Vol. 14, No. 2, 2019

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

Maria Cameron and Shuo Yang

Vol. 14 (2019), No. 2, 207–246
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.

Keywords
quasipotential, ordered line integral method, Lorenz'63, maximum likelihood transition path
Mathematical Subject Classification 2010
Primary: 65N99, 65P99, 58J65
Milestones
Received: 22 November 2018
Revised: 15 June 2019
Accepted: 16 July 2019
Published: 4 October 2019
Authors
Maria Cameron
Department of Mathematics
University of Maryland, College Park
College Park, MD
United States
Shuo Yang
Department of Mathematics
University of Maryland, College Park
College Park, MD
United States