Singularities and groups in bifurcation theory. Volume II
type: Item
instance of: scholarly article
title: Singularities and groups in bifurcation theory. Volume II
zbMATH Open document ID: 0691.58003
author: Martin Golubitsky
author: D. G. Schaeffer
author: Ian Stewart
published in: Applied Mathematical Sciences
publication date: 1993-06-05T00:00:00Z
review text: The aim of volume I (1985; Zbl 0607.35004) of this work was to study applications of singularity theory to bifurcation, showing that unfolding degenerate singularities illustrates complicated bifurcation diagrams. A very important new ingredient, namely symmetry, both in the equations and in their solutions, is introduced in volume II. Starting from the symmetry group G of an ordinary differential equation (ODE), the isotropy subgroup of a steady-state solution \(x_ 0\) is defined as \(\Sigma =\{g\in G:\) \(gx_ 0=x_ 0\}\). One of the main problems in this book is ``generically, for which isotropy subgroups \(\Sigma\) should we expect to find bifurcating branches of steady-states having \(\Sigma\) as their group of symmetries'' (p. 8). The fact that G- invariant solutions break into a branch of solutions with symmetry group \(\Sigma\) is named spontaneous symmetry-breaking. It is remarked that bifurcation problems become more complicated when symmetry is involved because this usually forces eigenvalues to have high (\(\geq 2)\) multiplicity, which implies, via the Lyapunov-Schmidt reduction, bifurcation equations with several variables. But, in turn, the very same symmetry complicating bifurcation provides techniques to deal with the corresponding difficulties. Multiplicity of eigenvalues is closely related with irreducible group actions, because all compact Lie groups, except 1 and \(Z_ 2=\{\pm 1\}\), have irreducible actions on vector spaces of dimension \(>1.\) An equivariant singularity theorem is developed in this volume in order to adapt the framework in vol. I to the presence of symmetry. Another main topic is mode interaction, i.e., complications created by several eigenvalues crossing simultaneously the imaginary axis. As a rule, the study of Hopf bifurcation (for ODEs) plays a very important role all along the book. Now we shall give a general idea of the contents of vol. II. More comments will follow at the end. Chapter 12 gives an introduction to Lie grou…
Mathematics Subject Classification ID: 58-02
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 20C32
zbMATH DE Number: 193465
zbMATH Keywords: isotropy group
zbMATH Keywords: irreducible representation
zbMATH Keywords: singularity theory
zbMATH Keywords: bifurcation
zbMATH Keywords: symmetry
zbMATH Keywords: spontaneous symmetry-breaking
zbMATH Keywords: equivariant singularity theory
zbMATH Keywords: equivariant Hopf bifurcation
reviewed by: Jesús Hernández
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 193465