Hopf bifurcation with broken circular symmetry

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Abstract
A symmetry-breaking Hopf bifurcation in an O(2)-symmetric system has eigenvalue 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 pi , 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 secondary steady-state and Hopf bifurcations, respectively. In contrast to the TW', the MW' only exist at small amplitudes, terminating on the TW' branch in either global or tertiary Hopf bifurcations. These solutions show remarkable resemblance to the states observed in recent experiments on binary fluid convection in large but finite containers.