Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids

TL;DR

This paper extends deterministic continuation methods to stochastic differential equations, enabling tracking of metastable equilibria using ellipsoids and Lyapunov equations, with applications to biological models.

Contribution

It combines probability, dynamical systems, and numerical analysis to develop a novel continuation algorithm for metastable states in SDEs, integrating ellipsoids and Lyapunov equations.

Findings

Successfully applied to neural and predator-prey models.

Efficient computation of ellipsoids and distances using iterative methods.

Incorporates global flow assumptions via Kramers' formula and Rayleigh iteration.

Abstract

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.