Symmetry-breaking perturbations on the global attractor of the Kuramoto--Sivashinsky equation

Abstract

We study symmetry-breaking of solutions on the global attractor of the Kuramoto--Sivashinsky equation. In our theory we prove that trajectories which result from small perturbations of a point on the global attractor stay close to the global attractor. In our numerics we exhibit a choice of parameters for the Kuramoto--Sivashinsky equation such that every $2\pi$-periodic initial condition (which isn't zero or periodic on some smaller domain) converges to a traveling wave solution and such that every $4\pi$-periodic initial condition converges to a distinctly different fixed point. Our main result is to compute a non-recurrent trajectory on the attractor, connecting the traveling wave to the fixed point, given as the limit of smaller and smaller symmetry-breaking perturbations.