Singularities and groups in bifurcation theory

Author(s)

• Golubitsky, Martin
• Stewart, Ian Nicholas
• Schaeffer, David G

Bibliographic Information

Singularities and groups in bifurcation theory

Martin Golubitsky, David G. Schaeffer

（Applied mathematical sciences, v. 51, 69）

Springer-Verlag, c1985-c1988

• v. 1 : us
• v. 1 : gw
• v. 2 : us
• v. 2 : gw

Note

Includes bibliography and indexes

Vol. 2 co-authored by Ian Stewart

Description and Table of Contents

Volume

v. 1 : us ISBN 9780387909998

Description

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.

Volume

v. 2 : us ISBN 9780387966526

Description

Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. Volume I discusses how singularity-theoretic techniques aid the understanding of transitions in multiparameter systems. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation, equicariant singularity theory, Hopf bifurcation with symmetry, and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications.

Table of Contents

of Volume II.- XI Introduction.- 0. Introduction.- 1. Equations with Symmetry.- 2. Techniques.- 3. Mode Interactions.- 4. Overview.- XII Group-Theoretic Preliminaries.- 0. Introduction.- 1. Group Theory.- 2. Irreducibility.- 3. Commuting Linear Mappings and Absolute Irreducibility.- 4. Invariant Functions.- 5. Nonlinear Commuting Mappings.- 6.* Proofs of Theorems in 4 and 5.- 7.* Tori.- XIII Symmetry-Breaking in Steady-State Bifurcation.- 0. Introduction.- 1. Orbits and Isotropy Subgroups.- 2. Fixed-Point Subspaces and the Trace Formula.- 3. The Equivariant Branching Lemma.- 4. Orbital Asymptotic Stability.- 5. Bifurcation Diagrams and DnSymmetry.- 6. Subgroups of SO(3).- 7. Representations of SO(3) and O(3): Spherical Harmonics.- 8. Symmetry-Breaking from SO(3).- 9. Symmetry-Breaking from O(3).- 10.* Generic Spontaneous Symmetry-Breaking.- Case Study 4 The Planar Benard Problem.- 0. Introduction.- 1. Discussion of the PDE.- 2. One-Dimensional Fixed-Point Subspaces.- 3. Bifurcation Diagrams and Asymptotic Stability.- XIV Equivariant Normal Forms.- 0. Introduction.- 1. The Recognition Problem.- 2.* Proof of Theorem 1.3.- 3. Sample Computations of RT(h, ?).- 4. Sample Recognition Problems.- 5. Linearized Stability and ?-equivalence.- 6. Intrinsic Ideals and Intrinsic Submodules.- 7. Higher Order Terms.- XV Equivariant Unfolding Theory.- 0. Introduction.- 1. Basic Definitions.- 2. The Equivariant Universal Unfolding Theorem.- 3. Sample Universal ?-unfoldings.- 4. Bifurcation with D3 Symmetry.- 5. The Spherical Benard Problem.- 6. Spherical Harmonics of Order 2.- 7.* Proof of the Equivariant Universal Unfolding Theorem.- 8.* The Equivariant Preparation Theorem.- Case Study 5 The Traction Problem for Mooney-Rivlin Material.- 0. Introduction.- 1. Reduction to D3 Symmetry in the Plane.- 2. Taylor Coefficients in the Bifurcation Equation.- 3. Bifurcations of the Rivlin Cube.- XVI Symmetry-Breaking in Hopf Bifurcation.- 0. Introduction.- 1. Conditions for Imaginary Eigenvalues.- 2. A Simple Hopf Theorem with Symmetry.- 3. The Circle Group Action.- 4. The Hopf Theorem with Symmetry.- 5. Birkhoff Normal Form and Symmetry.- 6. Floquet Theory and Asymptotic Stability.- 7. Isotropy Subgroups of ? x S1.- 8.* Dimensions of Fixed-Point Subspaces.- 9. Invariant Theory for ? x S1.- 10. Relationship Between Liapunov-Schmidt Reduction and Birkhoff Normal Form.- 11.* Stability in Truncated Birkhoff Normal Form.- XVII Hopf Bifurcation with O(2) Symmetry.- 0. Introduction.- 1. The Action of O(2) x S1.- 2. Invariant Theory for O(2) x S1.- 3. The Branching Equations.- 4. Amplitude Equations, D4 Symmetry, and Stability.- 5. Hopf Bifurcation with O(n) Symmetry.- 6. Bifurcation with D4 Symmetry.- 7. The Bifurcation Diagrams.- 8. Rotating Waves and SO(2) or Zn Symmetry.- XVIII Further Examples of Hopf Bifurcation with Symmetry.- 0. Introduction.- 1. The Action of Dn x S1.- 2. Invariant Theory for Dn x S1.- 3. Branching and Stability for Dn.- 4. Oscillations of Identical Cells Coupled in a Ring.- 5. Hopf Bifurcation with O(3) Symmetry.- 6. Hopf Bifurcation on the Hexagonal Lattice.- XIX Mode Interactions.- 0. Introduction.- 1. Hopf/Steady-State Interaction.- 2. Bifurcation Problems with Z2 Symmetry.- 3. Bifurcation Diagrams with Z2 Symmetry.- 4. Hopf/Hopf Interaction.- XX Mode Interactions with O(2) Symmetry.- 0. Introduction.- l. Steady-State Mode Interaction.- 2. Hopf/Steady-State Mode Interaction.- 3. Hopf/Hopf Mode Interaction.- Case Study 6 The Taylor-Couette System.- 0. Introduction.- 1. Detailed Overview.- 2. The Bifurcation Theory Analysis.- 3. Finite Length Effects.

Volume

v. 2 : gw ISBN 9783540966524

Description

Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation; equicariant singularity theory; Hopf bifurcation with symmetry; and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications.

Table of Contents

• Group theoretic preliminaries
• symmetry-breaking in steady-state bifurcation
• case study 4 - the planar Benard problem
• equivariant normal forms
• equivariant unfolding theory
• case study 5 - the traction problem for Mooney-Rivlin material
• symmetry-breaking in Hopf bifurcation
• Hopf bifurcation with O(2) symmetry
• further examples of Hopf bifurcation with symmetry
• mode interactions
• mode interactions with O(2) symmetry
• case study 6 - the Taylor-Couette system.

by "Nielsen BookData"

Details

• NCID: BA01160855
• ISBN:
  • 0387909990
  • 3540909990
  • 0387966528
  • 3540966528
• LCCN: 84001414
• Country Code: us
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: New York ; Tokyo
• Pages/Volumes: 2 v.
• Size: 25 cm
• Classification:
  • LCC : QA374
  • DC19 : 515.3/53
• Subject Headings:
  • LCSH : Bifurcation theory
  • LCSH : Groups, Theory of
• Parent Bibliography ID:
  • BA00020952