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The classification of reversible-equivariant steady-state bifurcations on self-dual spaces

Published online by Cambridge University Press:  01 September 2008

P. H. BAPTISTELLI  and
M. MANOEL
P. H. BAPTISTELLI*
Affiliation:
Departamento de Matemática, ICMC - USP, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil. e-mail: miriam@icmc.usp.br
M. MANOEL
Affiliation:
Departamento de Matemática, ICMC - USP, Caixa Postal 668, 13560-970, São Carlos-SP, Brazil. e-mail: miriam@icmc.usp.br
*
†Work supported by CNPq Grant 141672/2003-0. Email address: phbapt@icmc.usp.br
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Abstract

In this paper we apply singularity theory methods to the classification of reversible-equivariant steady-state bifurcations depending on one real parameter. We assume that the group of symmetries and reversing symmetries is a compact Lie group Γ, and the equivalence is defined in order to preserve these symmetries and reversing symmetries in the normal forms and their unfoldings. When the representation of Γ is self-dual, we show that the classification can be reduced to the standard equivariant context. In this case, we establish a one-to-one association between the classification of bifurcations in the reversible-equivariant context and the classification of purely equivariant bifurcations related to them. As an application of the results, we obtain the classification of self-dual representations of Z2 ⊕ Z2 and D4 on the plane.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 2 , September 2008 , pp. 379 - 401
Copyright
Copyright © Cambridge Philosophical Society 2008

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