Hyperbolicity and the effective dimension of spatially-extended dissipative systems

Abstract

We show, using covariant Lyapunov vectors, that the chaotic solutions of spatially extended dissipative systems evolve within a manifold spanned by a finite number of physical modes hyperbolically isolated from a set of residual degrees of freedom, themselves individually isolated from each other. In the context of dissipative partial differential equations, our results imply that a faithful numerical integration needs to incorporate at least all physical modes and that increasing the resolution merely increases the number of isolated modes.