Exploration of Order in Chaos with Replica Exchange Monte Carlo

TL;DR

This paper introduces a Replica Exchange Monte Carlo method to explore unstable structures in nonlinear dynamical systems, enabling efficient sampling of rare initial conditions and parameters for chaos analysis.

Contribution

It presents a novel application of REM for exploring unstable structures and bifurcations in nonlinear dynamical systems, improving search efficiency.

Findings

Successful sampling of unstable periodic orbits

Effective search for stable manifolds of unstable fixed points

Construction of global bifurcation diagrams

Abstract

A method for exploring unstable structures generated by nonlinear dynamical systems is introduced. It is based on the sampling of initial conditions and parameters by Replica Exchange Monte Carlo (REM), and efficient both for the search of rare initial conditions and for the combined search of rare initial conditions and parameters. Examples discussed here include the sampling of unstable periodic orbits in chaos and search for the stable manifold of unstable fixed points, as well as construction of the global bifurcation diagram of a map.