Hopf bifurcation on a sphere

Using the general theory of Hopf bifurcation with symmetry we study here the example where the group of symmetries is O(3), the rotations and reflections of a sphere. We make some amendments to previously published lists of C-axial isotropy subgroups of O(3) × S1 and list the isotropy subgroups with four-dimensional fixed-point subspaces. We then study the particular example where O(3) × S1 acts on the space V3 ⊕ V3 where V3 is the space of spherical harmonics of degree three. We find that in this case there are six C-axial isotropy subgroups of O(3) × S1. The equivariant Hopf theorem guarantees the existence of periodic solutions with each of these symmetries in O(3) × S1 equivariant differential equations. Three of the solutions are found to be standing waves and the other three are travelling waves. We compute conditions for each of these solution branches to be stable and by restricting the O(3) × S1 equivariant differential equations to four-dimensional invariant subspaces we are able to find additional periodic and quasiperiodic solutions