S. van Gils and M. Golubitsky
Journal of Dynamics and Differential Equations
Vol. 2, No. 2 (1990) 133-162.
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.