Complex dynamics in the Hopf bifurcation with broken translation symmetry

The dynamics of the normal form for the O(2)-equivariant Hopf bifurcation perturbed by linear translation symmetry-breaking terms are investigated. A variety of stable small amplitude complex behavior is uncovered even for parameters for which the unperturbed system has no stable small amplitude solutions. In some cases this behavior can be attributed to the proximity to global bifurcations of Sil'nikov and Sil'nikov-Hopf type. In other cases cascades of gluing bifurcations involving various two-frequency states are present. Since the equations studied can be obtained by center manifold reduction of the coupled complex Ginzburg-Landau (CCGL) equations in a finite container this behavior must also be present in the CCGL equations. The theory may explain the origin of erratic reversals in traveling wave convection and in spiral vortex flow.