Stability of Cellular States of the Kuramoto–Sivashinsky Equation

This paper is concerned with the instability property of a particular type of solution to the Kuramoto–Sivashinsky equation, namely, the symmetric, time-independent cellular states. Attention is focused on the dimension of their unstable manifolds. We show that as a control parameter varies, the dimension changes in an ordered way governed by certain properties of an associated ordinary differential equation. Further change in dimension is shown to be related to Hopf bifurcations from the cellular states. An asymptotic analysis explains the onset of these bifurcations and predicts approximate parameter values at which they occur.