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Numerical continuation methods for deterministic dynamical systems have beenone of the most successful tools in applied dynamical systems theory.Continuation techniques have been employed in all branches of the naturalsciences as well as in engineering to analyze ordinary, partial and delaydifferential equations. Here we show that the deterministic continuationalgorithm for equilibrium points can be extended to track information aboutmetastable equilibrium points of stochastic differential equations (SDEs). Westress that we do not develop a new technical tool but that we combine resultsand methods from probability theory, dynamical systems, numerical analysis,optimization and control theory into an algorithm that augments classicalequilibrium continuation methods. In particular, we use ellipsoids definingregions of high concentration of sample paths. It is shown that theseellipsoids and the distances between them can be efficiently calculated usingiterative methods that take advantage of the numerical continuation framework.We apply our method to a bistable neural competition model and a classicalpredator-prey system. Furthermore, we show how global assumptions on the flowcan be incorporated - if they are available - by relating numericalcontinuation, Kramers' formula and Rayleigh iteration.

Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids