Paper Info
Title
Deterministic Continuation of Stochastic Metastable Equilibria via Lyapunov Equations and Ellipsoids.
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. 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.
Year
DOI
Venue
2012
10.1137/110839874
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Keywords
Field
DocType
numerical continuation,bifurcation analysis,metastability,stochastic dynamics,covariance,Lyapunov equation,ellipsoids,iterative methods,neural competition,predator-prey system,Rayleigh iteration,Kramers' law
Lyapunov function,Lyapunov equation,Mathematical optimization,Mathematical analysis,Equilibrium point,Stochastic differential equation,Dynamical systems theory,Numerical analysis,Delay differential equation,Numerical continuation,Mathematics
Journal
Volume
Issue
ISSN
34
3
1064-8275
Citations 
PageRank 
References 
1
0.37
15
Authors
1
Name
Order
Citations
PageRank
Christian Kuehn19012.21