Computer assisted study of strange attractors of the Kuramoto-Sivashinsky equation

Abstract

Extensive numerical computations have been carried out using the Kuramoto-Sivashinsky equation which is one of the simplest PDE's which exhibit chaotic behavior. A period-doubling route to chaos has been located which conforms with the Feigenbaum scenario. The chaotic dynamics just beyond the accumulation point are calculated using highly accurate methods from which a picture of the attractor can be constructed. In particular we provide strong numerical evidence of the self-similarity of the attractor. In order to achieve this, magnifications of the attractor of order 106 are needed, showing the high accuracy requirements. As far as we know, this has not been done previously for an infinite-dimensional dynamical system.