Symmetry-breaking bifurcations of wreath product systems
type: Item
instance of: scholarly article
title: Symmetry-breaking bifurcations of wreath product systems
zbMATH Open document ID: 0973.37029
published in: Journal of Nonlinear Science
publication date: 2001-11-23T00:00:00Z
review text: The structure of matrices is investigated which are commuting with special linear representations of special finite groups. This research is motivated by the bifurcation analysis of dynamical systems describing coupled oscillators. Here a coupling is investigated which leads to a dynamical system which is equivariant with respect to a linear representation of a wreath product group. In the bifurcation analysis matrices commuting with a wreath product arise naturally. This paper uses the classification of bifurcation subgroups for absolutely irreducible representations of wreath products and of \(C\)-axial subgroups of complex irreducible representations of wreath products given by \textit{B. Dionne, M. Golubitsky} and \textit{I. N. Stewart} [Nonlinearity 9, No. 2, 559-574 (1996; Zbl 0894.58049)]. The two main results of this paper are based on Schur's lemma. The authors investigate the special form of matrices (a) commuting with a representation of an axial subgroup of a wreath product group which is derived by restricting an absolutely irreducible representation of the wreath product and (b) commuting with a linear representation of a \(C\)-axial subgroup of a wreath product derived by restricting a complex irreducible representation of the wreath product. The equivariant bifurcation lemma and the equivariant Hopf bifurcation are recalled for this context.
reviewed by: Karin Gatermann
Mathematics Subject Classification ID: 37G40
Mathematics Subject Classification ID: 20C35
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C15
zbMATH DE Number: 1385272
zbMATH Keywords: linear representation theory of wreath product groups
zbMATH Keywords: commuting matrices
zbMATH Keywords: coupled oscillators
zbMATH Keywords: equivariant bifurcation lemma
author: Ana Paula S. Dias
author: Ian Stewart
MaRDI profile type: Publication
full work available at URL: s003329900082
OpenAlex ID: W2085034489
DOI: 10.1007/S003329900082
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description: scientific article; zbMATH DE number 1385272