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E-raamat: Nonautonomous Dynamical Systems

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Nonautonomous Dynamical SystemsSuurem pilt
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The theory of nonautonomous dynamical systems in both of its formulations as processes and skew product flows is developed systematically in this book. The focus is on dissipative systems and nonautonomous attractors, in particular the recently introduced concept of pullback attractors. Linearization theory, invariant manifolds, Lyapunov functions, Morse decompositions and bifurcations for nonautonomous systems and set-valued generalizations are also considered as well as applications to numerical approximations, switching systems and synchronization. Parallels with corresponding theories of control and random dynamical systems are briefly sketched. With its clear and systematic exposition, many examples and exercises, as well as its interesting applications, this book can serve as a text at the beginning graduate level. It is also useful for those who wish to begin their own independent research in this rapidly developing area.
Preface vii
Chapter 1 Autonomous dynamical systems
1(22)
1 Introduction
1(3)
2 Local asymptotic behavior
4(8)
3 Global asymptotic behavior
12(5)
4 Dependence on parameters
17(6)
Chapter 2 Nonautonomous dynamical systems
23(14)
1 Processes formulation
23(3)
2 Skew product flow formulation
26(5)
3 Entire solutions and invariant sets
31(6)
Chapter 3 Attractors
37(32)
1 Attractors of processes
38(3)
2 Attractors of skew product flows
41(3)
3 Existence of pullback attractors
44(8)
4 Relationship between nonautonomous attractors
52(3)
5 Upper semi-continuous dependence on parameters
55(2)
6 Parametrically inflated pullback attractors
57(3)
7 Pullback attractors with continuous fibers
60(2)
8 Local attractors and repellers
62(7)
Chapter 4 Morse decompositions
69(10)
1 Attractor-repeller pairs
69(3)
2 Morse decompositions
72(3)
3 The one-dimensional case
75(4)
Chapter 5 Linear systems
79(26)
1 Exponential dichotomy
79(3)
2 Dichotomy spectrum
82(5)
3 Lyapunov spectrum
87(2)
4 Morse decompositions
89(16)
Chapter 6 Invariant manifolds
105(24)
1 Global invariant manifolds
105(7)
2 Local invariant manifolds
112(2)
3 Hierarchies of invariant manifolds
114(2)
4 Taylor approximation
116(7)
5 Reduction principle
123(6)
Chapter 7 Lyapunov functions
129(18)
1 Lyapunov functions for solutions
129(3)
2 Lyapunov functions for autonomous attractors
132(3)
3 Lyapunov functions for pullback attractors
135(8)
4 Lyapunov functions for Morse decompositions
143(4)
Chapter 8 Bifurcations
147(22)
1 Nonautonomous Bernoulli equations
147(2)
2 One-dimensional bifurcation patterns
149(8)
3 Higher-dimensional Bernoulli-like equations
157(6)
4 Further developments
163(6)
Chapter 9 Set-valued nonautonomous dynamical systems
169(16)
1 Set-valued processes
170(3)
2 Set-valued skew product flows
173(2)
3 Backward extension of autonomous semi-dynamical systems
175(3)
4 Proof of existence of nonautonomous invariant sets
178(7)
Chapter 10 Nonautonomous semi-dynamical systems
185(6)
1 Attractors of skew product semi-flows
185(4)
2 The twisted horseshoe mapping
189(2)
Chapter 11 Approximation and perturbation of attractors
191(14)
1 Nonautonomous perturbations of an autonomous system
191(2)
2 Numerical approximation of uniform attractors
193(4)
3 Perturbation of the driving system
197(8)
Chapter 12 Infinite-dimensional systems
205(8)
1 Squeezing and flattening properties: the autonomous case
205(2)
2 Pullback asymptotic compactness
207(6)
Chapter 13 Switching and control systems
213(14)
1 Switching systems
213(9)
2 Affine control systems
222(5)
Chapter 14 Random dynamical systems
227(8)
1 Random attractors
228(1)
2 The Ornstein-Uhlenbeck process
229(2)
3 Random attractors for stochastic differential equations
231(4)
Chapter 15 Synchronization
235(16)
1 Deterministic nonautonomous systems
235(7)
2 Synchronization of systems with additive noise
242(5)
3 Synchronization of systems with linear noise
247(4)
Appendix 251(2)
Bibliography 253(10)
Index 263
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