Identifying rare chaotic and regular trajectories in dynamical systems with Lyapunov weighted path sampling

TL;DR

This paper introduces a numerical method using Lyapunov weighted path sampling to identify rare trajectories in dynamical systems that exhibit atypical chaotic or regular behavior, applicable across systems with varying degrees of freedom.

Contribution

The paper presents a novel algorithm that efficiently samples rare trajectories with unusual chaoticity in dynamical systems, extending transition path sampling techniques.

Findings

Successfully identified rare pathways with atypical chaoticity in multiple systems.

Demonstrated the method's effectiveness on systems with up to hundreds of degrees of freedom.

Found rare reactive pathways in a model of isomerization under chaotic dynamics.

Abstract

Depending on initial conditions, individual finite time trajectories of dynamical systems can have very different chaotic properties. Here we present a numerical method to identify trajectories with atypical chaoticity, pathways that are either more regular or more chaotic than average. The method is based on the definition of an ensemble of trajectories weighted according to their chaoticity, the Lyapunov weighted path ensemble. This ensemble of trajectories is sampled using algorithms borrowed from transition path sampling, a method originally developed to study rare transitions between long-lived states. We demonstrate our approach by applying it to several systems with numbers of degrees of freedom ranging from one to several hundred and in all cases the algorithm found rare pathways with atypical chaoticity. For a double-well dimer embedded in a solvent, which can be viewed as simple model for an isomerizing molecule, rare reactive pathways were found for parameters strongly favoring chaotic dynamics.