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  • © 2015

Bifurcation without Parameters

Authors:

  • 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
  2. Preliminaries

    1. Front Matter

      Pages 1-1
    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
    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
  4. Codimension Two

    1. Front Matter

      Pages 55-55
    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
  5. Beyond Codimension Two

    1. Front Matter

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

      • Stefan Liebscher
      Pages 131-133

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.

Keywords

  • 34C23,34C20,34C37,37G99,35B32
  • Bifurcation without parameters
  • Equilibria
  • Manifolds
  • Nonlinear dynamical systems
  • ordinary differential equations
  • partial differential equations

Authors and Affiliations

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

    Stefan Liebscher

Bibliographic Information

  • Book Title: Bifurcation without Parameters

  • Authors: Stefan Liebscher

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-319-10777-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Softcover ISBN: 978-3-319-10776-9Published: 19 November 2014

  • eBook ISBN: 978-3-319-10777-6Published: 08 November 2014

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: XII, 142

  • Number of Illustrations: 5 b/w illustrations, 29 illustrations in colour

  • Topics: Differential Equations, Dynamical Systems

Buying options

eBook USD 39.99
Price excludes VAT (Canada)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 49.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions