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