Computing the quasipotential for highly dissipative and chaotic SDEs. An application to stochastic Lorenz'63

TL;DR

This paper introduces a new computational method for calculating the quasipotential in highly dissipative, chaotic stochastic systems, exemplified by the Lorenz'63 model, aiding understanding of rare noise-driven transitions.

Contribution

It develops a novel methodology combining 3D, 2D, and dimensional reduction techniques to compute quasipotentials in complex stochastic dynamical systems, with publicly available source code.

Findings

Successfully computed quasipotentials for Lorenz'63 with stochastic forcing.

Demonstrated the method's ability to handle high dissipation and chaos.

Provided accessible software tools for the scientific community.

Abstract

The study of noise-driven transitions occurring rarely on the time-scale of systems modeled by SDEs is of crucial importance for understanding such phenomena as genetic switches in living organisms and magnetization switches of the Earth. For a gradient SDE, the predictions for transition times and paths between its metastable states are done using the potential function. For a nongradient SDE, one needs to decompose its forcing into a gradient of the so-called quasipotential and a rotational component, which cannot be done analytically in general. We propose a methodology for computing the quasipotential for highly dissipative and chaotic systems built on the example of Lorenz'63 with an added stochastic term. It is based on the ordered line integral method, a Dijkstra-like quasipotential solver, and combines 3D computations in whole regions, a dimensional reduction technique, and 2D computations on radial meshes on manifolds or their unions. Our collection of source codes is available on M. Cameron's web page and on GitHub.