Ordinary differential equations and dynamical systems

Preface......Page 9
Part 1. Classical theory......Page 11
1.1. Newton's equations......Page 13
1.2. Classification of differential equations......Page 16
1.3. First order autonomous equations......Page 19
1.4. Finding explicit solutions......Page 23
1.5. Qualitative analysis of first-order equations......Page 30
1.6. Qualitative analysis of first-order periodic equations......Page 37
2.1. Fixed point theorems......Page 41
2.2. The basic existence and uniqueness result......Page 44
2.3. Some extensions......Page 47
2.4. Dependence on the initial condition......Page 50
2.5. Regular perturbation theory......Page 56
2.6. Extensibility of solutions......Page 58
2.7. Euler's method and the Peano theorem......Page 62
3.1. The matrix exponential......Page 67
3.2. Linear autonomous first-order systems......Page 74
3.3. Linear autonomous equations of order n......Page 81
3.4. General linear first-order systems......Page 87
3.5. Linear equations of order n......Page 94
3.6. Periodic linear systems......Page 98
3.7. Perturbed linear first order systems......Page 104
3.8. Appendix: Jordan canonical form......Page 110
4.1. The basic existence and uniqueness result......Page 119
4.2. The Frobenius method for second-order equations......Page 124
4.3. Linear systems with singularities......Page 138
4.4. The Frobenius method......Page 142
5.1. Introduction......Page 149
5.2. Compact symmetric operators......Page 153
5.3. Sturm–Liouville equations......Page 160
5.4. Regular Sturm–Liouville problems......Page 162
5.5. Oscillation theory......Page 170
5.6. Periodic Sturm–Liouville equations......Page 178
Part 2. Dynamical systems......Page 187
6.1. Dynamical systems......Page 189
6.2. The flow of an autonomous equation......Page 190
6.3. Orbits and invariant sets......Page 194
6.4. The Poincaré map......Page 198
6.5. Stability of fixed points......Page 200
6.6. Stability via Liapunov's method......Page 202
6.7. Newton's equation in one dimension......Page 204
7.1. Examples from ecology......Page 211
7.2. Examples from electrical engineering......Page 216
7.3. The Poincaré–Bendixson theorem......Page 222
8.1. Attracting sets......Page 229
8.2. The Lorenz equation......Page 233
8.3. Hamiltonian mechanics......Page 238
8.4. Completely integrable Hamiltonian systems......Page 242
8.5. The Kepler problem......Page 247
8.6. The KAM theorem......Page 248
9.1. Stability of linear systems......Page 253
9.2. Stable and unstable manifolds......Page 255
9.3. The Hartman–Grobman theorem......Page 262
9.4. Appendix: Integral equations......Page 268
Part 3. Chaos......Page 277
10.1. The logistic equation......Page 279
10.2. Fixed and periodic points......Page 282
10.3. Linear difference equations......Page 285
10.4. Local behavior near fixed points......Page 286
11.1. Period doubling......Page 291
11.2. Sarkovskii's theorem......Page 294
11.3. On the definition of chaos......Page 295
11.4. Cantor sets and the tent map......Page 298
11.5. Symbolic dynamics......Page 301
11.6. Strange attractors/repellors and fractal sets......Page 307
11.7. Homoclinic orbits as source for chaos......Page 311
12.1. Stability of periodic solutions......Page 315
12.2. The Poincaré map......Page 316
12.3. Stable and unstable manifolds......Page 319
12.4. Melnikov's method for autonomous perturbations......Page 321
12.5. Melnikov's method for nonautonomous perturbations......Page 327
13.1. The Smale horseshoe......Page 331
13.2. The Smale–Birkhoff homoclinic theorem......Page 333
13.3. Melnikov's method for homoclinic orbits......Page 334
Bibliographical notes......Page 337
Bibliography......Page 339
Glossary of notation......Page 341
Index......Page 343