Computational study of chaotic and ordered solutions of the Kuramoto-Sivashinsky equation

We report the results of extensive numerical experiments on the Kuramoto-Sivashinsky equation in the strongly chaotic regime as the viscosity parameter is decreased and increasingly more linearly unstable modes enter the dynamics. General initial conditions are used and evolving states do not assume odd-parity. A large number of numerical experiments are employed in order to obtain quantitative characteristics of the dynamics. We report on di erent routes to chaos and provide numerical evidence and construction of strange attractors with self-similar characteristics. As the \viscosity" parameter decreases the dynamics becomes increasingly more complicated and chaotic. In particular it is found that regular behavior in the form of steady state or steady state traveling waves is supported amidst the time-dependent and irregular motions. We show that multimodal steady states emerge and are supported on decreasing windows in parameter space. In addition we invoke a self-similarity property of the equation, to show that these pro les are obtainable from global xed point attractors of the Kuramoto-Sivashinsky equation at much larger values of the viscosity. The work of Y.S. Smyrlis was supported by NATO Grant CRG 920097. The work of D.T. Papageorgiou was supported by the National Science Foundation Grant NSF-DMS-9003227 and by NATO Grant CRG 920097. Additional support was also provided by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while he was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23681-0001.