Ordinary Differential Equations and Dynamical Systems

✍ Scribed by Gerald Teschl

• Publisher: Institut f¨ur Mathematik, Universit¨at Wien, Austria
• Year: 2004
• Tongue: English
• Leaves: 237
• Category: Library

✦ Table of Contents

Preface......Page 8
Part 1. Classical theory......Page 10
1.1. Newton's equations......Page 12
1.2. Classification of differential equations......Page 14
1.3. First order autonomous equations......Page 17
1.4. Finding explicit solutions......Page 20
1.5. Qualitative analysis of first order equations......Page 25
2.1. Fixed point theorems......Page 30
2.2. The basic existence and uniqueness result......Page 32
2.3. Dependence on the initial condition......Page 35
2.4. Extensibility of solutions......Page 38
2.5. Euler's method and the Peano theorem......Page 41
2.6. Appendix: Volterra integral equations......Page 43
3.1. Preliminaries from linear algebra......Page 50
3.2. Linear autonomous first order systems......Page 56
3.3. General linear first order systems......Page 59
3.4. Periodic linear systems......Page 63
4.1. The basic existence and uniqueness result......Page 70
4.2. Linear equations......Page 72
4.3. The Frobenius method......Page 76
4.4. Second order equations......Page 79
5.1. Introduction......Page 86
5.2. Symmetric compact operators......Page 89
5.3. Regular Sturm-Liouville problems......Page 94
5.4. Oscillation theory......Page 99
Part 2. Dynamical systems......Page 106
6.1. Dynamical systems......Page 108
6.2. The flow of an autonomous equation......Page 109
6.3. Orbits and invariant sets......Page 112
6.4. Stability of fixed points......Page 116
6.5. Stability via Liapunov's method......Page 118
6.6. Newton's equation in one dimension......Page 119
7.1. Stability of linear systems......Page 124
7.2. Stable and unstable manifolds......Page 127
7.3. The Hartman-Grobman theorem......Page 132
7.4. Appendix: Hammerstein integral equations......Page 136
8.1. The Poincaré--Bendixson theorem......Page 138
8.2. Examples from ecology......Page 142
8.3. Examples from electrical engineering......Page 146
9.1. Attracting sets......Page 152
9.2. The Lorenz equation......Page 155
9.3. Hamiltonian mechanics......Page 159
9.4. Completely integrable Hamiltonian systems......Page 163
9.5. The Kepler problem......Page 168
9.6. The KAM theorem......Page 169
Part 3. Chaos......Page 174
10.1. The logistic equation......Page 176
10.2. Fixed and periodic points......Page 179
10.3. Linear difference equations......Page 181
10.4. Local behavior near fixed points......Page 183
11.1. Stability of periodic solutions......Page 186
11.2. The Poincaré map......Page 187
11.3. Stable and unstable manifolds......Page 189
11.4. Melnikov's method for autonomous perturbations......Page 192
11.5. Melnikov's method for nonautonomous perturbations......Page 197
12.1. Period doubling......Page 200
12.2. Sarkovskii's theorem......Page 203
12.3. On the definition of chaos......Page 204
12.4. Cantor sets and the tent map......Page 207
12.5. Symbolic dynamics......Page 210
12.6. Strange attractors/repellors and fractal sets......Page 214
12.7. Homoclinic orbits as source for chaos......Page 218
13.1. The Smale horseshoe......Page 222
13.2. The Smale-Birkhoff homoclinic theorem......Page 224
13.3. Melnikov's method for homoclinic orbits......Page 225
Bibliography......Page 228
Glossary of notations......Page 230
Index......Page 232