The Kuramoto-Sivashinsky Equation: Spatio-Temporal Chaos and Intermittencies for a Dynamical System

Abstract

During the last decade we have seen a number of major developments which show that the long-time behavior of solutions of a very large class of partial differential equations (PDEs) possess a striking resemblance to the behavior of solution of finite dimensional dynamical systems, or ordinary differential equations (ODEs). The first of these advances was the discovery (by a number of researchers) that a dissipative PDE has a compact, maximal attractor X with finite Hausdorff and fractal dimensions. More recently [6, 12–15] it was shown that some of these PDEs possess a finite dimensional inertial manifold, i.e., an invariant manifold that contains the attractor X. For the later equation, the connection with ODEs is no longer a mere resemblance, instead it has become a striking reality! The reason for this is that where one restricts the PDE to the inertial manifold one obtains an ODE, which we call an inertial form for the given PDE. Since an inertial manifold contains the universal attractor, this means that the long-time behavior of solutions of a PDE with an inertial manifold is completely determined by the inertial form.KeywordsUnstable ManifoldHomoclinic OrbitHigh Frequency OscillationChaotic RegimeChaotic Time SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.