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
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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 -