Importance sampling with imperfect cloning for the computation of generalized Lyapunov exponents

TL;DR

This paper introduces an alternative importance sampling method using imperfect cloning to compute generalized Lyapunov exponents in deterministic systems, maintaining accuracy while preserving deterministic dynamics.

Contribution

It proposes and compares an imperfect cloning importance sampling method with the standard noisy approach for calculating Lyapunov exponents, demonstrating comparable performance.

Findings

Imperfect cloning performs as well as standard noisy methods.

The method preserves the deterministic nature of the system.

Validated on various dynamical systems including maps and coupled systems.

Abstract

We revisit the numerical calculation of generalized Lyapunov exponents, $L$($q$), in deterministic dynamical systems. The standard method consists of adding noise to the dynamics in order to use importance sampling algorithms. Then $L$($q$) is obtained by taking the limit noise-amplitude $\to$0 after the calculation. We focus on a particular method that involves periodic cloning and pruning of a set of trajectories. However, instead of considering a noisy dynamics, we implement an imperfect (noisy) cloning. This alternative method is compared with the standard one and, when possible, with analytical results. As a workbench, we use the asymmetric tent map, the standard map, and a system of coupled symplectic maps. The general conclusion of this study is that the imperfect-cloning method performs as well as the standard one, with the advantage of preserving the deterministic dynamics.