Differential Equation Ordinary Differential Equations

Preface......Page 8
Part 1. Classical theory......Page 10
1.1. Newton's equations......Page 12
1.2. Classification of differential equations......Page 15
1.3. First order autonomous equations......Page 17
1.4. Finding explicit solutions......Page 22
1.5. Qualitative analysis of first-order equations......Page 28
1.6. Qualitative analysis of first-order periodic equations......Page 35
2.1. Fixed point theorems......Page 40
2.2. The basic existence and uniqueness result......Page 42
2.3. Some extensions......Page 45
2.4. Dependence on the initial condition......Page 48
2.5. Extensibility of solutions......Page 53
2.6. Euler's method and the Peano theorem......Page 56
3.1. The matrix exponential......Page 60
3.2. Linear autonomous first-order systems......Page 65
3.3. Linear autonomous equations of order n......Page 71
3.4. General linear first-order systems......Page 78
3.5. Periodic linear systems......Page 84
3.6. Appendix: Jordan canonical form......Page 89
4.1. The basic existence and uniqueness result......Page 96
4.2. The Frobenius method for second-order equations......Page 99
4.3. Linear systems with singularities......Page 109
4.4. The Frobenius method......Page 114
5.1. Introduction......Page 120
5.2. Compact symmetric operators......Page 123
5.3. Regular Sturm-Liouville problems......Page 129
5.4. Oscillation theory......Page 136
5.5. Periodic operators......Page 142
Part 2. Dynamical systems......Page 150
6.1. Dynamical systems......Page 152
6.2. The flow of an autonomous equation......Page 153
6.3. Orbits and invariant sets......Page 156
6.4. The Poincaré map......Page 161
6.5. Stability of fixed points......Page 162
6.6. Stability via Liapunov's method......Page 163
6.7. Newton's equation in one dimension......Page 165
7.1. Stability of linear systems......Page 170
7.2. Stable and unstable manifolds......Page 172
7.3. The Hartman-Grobman theorem......Page 179
7.4. Appendix: Integral equations......Page 184
8.1. The Poincaré--Bendixson theorem......Page 192
8.2. Examples from ecology......Page 196
8.3. Examples from electrical engineering......Page 201
9.1. Attracting sets......Page 206
9.2. The Lorenz equation......Page 210
9.3. Hamiltonian mechanics......Page 215
9.4. Completely integrable Hamiltonian systems......Page 219
9.5. The Kepler problem......Page 223
9.6. The KAM theorem......Page 225
Part 3. Chaos......Page 230
10.1. The logistic equation......Page 232
10.2. Fixed and periodic points......Page 235
10.3. Linear difference equations......Page 237
10.4. Local behavior near fixed points......Page 239
11.1. Period doubling......Page 242
11.2. Sarkovskii's theorem......Page 245
11.3. On the definition of chaos......Page 246
11.4. Cantor sets and the tent map......Page 249
11.5. Symbolic dynamics......Page 252
11.6. Strange attractors/repellors and fractal sets......Page 257
11.7. Homoclinic orbits as source for chaos......Page 261
12.1. Stability of periodic solutions......Page 266
12.2. The Poincaré map......Page 267
12.3. Stable and unstable manifolds......Page 269
12.4. Melnikov's method for autonomous perturbations......Page 272
12.5. Melnikov's method for nonautonomous perturbations......Page 277
13.1. The Smale horseshoe......Page 280
13.2. The Smale-Birkhoff homoclinic theorem......Page 282
13.3. Melnikov's method for homoclinic orbits......Page 283
Bibliography......Page 286
Glossary of notation......Page 288
Index......Page 290