Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study

Document Type

Article

Publication Date

1-1-1991

Abstract

The results of extensive computations are presented to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular we follow the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos. As many as 13 period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a self-similar behavior that is demonstrated and used to compute a universal scaling factor, which arises also from the theory of nonlinear maps and can enable continuation of the solution into a chaotic regime. Aperiodic solutions alternate with periodic ones after chaos sets in, and we show the existence of a period six solution separated by chaotic regions.

Identifier

0026325935 (Scopus)

Publication Title

Proceedings of the National Academy of Sciences of the United States of America

ISSN

00278424

PubMed ID

11607246

First Page

11129

Last Page

11132

Issue

24

Volume

88

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