikov's method for homoclinic orbits 276

Bibliography 281

Glossary of notation 283

Inde 285