Searching chaotic saddles in high dimensions

TL;DR

This paper introduces two novel numerical methods to efficiently approximate non-attracting chaotic sets in high-dimensional systems, outperforming previous methods especially when multiple positive Lyapunov exponents are present.

Contribution

The paper presents anisotropic search domain methods that improve the efficiency of locating chaotic saddles in high-dimensional systems, independent of escape time.

Findings

Both methods outperform the Stagger-and-Step method.

Anisotropic method achieves efficiency independent of escape time.

Simulations in 24-dimensional coupled Hénon maps demonstrate effectiveness.

Abstract

We propose new methods to numerically approximate non-attracting sets governing transiently-chaotic systems. Trajectories starting in a vicinity $\Omega$ of these sets escape $\Omega$ in a finite time $\tau$ and the problem is to find initial conditions ${\bf x} \in \Omega$ with increasingly large $\tau= \tau({\bf x})$. We search points ${\bf x}'$ with $\tau({\bf x}')>\tau({\bf x})$ in a {\it search domain} in $\Omega$. Our first method considers a search domain with size that decreases exponentially in $\tau$, with an exponent proportional to the largest Lyapunov exponent $\lambda_1$. Our second method considers anisotropic search domains in the {\it tangent} unstable manifold, where each direction scale as the inverse of the corresponding {\it expanding} singular value of the Jacobian matrix of the iterated map. We show that both methods outperform the state-of-the-art {\it Stagger-and-Step} method (Sweet, Nusse, and York, Phys. Rev. Lett. {\bf 86}, 2261, 2001) but that only the anisotropic method achieves an efficiency independent of $\tau$ for the case of high-dimensional systems with multiple positive Lyapunov exponents. We perform simulations in a chain of coupled H\'enon maps in up to 24 dimensions ($12$ positive Lyapunov exponents). This suggests the possibility of characterizing also non-attracting sets in spatio-temporal systems.