Nonautonomous Bifurcation Theory: Concepts and Tools

Description

Bifurcation theory is a major topic in dynamical systems theory with profound applications. However, in contrast to autonomous dynamical systems, it is not clear what a bifurcation of a nonautonomous dynamical system actually is, and so far, various different approaches to describe qualitative changes have been suggested in the literature. The aim of this book is to provide a concise survey of the area and equip the reader with suitable tools to tackle nonautonomous problems. A review, discussion and comparison of several concepts of bifurcation is provided, and these are formulated in a unified notation and illustrated by means of comprehensible examples. Additionally, certain relevant tools needed in a corresponding analysis are presented. Preface Notation Contents 1 Introduction 1.1 Nonautonomous Dynamical Systems 1.1.1 Processes 1.1.2 Skew Product Flows 1.2 Examples of Nonautonomous Bifurcations 1.3 Topological Equivalence 1.4 Neglected Topics Part I Nonautonomous Differential Equations 2 Spectral Theory, Stability and Continuation 2.1 Spectral Theory 2.1.1 Dichotomy Spectrum 2.1.2 Lyapunov Spectrum 2.2 Stability 2.3 Continuation 3 Nonautonomous Bifurcation 3.1 Attractor Bifurcation 3.2 Solution Bifurcation 3.2.1 One-Dimensional Solution Bifurcations 3.2.2 Shovel Bifurcation 3.2.3 Rate-Induced Tipping 3.3 Bifurcation of Minimal Sets 3.3.1 Scalar Equations 3.3.2 Hopf Bifurcation 4 Reduction Techniques 4.1 Centre Integral Manifolds 4.2 Normal Forms Part II Nonautonomous Difference Equations 5 Spectral Theory, Stability and Continuation 5.1 Spectral Theory 5.1.1 Dichotomy Spectrum 5.1.2 Lyapunov Spectrum 5.2 Stability 5.3 Continuation 5.3.1 Continuation of Entire Solutions 5.3.2 Continuation on Half Lines: Invariant Fibre Bundles 6 Nonautonomous Bifurcation 6.1 Attractor Bifurcation 6.1.1 Transcritical and Pitchfork Bifurcation 6.1.2 Sacker–Neimark Bifurcation 6.2 Solution Bifurcation 6.2.1 Fold Bifurcation 6.2.2 Crossing Curve Bifurcation 6.2.3 Shovel Bifurcation 6.3 Bifurcation of Minimal Sets and Invariant Graphs 6.3.1 Fold Bifurcations of Invariant Graphs 6.3.2 Fold Bifurcation in the Logistic Equation 7 Reduction Techniques 7.1 Centre Fibre Bundles 7.2 Normal Forms Appendix A.1 Stability Theory A.2 Bohl and Lyapunov Exponents References Index