Non-hyperbolicity at large scales of a high-dimensional chaotic system

TL;DR

This paper provides numerical evidence that large-scale dynamics in certain high-dimensional chaotic systems are non-hyperbolic, challenging the widely held chaotic hypothesis and indicating the need for a nuanced understanding of such systems.

Contribution

The study introduces a numerical approach to detect non-hyperbolic structures in high-dimensional systems, demonstrating the failure of the chaotic hypothesis in a mean-field coupled system.

Findings

Existence of non-hyperbolic large-scale structures

Failure of uniform hyperbolicity in the studied system

Robust non-hyperbolic behavior under perturbations

Abstract

The dynamics of many important high-dimensional dynamical systems are both chaotic and complex, meaning that strong reducing hypotheses are required to understand the dynamics. The highly influential chaotic hypothesis of Gallavotti and Cohen states that the large-scale dynamics of high-dimensional systems are effectively uniformly hyperbolic, which implies many felicitous statistical properties. We obtain direct and reliable numerical evidence, contrary to the chaotic hypothesis, of the existence of non-hyperbolic large-scale dynamical structures in a mean-field coupled system. To do this we reduce the system to its thermodynamic limit, which we approximate numerically with a Chebyshev basis transfer operator discretisation. This enables us to obtain a high precision estimate of a homoclinic tangency, implying a failure of uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under perturbation. As a result, the chaotic hypothesis should not be {\it a priori} assumed to hold in all systems, and a better understanding of the domain of its validity is required.