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Bifurcation without Parameters

  • Book
  • © 2015

Overview

Authors:
  1. Stefan Liebscher

    1. Institur für Mathematik, Freie Universität Berlin, Berlin, Germany

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  • This is the first systematic treatment of the topic
  • Numerous figures augment the analytic treatment and assist in understanding
  • The abstract theory is complemented by many applications

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2117)

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Table of contents (16 chapters)

  1. Front Matter

    Pages i-xii
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  2. Preliminaries

    1. Front Matter

      Pages 1-1
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    2. Introduction

      • Stefan Liebscher
      Pages 3-12

    3. Methods and Concepts

      • Stefan Liebscher
      Pages 13-19

    4. Cosymmetries

      • Stefan Liebscher
      Pages 21-23

  3. Codimension One

    1. Front Matter

      Pages 25-25
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    2. Transcritical Bifurcation

      • Stefan Liebscher
      Pages 27-34

    3. Poincaré-Andronov-Hopf Bifurcation

      • Stefan Liebscher
      Pages 35-41

    4. Application: Decoupling in Networks

      • Stefan Liebscher
      Pages 43-47

    5. Application: Oscillatory Profiles in Systems of Hyperbolic Balance Laws

      • Stefan Liebscher
      Pages 49-54

  4. Codimension Two

    1. Front Matter

      Pages 55-55
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    2. Degenerate Transcritical Bifurcation

      • Stefan Liebscher
      Pages 57-65

    3. Degenerate Poincaré-Andronov-Hopf Bifurcation

      • Stefan Liebscher
      Pages 67-79

    4. Bogdanov-Takens Bifurcation

      • Stefan Liebscher
      Pages 81-102

    5. Zero-Hopf Bifurcation

      • Stefan Liebscher
      Pages 103-108

    6. Double-Hopf Bifurcation

      • Stefan Liebscher
      Pages 109-113

    7. Application: Cosmological Models of Bianchi Type, the Tumbling Universe

      • Stefan Liebscher
      Pages 115-118

    8. Application: Fluid Flow in a Planar Channel, Spatial Dynamics with Reversible Bogdanov-Takens Bifurcation

      • Stefan Liebscher
      Pages 119-128

  5. Beyond Codimension Two

    1. Front Matter

      Pages 129-129
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    2. Codimension-One Manifolds of Equilibria

      • Stefan Liebscher
      Pages 131-133
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Keywords

About this book

Targeted at mathematicians having at least a basic familiarity with classical bifurcation theory, this monograph provides a systematic classification and analysis of bifurcations without parameters in dynamical systems. Although the methods and concepts are briefly introduced, a prior knowledge of center-manifold reductions and normal-form calculations will help the reader to appreciate the presentation. Bifurcations without parameters occur along manifolds of equilibria, at points where normal hyperbolicity of the manifold is violated. The general theory, illustrated by many applications, aims at a geometric understanding of the local dynamics near the bifurcation points.

Authors and Affiliations

  • Institur für Mathematik, Freie Universität Berlin, Berlin, Germany

    Stefan Liebscher

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