# Taming chaos to sample rare events: the effect of weak chaos

## TL;DR

This paper explores strategies to efficiently sample rare events in weakly chaotic dynamical systems by controlling non-hyperbolicities, extending previous methods used for strongly chaotic systems.

## Contribution

It introduces new approaches to overcome limitations posed by non-hyperbolicities in weakly chaotic systems within a Monte Carlo framework.

## Key findings

- Identified how non-hyperbolicities hinder sampling efficiency.
- Proposed strategies to improve sampling in weakly chaotic systems.
- Validated methods on low-dimensional chaotic maps.

## Abstract

Rare events in non-linear dynamical systems are difficult to sample because of the sensitivity to perturbations of initial conditions and of complex landscapes in phase space. Here we discuss strategies to control these difficulties and succeed in obtainining an efficient sampling within a Metropolis-Hastings Monte Carlo framework. After reviewing previous successes in the case of strongly chaotic systems, we discuss the case of weakly chaotic systems. We show how different types of non-hyperbolicities limit the efficiency of previously designed sampling methods and we discuss strategies how to account for them. We focus on paradigmatic low-dimensional chaotic systems such as the logistic map, the Pomeau-Maneville map, and area-preserving maps with mixed phase space.