Attractivity and Bifurcation for Nonautonomous Dynamical Systems

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Springer Science & Business Media, 08.06.2007 - 217 Seiten

Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two different approaches are developed which are based on special definitions of local attractivity and repulsivity. It is shown that these notions lead to nonautonomous Morse decompositions, which are useful to describe the global asymptotic behavior of systems on compact phase spaces. Furthermore, methods from the qualitative theory for linear and nonlinear systems are derived, and nonautonomous counterparts of the classical one-dimensional autonomous bifurcation patterns are developed.

 

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Inhalt

Introduction
1
Notions of Attractivity and Bifurcation
7
22 Nonautonomous Dynamical Systems
9
23 Attractivity and Repulsivity
12
232 The Noninvertible Case
21
233 Radii of Attraction and Repulsion
22
234 Domains of Attraction and Repulsion
23
235 Properties of Time Reversal
28
Linear Systems
81
42 Dichotomy Spectra
94
43 Lyapunov Spectra
106
44 Spectra of Autonomous Linear Systems
108
45 Roughness
112
Nonlinear Systems
114
51 Invariant Manifolds
116
52 An Application to Bifurcation Theory
124

236 Existence and Uniqueness
29
24 Other Notions of Attractivity and Repulsivity
39
242 Autonomous Attractors and Repellers
40
243 Nonautonomous Attractors
41
25 Bifurcation and Transition
42
252 Examples
45
26 Other Notions of Bifurcation and Transition
47
262 Topological Skew Product Flows
48
263 Random Dynamical Systems
49
264 General Nonautonomous Dynamical Systems
50
Nonautonomous Morse Decompositions
51
32 Morse Decompositions
57
33 Lyapunov Functions
62
34 Morse Decompositions in Dimension One
64
35 Morse Decompositions of Linear Systems
67
53 Linearized Attractivity and Repulsivity
126
54 Bifurcation Theory of Adiabatic Systems
130
Bifurcations in Dimension One
137
62 Nonautonomous Pitchfork Bifurcation
144
Bifurcations of Asymptotically Autonomous Systems
153
71 Basic Properties of Asymptotically Autonomous Systems
154
72 Bifurcations in Dimension One
168
73 Bifurcations in Dimension Two
181
Appendix
192
A2 Useful Lemmata
195
A3 Projective Spaces
196
References
199
Index
209
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