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

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.

💡 Research Summary

The paper introduces a novel computational framework for locating trajectories in dynamical systems that exhibit atypical chaotic behavior—either unusually regular or unusually chaotic compared with the typical ensemble. The core idea is to construct a “Lyapunov weighted path ensemble” (LWPE) in which each finite‑time trajectory is assigned a statistical weight that depends exponentially on its finite‑time Lyapunov exponent (FTLE). Specifically, the probability density of a trajectory (x(t)) is taken as (P