Hyperbolicity and the effective dimension of spatially-extended dissipative systems

TL;DR

This paper demonstrates that chaotic solutions in spatially extended dissipative systems are confined to a finite set of hyperbolically isolated physical modes, influencing numerical integration and system complexity understanding.

Contribution

It reveals the hyperbolic isolation of physical modes in dissipative systems using covariant Lyapunov vectors, providing insights into their effective dimension and numerical modeling.

Findings

Chaotic solutions evolve within a finite set of physical modes.

Physical modes are hyperbolically isolated from residual degrees of freedom.

Increasing resolution adds more isolated modes, not changing core dynamics.

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.