# Importance Sampling of Rare Events in Chaotic Systems

## TL;DR

This paper introduces a novel Monte Carlo sampling method for efficiently sampling rare trajectories in chaotic dynamical systems, enabling the computation of distributions of rare events like Lyapunov exponents and escape times.

## Contribution

It develops a Metropolis-Hastings Monte Carlo framework tailored for chaotic systems to sample exponentially rare events with polynomial computational effort.

## Key findings

- Successfully computes distributions of finite-time Lyapunov exponents.
- Efficiently samples escape times in transient-chaos scenarios.
- Provides open-source software for implementation.

## Abstract

Finding and sampling rare trajectories in dynamical systems is a difficult computational task underlying numerous problems and applications. In this paper we show how to construct Metropolis- Hastings Monte Carlo methods that can efficiently sample rare trajectories in the (extremely rough) phase space of chaotic systems. As examples of our general framework we compute the distribution of finite-time Lyapunov exponents (in different chaotic maps) and the distribution of escape times (in transient-chaos problems). Our methods sample exponentially rare states in polynomial number of samples (in both low- and high-dimensional systems). An open-source software that implements our algorithms and reproduces our results can be found in https://github.com/jorgecarleitao/chaospp