M. Golubitsky and I. Stewart
An algebraic criterion for symmetric Hopf bifurcation
Proc. R. Soc. London. 440 (1993) 727-732.
In the presence of symmetry eigenvalues of high multiplicity may be expected to cross the imaginary axis as parameters are varied. 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. As stated this hypothesis is an assumption about the specific representation of the group of symmetries on the (real) eigenspace associated with these critical eigenvalues.
In this paper 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. This group-theoretic restriction is in many cases computable and substantially simplifies the calculations needed to find solutions supported on two-dimensional fixed-point subspaces. We illustrate this assertion by determining all of the periodic solutions supported on two-dimensional fixed-point subspaces for all of the (l even) irreducible representations of SO(3).