Abstract
A symmetry-breaking Hopf bifurcation in an O(2)-equivariant system generally produces a branch of standing waves and two branches of oppositely propagating travelling waves. This generic bifurcation assumes three non-degeneracy conditions on the cubic terms of the Poincare-Birkhoff normal form. When these conditions fail more complicated behaviour accompanies the bifurcation; in particular one finds secondary bifurcations of quasiperiodic waves. For these degenerate bifurcations, the effects of perturbations which break the reflection symmetry are considered. The perturbed system retains a residual SO(2) symmetry. Qualitatively these perturbations have three effects: (1) they split the double multiplicity eigenvalues to that the travelling waves bifurcate separately, (2) they perturb the primary standing wave branches to secondary branches of modulated waves and (3) they produce new steady-state bifurcations along the modulated wave branches.