Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry

We perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a PDE with O(2) symmetry to one with SO(2) symmetry. Taking particular care of the consequences of the continuous group action, we concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. In particular, we note that although Chossat (1993) has proved that generic perturbations cause the appearance of quasiperiodic orbits, for the simplest possible flux terms this is not the case. We compare these results with numerical simulations of a Galerkin approximation of the equations.