# 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.

## Key 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.