The route to chaos for the KuramotoSivashinsky equation
Abstract
The results of extensive numerical experiments of the spatially periodic initial value problem for the KuramotoSivashinsky equation. This paper is concerned with the asymptotic nonlinear dynamics at the dissipation parameter decreases and spatiotemporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic attractor, to another steady fully modal attractor, to another periodic attractor, to a steady tetramodal attractor, to another periodic attractor having a full sequence of perioddoublings (in parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the perioddoubling cascades which precede chaos for this infinitedimensional dynamical system. These results permit a computation of the length of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period doubling sequences found here for the KuramotoSivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4,669201609... . Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.
 Publication:

Final Report Institute for Computer Applications in Science and Engineering
 Pub Date:
 October 1990
 Bibcode:
 1990icas.reptS....P
 Keywords:

 Branching (Mathematics);
 Chaos;
 Dynamical Systems;
 Nonlinear Systems;
 Numerical Analysis;
 Period Doubling;
 Boundary Value Problems;
 Strange Attractors;
 Turbulent Flow;
 Viscosity;
 Windows;
 Fluid Mechanics and Heat Transfer