Symmetry-breaking Hopf bifurcation in anisotropic systems

Symmetry-breaking Hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered. This work is motivated by the experimental observation of a Hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals. Symmetry forces four traveling rolls to lose stability simultaneously. Four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory. Six branches of periodic solutions always bifurcate from the trivial state at the Hopf bifurcation. These correspond to traveling and standing wave patterns. In an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution. The Hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable, structurally stable heteroclinic cycle. The theory is applied to a model for the transition from normal to oblique traveling rolls.