Branching of rotating waves in a one-parameter problem of steady-state bifurcation with spherical symmetry

Abstract

The author shows that in steady-state bifurcation problems which are invariant by an irreducible representation of O(3) of dimension 11, branches of limit cycles with long period can bifurcate from the trivial solution. This bifurcation, which does not occur in the lower dimensional cases (except possibly in dimension 9), results from a 'natural' cubic degeneracy of the stability of certain branches of equilibria. This degeneracy leads to the existence of critical orbits of solutions 'close' to these orbits of equilibria, and an argument of bifurcation from a group orbit shows the presence of a non-trivial flow along these critical orbits.