1. Introduction ................................................ 1
2. Notions of Attractivity and Bifurcation ..................... 7
2.1. Preliminary Definitions ............................... 7
2.2. Nonautonomous Dynamical Systems ....................... 9
2.3. Attractivity and Repulsivity ......................... 12
2.3.1. Definitions .................................. 12
2.3.2. The Noninvertible Case ....................... 21
2.3.3. Radii of Attraction and Repulsion ............ 22
2.3.4. Domains of Attraction and Repulsion .......... 23
2.3.5. Properties of Time Reversal .................. 28
2.3.6. Existence and Uniqueness ..................... 29
2.4. Other Notions of Attractivity and Repulsivity ........ 39
2.4.1. Stability in the Sense of Lyapunov ........... 39
2.4.2. Autonomous Attractors and Repellers .......... 40
2.4.3. Nonautonomous Attractors ..................... 41
2.5. Bifurcation and Transition ........................... 42
2.5.1. Definitions .................................. 42
2.5.2. Examples ..................................... 45
2.6. Other Notions of Bifurcation and Transition .......... 47
2.6.1. The Autonomous Case .......................... 47
2.6.2. Topological Skew Product Flows ............... 48
2.6.3. Random Dynamical Systems ..................... 49
2.6.4 General Nonautonomous Dynamical Systems ....... 50
3. Nonautonomous Morse Decompositions ......................... 51
3.1. Attractor-Repeller Pairs ............................. 51
3.2. Morse Decompositions ................................. 57
3.3. Lyapunov Functions ................................... 62
3.4. Morse Decompositions in Dimension One ................ 64
3.5. Morse Decompositions of Linear Systems ............... 67
4. Linear Systems ............................................. 81
4.1. Notions of Dichotomy ................................. 81
4.2. Dichotomy Spectra .................................... 94
4.3. Lyapunov Spectra .................................... 106
4.4. Spectra of Autonomous Linear Systems ................ 108
4.5. Roughness ........................................... 112
5. Nonlinear Systems ......................................... 115
5.1. Invariant Manifolds ................................. 116
5.2. An Application to Bifurcation Theory ................ 124
5.3. Linearized Attractivity and Repulsivity ............. 126
5.4. Bifurcation Theory of Adiabatic Systems ............. 130
6. Bifurcations in Dimension One ............................. 137
6.1. Nonautonomous Transcritical Bifurcation ............. 137
6.2. Nonautonomous Pitchfork Bifurcation ................. 144
7. Bifurcations of Asymptotically Autonomous Systems ......... 153
7.1. Basic Properties of Asymptotically Autonomous
Systems ............................................. 154
7.2. Bifurcations in Dimension One ....................... 168
7.3. Bifurcations in Dimension Two ....................... 181
A Appendix .................................................. 193
Appendix ...................................................... 193
A.l. Ordinary Differential Equations ...................... 193
A.2. Useful Lemmata ....................................... 195
А.З. Projective Spaces .................................... 196
References .................................................... 199
Index ......................................................... 209
| 
