S.A. van Gils and M. Golubitsky
A torus bifurcation theorem in the presence of symmetry
Dyn. Diff. Eqn. 2 (2) (1990) 133-163.
A general theory for the study of degenerate Hopf bifurcation in the presence of symmetry has been carried out only in situations where the normal form equations decouple into phase/amplitude equations. In this paper we prove a theorem showing that in general we expect such degeneracies to lead to secondary torus bifurcations. We then apply this theorem to the case of degenerate Hopf bifurcation with triangular (D_3) symmetry, proving that in codimension two there exist regions of parameter space where two branches of asymptoticaly stable 2-tori coexist but where no stable periodic solutions are present. Although this study does not lead to a theory for degenerate Hopf bifurcations in the presence of symmetry, it does present examples that would have to be accounted for by any such general theory.