Royal Society Publishing

An Algebraic Criterion for Symmetric Hopf Bifurcation

Martin Golubitsky, Ian Stewart


The equivariant Hopf bifurcation theorem states that bifurcating branches of periodic solutions with certain symmetries exist when the fixed-point subspace of that subgroup of symmetries is two dimensional. We show that there is a group-theoretic restriction on the subgroup of symmetries in order for that subgroup to have a two-dimensional fixed-point subspace in any representation. We illustrate this technique for all irreducible representations of SO(3) on the space V$_{l}$ of spherical harmonics for l even.