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