PORTUGALIAE MATHEMATICA Vol. 63, No. 2, pp. 127155 (2006) 

Hopf bifurcation with ${\bf S}_3$symmetryAna Paula S. Dias and Rui C. PaivaDep. de Matemática Pura, Centro de Matemática, Universidade do Porto,Rua do Campo Alegre, 687, 4169007 Porto  PORTUGAL Email: apdias@fc.up.pt Dep. de Matemática Aplicada, Universidade do Porto, Rua do Campo Alegre, 687, 4169007 Porto  PORTUGAL Email: rui.paiva@fc.up.pt Abstract: The aim of this paper is to study Hopf bifurcation with ${\bf S}_3$symmetry assuming Birkhoff normal form. We consider the standard action of ${\bf S}_3$ on $\R^2$ obtained from the action of ${\bf S}_3$ on $\R^3$ by permutation of coordinates. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on $\R^2\oplus\R^2$. Golubitsky, Stewart and Schaeffer ({\em Singularities and Groups in Bifurcation Theory: Vol.2.} Applied Mathematical Sciences {\bf 69}, SpringerVerlag, New York 1988) and Wood (Hopf bifurcations in three coupled oscillators with internal ${\bf Z}_2$ symmetries, {\em Dynamics and Stability of Systems} \textbf{13}, 5593, 1998) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with ${\bf S}_3$symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We describe the most general possible form of a ${\bf S}_3\times{\bf S}^1$equivariant mapping (assuming Birkhoff normal form) for the standard ${\bf S}_3$simple action on $\R^2\oplus\R^2$. Moreover, we prove that generically these are the \textsl{only} branches of periodic solutions that bifurcate from the trivial solution. Keywords: bifurcation; periodic solution; spatiotemporal symmetry. Classification (MSC2000): 37G40, 34C23, 34C25. Full text of the article:
Electronic version published on: 7 Mar 2008.
© 2006 Sociedade Portuguesa de Matemática
