Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben

2023 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Nonautonomous Bifurcation Theory

Concepts and Tools

verfasst von: Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen

Verlag: Springer Nature Switzerland

Buchreihe : Frontiers in Applied Dynamical Systems: Reviews and Tutorials

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
We first introduce our basic terminology, as well as processes and skew product flows as basic concepts to describe nonautonomous dynamics. Several examples are presented illustrating different types of nonautonomous bifurcations, which set the stage for our further analysis. Finally, we remark on topological equivalence and describe several neglected topics.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen

Nonautonomous Differential Equations

Frontmatter
Chapter 2. Spectral Theory, Stability and Continuation
Abstract
We review concepts to describe stability and hyperbolicity of nonautonomous ordinary differential equations based on their linearisation, namely exponential dichotomies, the resulting dichotomy (Sacker-Sell) spectrum and the Spectral Theorem, as well as the Lyapunov spectrum. Moreover, it is shown that hyperbolic bounded entire solutions can be continued in parameters and therefore nonhyperbolicity is necessary for bifurcation.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen
Chapter 3. Nonautonomous Bifurcation
Abstract
Three classes of nonautonomous bifurcations are introduced for ordinary differential equations, namely attractor bifurcation, solution bifurcation and bifurcation of minimal sets. We review sufficient conditions for such bifurcations and present suitable examples. Rate-induced tipping is understood as a special case of solution bifurcation. The chapter closes with some remarks on nonautonomous Hopf bifurcations.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen
Chapter 4. Reduction Techniques
Abstract
We review nonautonomous versions of the two classical methods to simplify bifurcation problems: (1) centre integral manifolds allow a reduction in dimension and we describe how to obtain a Taylor approximation of integral manifolds and (2) under suitable nonresonance conditions, normal form theory yields algebraic simplification. Both approaches are based on suitable assumptions on the dichotomy spectrum.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen

Nonautonomous Difference Equations

Frontmatter
Chapter 5. Spectral Theory, Stability and Continuation
Abstract
In the framework of nonautonomous difference equations, we review tools to describe stability and hyperbolicity based on their linearisation, namely exponential dichotomies and Lyapunov exponents. We comment on the fine structure of the resulting dichotomy (Sacker–Sell) spectrum and formulae its Spectral Theorem. For the Lyapunov spectrum, a version of the Multiplicative Ergodic Theorem is formulated. It is discussed that hyperbolic bounded entire solutions can be continued in parameters, how Taylor approximations of the perturbed solutions are obtained. Furthermore, solutions hyperbolic on semisaxes persist as stabile and unstable fibre bundles.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen
Chapter 6. Nonautonomous Bifurcation
Abstract
As in the continuous time situation, bifurcations in nonautonomous difference equations are classified as attractor bifurcation, solution bifurcation and bifurcation of minimal sets and invariant graphs. In reviewing sufficient conditions for such bifurcations, we try to complement those results covered for ordinary differential equations already. A nonautonomous Sacker–Neimark bifurcation is understood as an attractor bifurcation.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen
Chapter 7. Reduction Techniques
Abstract
We review nonautonomous versions of centre manifolds, namely centre fibre bundles, their Taylor approximation and the use of the Reduction Principle in bifurcation theory. In addition, nonautonomous normal form theory is discussed.
Vasso Anagnostopoulou, Christian Pötzsche, Martin Rasmussen
Backmatter
Metadaten
Titel
Nonautonomous Bifurcation Theory
verfasst von
Vasso Anagnostopoulou
Christian Pötzsche
Martin Rasmussen
Copyright-Jahr
2023
Electronic ISBN
978-3-031-29842-4
Print ISBN
978-3-031-29841-7
DOI
https://doi.org/10.1007/978-3-031-29842-4