Geometry of inertial manifolds probed via a Lyapunov projection method

TL;DR

This paper introduces a Lyapunov projection method to determine the dimension and geometry of inertial manifolds in dissipative dynamical systems, enabling experimental analysis of their structure.

Contribution

It presents a new, straightforward approach using Lyapunov vectors to identify inertial manifold dimensions and geometry in extended systems.

Findings

Sharp transition in projection error indicates inertial manifold dimension

Method works with standard orthogonal Lyapunov vectors

Applicable to experimental data for manifold characterization

Abstract

A method for determining the dimension and state space geometry of inertial manifolds of dissipative extended dynamical systems is presented. It works by projecting vector differences between reference states and recurrent states onto local linear subspaces spanned by the Lyapunov vectors. A sharp characteristic transition of the projection error occurs as soon as the number of basis vectors is increased beyond the inertial manifold dimension. Since the method can be applied using standard orthogonal Lyapunov vectors, it provides a simple way to determine also experimentally inertial manifolds and their geometric characteristics.