×

Hopf bifurcation with broken circular symmetry. (English) Zbl 0731.58051

A symmetry-breaking Hopf 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 analyzed in detail. It is found that the perturbation selects the phase of the standing wave (SW) solutions and two SW branches, differing in phase by \(\pi\), bifurcate from the trivial solution in succession. Pure traveling waves are no longer possible solutions. Instead two new solution branches denoted by \(TW'\) an \(MW'\) bifurcate from the SW branches in secondary steady-state and Hopf bifurcations, respectively. In contrast to \(TW'\), 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 finite containers.

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37C80 Symmetries, equivariant dynamical systems (MSC2010)
76B25 Solitary waves for incompressible inviscid fluids
PDFBibTeX XMLCite
Full Text: DOI Link