ODE and dynamical systems (lecture notes)

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 21
1.5. Qualitative analysis of first order equations......Page 27
2.1. Fixed point theorems......Page 34
2.2. The basic existence and uniqueness result......Page 37
2.3. Dependence on the initial condition......Page 40
2.4. Extensibility of solutions......Page 43
2.5. Euler's method and the Peano theorem......Page 46
2.6. Appendix: Volterra integral equations......Page 49
3.1. The matrix exponential......Page 54
3.2. Linear autonomous first order systems......Page 59
3.3. Linear autonomous equations of order n......Page 62
3.4. General linear first order systems......Page 67
3.5. Periodic linear systems......Page 72
3.6. Appendix: Jordan canonical form......Page 76
4.1. The basic existence and uniqueness result......Page 82
4.2. The Frobenius method for second order equations......Page 85
4.3. Linear systems with singularities......Page 95
4.4. The Frobenius method......Page 98
5.1. Introduction......Page 106
5.2. Symmetric compact operators......Page 109
5.3. Regular Sturm-Liouville problems......Page 114
5.4. Oscillation theory......Page 119
Part 2. Dynamical systems......Page 126
6.1. Dynamical systems......Page 128
6.2. The flow of an autonomous equation......Page 129
6.3. Orbits and invariant sets......Page 132
6.4. The Poincaré map......Page 136
6.5. Stability of fixed points......Page 137
6.6. Stability via Liapunov's method......Page 139
6.7. Newton's equation in one dimension......Page 141
7.1. Stability of linear systems......Page 146
7.2. Stable and unstable manifolds......Page 148
7.3. The Hartman-Grobman theorem......Page 153
7.4. Appendix: Hammerstein integral equations......Page 157
8.1. The Poincaré--Bendixson theorem......Page 160
8.2. Examples from ecology......Page 164
8.3. Examples from electrical engineering......Page 168
9.1. Attracting sets......Page 174
9.2. The Lorenz equation......Page 177
9.3. Hamiltonian mechanics......Page 181
9.4. Completely integrable Hamiltonian systems......Page 185
9.5. The Kepler problem......Page 190
9.6. The KAM theorem......Page 192
Part 3. Chaos......Page 196
10.1. The logistic equation......Page 198
10.2. Fixed and periodic points......Page 201
10.3. Linear difference equations......Page 203
10.4. Local behavior near fixed points......Page 205
11.1. Stability of periodic solutions......Page 208
11.2. The Poincaré map......Page 209
11.3. Stable and unstable manifolds......Page 211
11.4. Melnikov's method for autonomous perturbations......Page 214
11.5. Melnikov's method for nonautonomous perturbations......Page 219
12.1. Period doubling......Page 222
12.2. Sarkovskii's theorem......Page 225
12.3. On the definition of chaos......Page 226
12.4. Cantor sets and the tent map......Page 229
12.5. Symbolic dynamics......Page 232
12.6. Strange attractors/repellors and fractal sets......Page 237
12.7. Homoclinic orbits as source for chaos......Page 241
13.1. The Smale horseshoe......Page 246
13.2. The Smale-Birkhoff homoclinic theorem......Page 248
13.3. Melnikov's method for homoclinic orbits......Page 249
Bibliography......Page 252
Glossary of notations......Page 254
Index......Page 256