A NOTE ON HOPF BIFURCATION WITH DIHEDRAL GROUP SYMMETRY

Abstract

We consider the standard action of the dihedral group $\bf{D}_n$ of order $2n$ on $\bf{C}$. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on $\bf{C} \oplus \bf{C}$. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series, M. Golubitsky and J. Guckenheimer, eds., Contemporary Mathematics 46, Am. Math. Soc., Providence, R.I. 1986, 131–173) and van Gils and Valkering (Hopf bifurcation and symmetry: standing and travelling waves in a circular chain. Japan J. Appl. Math.3, 207–222, 1986) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with $\bf{D}_n$-symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We prove that generically, when $n\neq 4$ and assuming Birkhoff normal form, these are the only branches of periodic solutions that bifurcate from the trivial solution.