Estimating dimension of inertial manifold from unstable periodic orbits

TL;DR

This paper demonstrates that the inertial manifold dimension of a chaotic dissipative system can be numerically estimated using only unstable periodic orbits, aligning with previous hyperbolicity-based measurements.

Contribution

It introduces a method to construct the inertial manifold dimension solely from unstable periodic orbits, validated on the Kuramoto-Sivashinsky system.

Findings

Inertial manifold dimension matches previous hyperbolicity-based estimates.

Numerical evidence supports the construction of inertial manifolds from periodic orbits.

The approach applies to chaotic dissipative systems like Kuramoto-Sivashinsky.

Abstract

We provide numerical evidence that a finite-dimensional inertial manifold on which the dynamics of a chaotic dissipative dynamical system lives can be constructed solely from the knowledge of a set of unstable periodic orbits. In particular, we determine the dimension of the inertial manifold for Kuramoto-Sivashinsky system, and find it to be equal to the `physical dimension' computed previously via the hyperbolicity properties of covariant Lyapunov vectors.