Hopf bifurcation with ${\bf S}_3$-symmetry

Ana Paula S. Dias and Rui C. Paiva

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}, Springer-Verlag, 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}, 55--93, 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; spatio-temporal symmetry.

Classification (MSC2000): 37G40, 34C23, 34C25.

Full text of the article:

• Compressed PostScript file (228 kilobytes)
• PDF file (276 kilobytes)