The symmetry perspective. From equilibrium to chaos in phase space and physical space
type: Item
instance of: scholarly article
title: The symmetry perspective. From equilibrium to chaos in phase space and physical space
zbMATH Open document ID: 1031.37001
author: Martin Golubitsky
author: Ian Stewart
published in: Progress in Mathematics
publication date: 2002-10-28T00:00:00Z
review text: This skillfully written monograph has been awarded the Ferran Sunyer i Balaguer 2001 prize for a mathematical text of expository nature. Its principal goal is to demonstrate the important role of symmetry in pattern-formation and the ways symmetry can be used for exploring mechanisms by which patterns form. The book is divided into ten chapters and concludes with the exhaustive list of 528 references. The first chapter introduces two important notions: symmetry of a system of ODEs and symmetry of a solution of a system of ODEs which, in general, need not be the same as the symmetry of the system. Then the phenomenon of symmetry-breaking in which solutions have less symmetry than equations is considered. Two key results, Lyapunov-Schmidt reduction which reduces the dimension of a bifurcation problem, and the Equivariant Branching Lemma which provides sufficient conditions for the existence of a bifurcating branch of equilibria with a particular type of symmetry group, called axial, are presented. Then this lemma is applied to the specification model, and a number of important issues on the relevance of symmetry assumptions in modeling are discussed. Chapter 2 deals with the stability of equilibria, in particular with those whose existence follows from the Equivariant Branching Lemma. It is demonstrated that symmetry can be exploited systematically to simplify computations of stabilities, and that in some cases symmetry constraints force instability of all bifurcating branches, found by using the Equivariant Branching Lemma, near the bifurcation point. Animal locomotion, which uses different time-periodic patterns having spatial regularity corresponding to permutations of the legs, is the main topic of Chapter 3, where classification of all possible spatio-temporal symmetry groups for time-periodic states of systems with a given spatial symmetry group is provided. Thorough analysis of the Hopf bifurcation, which is one of the most common routes to time periodicity, is…
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 37C80
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 70H33
zbMATH DE Number: 1821147
zbMATH Keywords: symmetry
zbMATH Keywords: bifurcation
zbMATH Keywords: equivariant dynamical systems
zbMATH Keywords: invariants
zbMATH Keywords: reduction
zbMATH Keywords: equivariant bifurcation theory
zbMATH Keywords: heteroclinic cycles
zbMATH Keywords: stability
zbMATH Keywords: equilibria
zbMATH Keywords: symmetry groups
zbMATH Keywords: pattern formation
zbMATH Keywords: nonlinear dynamics
reviewed by: Yuriy V. Rogovchenko
describes a project that uses: Symmetry in Chaos
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1821147