Attractivity and bifurcation for nonautonomous dynamical systems / Martin Rasmussen.

Format:

Book

Author/Creator:

Rasmussen, Martin, 1975-

Series:

Lecture notes in mathematics (Springer-Verlag) ; 1907.

Lecture notes in mathematics, 0075-8434 ; 1907

Language:

English

Subjects (All):

Differentiable dynamical systems.

Differential equations, Linear.

Bifurcation theory.

Physical Description:

xi, 212 pages : illustrations ; 24 cm.

Place of Publication:

Berlin ; New York : Springer, [2007]

Summary:

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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.

Contents:

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.1 Ordinary Differential Equations 193

A.2 Useful Lemmata 195

A.3 Projective Spaces 196.

Notes:

Includes bibliographical references and index.

ISBN:

3540712240

9783540712244

OCLC:

141385299