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 20
1.5. Qualitative analysis of first order equations......Page 26
2.1. Fixed point theorems......Page 32
2.2. The basic existence and uniqueness result......Page 35
2.3. Dependence on the initial condition......Page 38
2.4. Extensibility of solutions......Page 41
2.5. Euler's method and the Peano theorem......Page 43
2.6. Appendix: Volterra integral equations......Page 46
3.1. Preliminaries from linear algebra......Page 52
3.2. Linear autonomous first order systems......Page 59
3.3. General linear first order systems......Page 64
3.4. Periodic linear systems......Page 68
4.1. The basic existence and uniqueness result......Page 74
4.2. Linear equations......Page 76
4.3. The Frobenius method......Page 79
4.4. Second order equations......Page 83
5.1. Introduction......Page 92
5.2. Symmetric compact operators......Page 95
5.3. Regular Sturm-Liouville problems......Page 100
5.4. Oscillation theory......Page 105
Part 2. Dynamical systems......Page 112
6.1. Dynamical systems......Page 114
6.2. The flow of an autonomous equation......Page 115
6.3. Orbits and invariant sets......Page 118
6.4. Stability of fixed points......Page 122
6.5. Stability via Liapunov's method......Page 124
6.6. Newton's equation in one dimension......Page 125
7.1. Stability of linear systems......Page 130
7.2. Stable and unstable manifolds......Page 132
7.3. The Hartman-Grobman theorem......Page 137
7.4. Appendix: Hammerstein integral equations......Page 141
8.1. The Poincaré--Bendixson theorem......Page 144
8.2. Examples from ecology......Page 148
8.3. Examples from electrical engineering......Page 152
9.1. Attracting sets......Page 158
9.2. The Lorenz equation......Page 161
9.3. Hamiltonian mechanics......Page 165
9.4. Completely integrable Hamiltonian systems......Page 169
9.5. The Kepler problem......Page 174
9.6. The KAM theorem......Page 175
Part 3. Chaos......Page 180
10.1. The logistic equation......Page 182
10.2. Fixed and periodic points......Page 185
10.3. Linear difference equations......Page 187
10.4. Local behavior near fixed points......Page 189
11.1. Stability of periodic solutions......Page 192
11.2. The Poincaré map......Page 193
11.3. Stable and unstable manifolds......Page 195
11.4. Melnikov's method for autonomous perturbations......Page 198
11.5. Melnikov's method for nonautonomous perturbations......Page 203
12.1. Period doubling......Page 206
12.2. Sarkovskii's theorem......Page 209
12.3. On the definition of chaos......Page 210
12.4. Cantor sets and the tent map......Page 213
12.5. Symbolic dynamics......Page 216
12.6. Strange attractors/repellors and fractal sets......Page 220
12.7. Homoclinic orbits as source for chaos......Page 224
13.1. The Smale horseshoe......Page 228
13.2. The Smale-Birkhoff homoclinic theorem......Page 230
13.3. Melnikov's method for homoclinic orbits......Page 231
Bibliography......Page 234
Glossary of notations......Page 236
Index......Page 238