Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry
This paper concerns a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a partial differential equation (PDE) with O(2) symmetry to one with SO(2) symmetry. The authors concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles under perturbations breaking the symmetries is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. As \textit{P. Chossat} [Nonlinearity 6, No. 5, 723-731 (1993; Zbl 0784.58052)] has shown if the reflectional symmetry is broken then such a homoclinic orbit generically bifurcates to a quasiperiodic flow on a 2-torus. Comparisons of these results with numerical simulations of a Galerkin approximation of the equations are presented.
scientific article; zbMATH DE number 908446
scientific article