TY - CHAP

T1 - Group-theoretic bifurcation theory

AU - Ikeda, Kiyohiro

AU - Murota, Kazuo

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Bifurcation

KW - Bifurcation equation

KW - Block-diagonalization

KW - Critical point

KW - Equivariant branching lemma

KW - Group equivariance

KW - Group representation

KW - Group-theoretic bifurcation theory

KW - Imperfection sensitivity matrix

KW - Liapunov-Schmidt reduction

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=85073146066&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073146066&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-21473-9_8

DO - 10.1007/978-3-030-21473-9_8

M3 - Chapter

AN - SCOPUS:85073146066

T3 - Applied Mathematical Sciences (Switzerland)

SP - 201

EP - 235

BT - Applied Mathematical Sciences (Switzerland)

PB - Springer

ER -