An improved approach for estimating the dimension of inertial manifolds in chaotic distributed dynamical systems via analysis of angles between tangent subspaces

TL;DR

This paper introduces a more efficient method for estimating the dimension of inertial manifolds in chaotic systems by analyzing angles between tangent subspaces, reducing computational resources compared to previous approaches.

Contribution

The authors develop a faster, less resource-intensive technique for determining inertial manifold dimension without explicit covariant Lyapunov vector computation.

Findings

Accurately estimates inertial manifold dimension in Ginzburg-Landau system

Identifies absence of low-dimensional inertial manifold in Lorenz oscillator chain

Reduces computational effort compared to previous methods

Abstract

While a previously proposed method for estimating inertial manifold dimension, based on explicitly computing angles between pairs of covariant Lyapunov vectors (CLVs), employs efficient algorithms, it remains computationally demanding due to its substantial resource requirements. In this work, we introduce an improved method to determine this dimension by analyzing the angles between tangent subspaces spanned by the CLVs. This approach builds upon a fast numerical technique for assessing chaotic dynamics hyperbolicity. Crucially, the proposed method requires significantly less computational effort and minimizes memory usage by eliminating the need for explicit CLV computation. We test our method on two canonical systems: the complex Ginzburg-Landau equation and a diffusively coupled chain of Lorenz oscillators. For the former, the results confirm the accuracy of the new approach by matching prior dimension estimates. For the latter, the analysis demonstrates the absence of a low-dimensional inertial manifold, highlighting a complex regime that merits further investigation. The presented method offers a practical and efficient tool for characterizing high-dimensional attractors in extended dynamical systems.