Multicanonical Sampling of Rare Trajectories in Chaotic Dynamical Systems

TL;DR

This paper introduces a multicanonical Monte Carlo method to efficiently identify and estimate the probabilities of rare trajectories with specific chaotic properties in high-dimensional chaotic systems.

Contribution

It develops a novel quantitative approach for locating rare trajectories in chaotic systems, capable of estimating extremely small probabilities.

Findings

Successfully tested on four-dimensional coupled standard maps

Estimated probabilities as low as 10^{-14}

Demonstrated effectiveness in identifying rare chaotic trajectories

Abstract

In chaotic dynamical systems, a number of rare trajectories with low level of chaoticity are embedded in chaotic sea, while extraordinary unstable trajectories can exist even in weakly chaotic regions. In this study, a quantitative method for searching these rare trajectories is developed; the method is based on multicanonical Monte Carlo and can estimate the probability of initial conditions that lead to trajectory fragments of a given level of chaoticity. The proposed method is successfully tested with four-dimensional coupled standard maps, where probabilities as small as $10^{-14}$ are estimated.