Hopf bifurcation with broken circular symmetry

A symmetry-breaking Hopi bifurcation in an O(2)-symmetric system has eigenvalues of multiplicity two. When the circular symmetry is broken these eigenvalues split into two pairs. The consequences of this splitting in the nonlinear regime are analysed in detail. It is found that the perturbation selects the phase of the standing wave (sw) solutions and that two sw branches, differing in phase by n, bifurcate from the trivial solution in succession. Pure travelling waves (TW) are no longer possible. Instead two new solution branches denoted by TW' and MW' bifurcate from the sw branches in acondary steady-state and Hopf bifurcations, respectively. In contrast to the TW', the MW' only exist at small amplitudes, terminating on the rw' branch in either global or tertiary Hapf bifurcations. These solutions show remarkable resemblance to the stales observed in recent experiments on binary fluid convection in large but finite containers. PACS numbers: 0340K. 4425 i