Nonlinear ordinary differential equations

Grimshaw, R. Nonlinear ordinary differential equations AMEST vol.2(BSP,1990) ......Page 4
Copyright ......Page 5
Contents ......Page 7
Preface vii 8......Page 8
1.1 Preliminary notions, 1......Page 10
1.2 First-order systems, 3......Page 12
1.3 Uniqueness and existence theorems, 7......Page 16
1.4 Dependence on parameters, and continuation, 15......Page 24
Problems, 20......Page 29
2.1 Uniqueness and existence theorem for a linear system, 23......Page 32
2.2 Homogeneous linear systems, 25......Page 34
2.3 Inhomogeneous linear systems, 30......Page 39
2.4 Second-order linear equations, 32......Page 41
2.5 Linear equations with constant coefficients, 35......Page 44
Problems, 44......Page 53
3.1 Floquet theory, 47......Page 56
3.2 Parametric resonance, 56......Page 65
3.3 Perturbation methods for the Mathieu equation, 62......Page 71
3.4 The Mathieu equation with damping, 73......Page 82
Problems, 79......Page 88
4.1 Preliminary definitions, 83......Page 92
4.2 Stability for linear systems, 86......Page 95
4.3 Principle of linearized stability, 92......Page 101
4.4 Stability for autonomous systems, 98......Page 107
4.5 Liapunov functions, 102......Page 111
Problems, 108......Page 117
5.1 Critical points, 112......Page 121
5.2 Linear plane, autonomous systems, 115......Page 124
5.3 Nonlinear perturbations of plane, autonomous systems, 125......Page 134
Problems, 140......Page 149
6.1 Preliminary results, 142......Page 151
6.2 The index of a critical point, 149......Page 158
6.3 Van der Pol equation, 153......Page 162
6.4 Conservative systems, 165......Page 174
Problems, 169......Page 178
7.1 Poincare-Lindstedt method, 173......Page 182
7.2 Poincare-Lindstedt method, continued, 177......Page 186
7.3 Stability, 185......Page 194
Problems, 187......Page 196
8.1 Non-resonant case, 190......Page 199
8.2 Non-resonant case, continued, 192......Page 201
8.3 Resonant case, 198......Page 207
8.4 Resonant oscillations for Duffing’s equation, 205......Page 214
8.5 Resonant oscillations for Van der Pol’s equation, 209......Page 218
Problems, 211......Page 220
9.1 Averaging methods for autonomous equations, 216......Page 225
9.2 Averaging methods for forced oscillations, 223......Page 232
9.3 Adiabatic invariance, 233......Page 242
9.4 Multi-scale methods, 236......Page 245
Problems, 240......Page 249
10.1 Preliminary notions, 245......Page 254
10.2 One-dimensional bifurcations, 247......Page 256
10.3 One-dimensional bifurcations, continued, 255......Page 264
10.4 Hopf bifurcation, 261......Page 270
Problems, 273......Page 282
11.1 Hamiltonian and Lagrangian dynamics, 276......Page 285
11.2 Liouville’s theorem, 285......Page 294
11.3 Integral invariants and canonical transformations, 294......Page 303
11.4 Action-angle variables, 303......Page 312
11.5 Action-angle variables: perturbation theory, 308......Page 317
Problems, 315......Page 324
ANSWERS TO SELECTED PROBLEMS, 319......Page 328
REFERENCES 325......Page 334
INDEX 326......Page 335
cover......Page 1