Searching for fixed points of dynamical systems by the Explorative Relaxation Redistribution Method

TL;DR

This paper introduces the Explorative Relaxation Redistribution Method (ERRM), a reformulation of RRM, to efficiently locate saddle and fixed points in dynamical systems, including stochastic simulators, with promising benchmark results.

Contribution

The paper presents ERRM, a novel reformulation of RRM, enabling effective fixed point detection in stochastic dynamical systems, addressing efficiency and implementation issues.

Findings

Successful application to Mueller-Brown potential

Effective in locating saddle points in stochastic systems

Improved computational efficiency over existing methods

Abstract

Being able to effectively locate saddle (and other fixed) points in dynamical systems holds tremendous implications in a number of applications in engineering and science, among which the study of rare events in molecular simulations stands as one of the most prominent field of interest. Although there is a vast literature on methods aiming at addressing this challenge, open issues still remain in several aspects such as computational efficiency and easy implementation. In this work, we suggest that the Relaxation Redistribution Method (RRM) - formerly introduced to solve the invariance equation in the context of stiff dynamical systems ruling detailed chemical kinetics - can be reformulated to locate saddle and other fixed points even for stochastic simulators driven by effective energy gradients. This new formulation of the RRM is referred to as the Explorative Relaxation Redistribution Method (ERRM). Benchmarks based on the popular Mueller-Brown and other potentials are used to test the ERRM.