Group-theoretic bifurcation theory

Kiyohiro Ikeda; Kazuo Murota

doi:10.1007/978-3-030-21473-9_8

Group-theoretic bifurcation theory

Kiyohiro Ikeda, Kazuo Murota

ENG - Civil and Environmental Engineering

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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 language	English
Title of host publication	Applied Mathematical Sciences (Switzerland)
Publisher	Springer
Pages	201-235
Number of pages	35
DOIs	https://doi.org/10.1007/978-3-030-21473-9_8
Publication status	Published - 2019 Jan 1

Publication series

Name	Applied Mathematical Sciences (Switzerland)
Volume	149
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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10.1007/978-3-030-21473-9_8

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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.",

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KW - Block-diagonalization

KW - Critical point

KW - Equivariant branching lemma

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KW - Group representation

KW - Group-theoretic bifurcation theory

KW - Imperfection sensitivity matrix

KW - Liapunov-Schmidt reduction

KW - Symmetry

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