Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation

A numerical and analytical study of the Kuramoto–Sivashinsky partial differential equation (PDE) in one spatial dimension with periodic boundary conditions is presented. The structure, stability, and bifurcation characteristics of steady state and time-dependent solutions of the PDE for values of the parameter $\alpha $ less than 40 are examined. The numerically observed primary and secondary bifurcations of steady states, as well as bifurcations to constant speed traveling waves (limit cycles), are analytically verified. Persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of $O( 2 )$. Their effect on the system dynamics is discussed, and several tertiary bifurcations, observed numerically, are presented.

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