Preface ....................................................... vii
Chapter 1 Autonomous dynamical systems ......................... 1
1 Introduction ................................................. 1
2 Local asymptotic behavior .................................... 4
3 Global asymptotic behavior .................................. 12
4 Dependence on parameters .................................... 17
Chapter 2 Nonautonomous dynamical systems ..................... 23
1 Processes formulation ....................................... 23
2 Skew product flow formulation ............................... 26
3 Entire solutions and invariant sets ......................... 31
Chapter 3 Attractors .......................................... 37
1 Attractors of processes ..................................... 38
2 Attractors of skew product flows ............................ 41
3 Existence of pullback attractors ............................ 44
4 Relationship between nonautonomous attractors ............... 52
5 Upper semi-continuous dependence on parameters .............. 55
6 Parametrically inflated pullback attractors ................. 57
7 Pullback attractors with continuous fibers .................. 60
8 Local attractors and repellers .............................. 62
Chapter 4 Morse decompositions ................................ 69
1 Attractor-repeller pairs .................................... 69
2 Morse decompositions ........................................ 72
3 The one-dimensional case .................................... 75
Chapter 5 Linear systems ...................................... 79
1 Exponential dichotomy ....................................... 79
2 Dichotomy spectrum .......................................... 82
3 Lyapunov spectrum ........................................... 87
4 Morse decompositions ........................................ 89
Chapter 6 Invariant manifolds ................................ 105
1 Global invariant manifolds ................................. 105
2 Local invariant manifolds .................................. 112
3 Hierarchies of invariant manifolds ......................... 114
4 Taylor approximation ....................................... 116
5 Reduction principle ........................................ 123
Chapter 7 Lyapunov functions ................................. 129
1 Lyapunov functions for solutions ........................... 129
2 Lyapunov functions for autonomous attractors ............... 132
3 Lyapunov functions for pullback attractors ................. 135
4 Lyapunov functions for Morse decompositions ................ 143
Chapter 8 Bifurcations ....................................... 147
1 Nonautonomous Bernoulli equations .......................... 147
2 One-dimensional bifurcation patterns ....................... 149
3 Higher-dimensional Bernoulli-like equations ................ 157
4 Further developments ....................................... 163
Chapter 9 Set-valued nonautonomous dynamical systems ......... 169
1 Set-valued processes ....................................... 170
2 Set-valued skew product flows .............................. 173
3 Backward extension of autonomous semi-dynamical systems .... 175
4 Proof of existence of nonautonomous invariant sets ......... 178
Chapter 10 Nonautonomous semi-dynamical systems ............... 185
1 Attractors of skew product semi-flows ...................... 185
2 The twisted horseshoe mapping .............................. 189
Chapter 11 Approximation and perturbation of attractors ....... 191
1 Nonautonomous perturbations of an autonomous system ........ 191
2 Numerical approximation of uniform attractors .............. 193
3 Perturbation of the driving system ......................... 197
Chapter 12 Infinite-dimensional systems ....................... 205
1 Squeezing and flattening properties: the autonomous case ... 205
2 Pullback asymptotic compactness ............................ 207
Chapter 13 Switching and control systems ..................... 213
1 Switching systems .......................................... 213
2 Affine control systems ..................................... 222
Chapter 14 Random dynamical systems .......................... 227
1 Random attractors .......................................... 228
2 The Ornstein-Uhlenbeck process ............................. 229
3 Random attractors for stochastic differential equations .... 231
Chapter 15 Synchronization ................................... 235
1 Deterministic nonautonomous systems ........................ 235
2 Synchronization of systems with additive noise ............. 242
3 Synchronization of systems with linear noise ............... 247
Appendix ...................................................... 251
Bibliography .................................................. 253
Index ......................................................... 263
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