Group-theoretic bifurcation theory

Kiyohiro Ikeda, Kazuo Murota

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Group-theoretic bifurcation theory is introduced as a means to describe qualitative aspects of symmetry-breaking bifurcation, such as possible types of critical points and the symmetry of bifurcating solutions. We advance a series of mathematical concepts and tools, including: group equivariance, Liapunov–Schmidt reduction, equivariant branching lemma, and block-diagonalization. The theory of linear representations of finite groups in Chap. 7 forms a foundation of this chapter. This chapter is an extension of Chap. 2 to a system with symmetry and a prerequisite to the study of structures and materials with dihedral symmetry in Chaps. 9 – 13 and larger symmetries in Chaps. 14 – 17.

Original languageEnglish
Title of host publicationApplied Mathematical Sciences (Switzerland)
PublisherSpringer
Pages201-235
Number of pages35
DOIs
Publication statusPublished - 2019 Jan 1

Publication series

NameApplied Mathematical Sciences (Switzerland)
Volume149
ISSN (Print)0066-5452
ISSN (Electronic)2196-968X

Keywords

  • Bifurcation
  • Bifurcation equation
  • Block-diagonalization
  • Critical point
  • Equivariant branching lemma
  • Group equivariance
  • Group representation
  • Group-theoretic bifurcation theory
  • Imperfection sensitivity matrix
  • Liapunov-Schmidt reduction
  • Symmetry

ASJC Scopus subject areas

  • Applied Mathematics

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