Fast numerical test of hyperbolic chaos

TL;DR

The paper introduces a fast, memory-efficient numerical method for testing hyperbolicity in chaotic systems, avoiding explicit computation of covariant Lyapunov vectors and suitable for high-dimensional dynamics.

Contribution

It develops a novel numerical approach that simplifies hyperbolicity testing by using a characteristic value distribution, reducing computational complexity.

Findings

The method accurately detects tangencies indicating non-hyperbolicity.

It significantly reduces computational time compared to traditional methods.

Applicable to high-dimensional chaotic systems with less memory usage.

Abstract

The effective numerical method is developed performing the test of the hyperbolicity of chaotic dynamics. The method employs ideas of algorithms for covariant Lyapunov vectors but avoids their explicit computation. The outcome is a distribution of a characteristic value which is bounded within the unit interval and whose zero indicate the presence of tangency between expanding and contracting subspaces. To perform the test one needs to solve several copies of equations for infinitesimal perturbations whose amount is equal to the sum of numbers of positive and zero Lyapunov exponents. Since for high-dimensional system this amount is normally much less then the full phase space dimension, this method provide the fast and memory saving way for numerical hyperbolicity test of such systems.