Kuramoto-Sivashinsky dynamics on the center-unstable manifold

TLDR: In this paper, the authors studied the dynamical behavior of solutions of the Kuramoto-Sivashinsky partial differential equation with periodic boundary conditions on a spatial interval, where the length h is the bifurcation parameter.

Abstract: This paper studies the dynamical behavior of solutions of the Kuramoto–Sivashinsky partial differential equation with periodic boundary conditions on a spatial interval $[ 0,h ]$. The length h is the bifurcation parameter and reduction is made to a two-(complex-)dimensional system on a local center-unstable manifold near the second bifurcation point $h_2 $ from the trivial solution. The resulting $O( 2 )$-equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point. In particular, bifurcation sequences to stable traveling waves, unstable modulated traveling waves, and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within $1\%$ in the parameter range $h_2 \pm 20\% $. A clear understanding of the global dynamical behavior in this region is thus obtained.