Continuation of Probability Density Functions using a Generalized Lyapunov Approach

Techniques from numerical bifurcation theory are very useful to studytransitions between steady fluid flow patterns and the instabilities involved.Here, we provide computational methodology to use parameter continuationin determining probability density functions of systems of stochasticpartial differential equations near fixed points, under a small noise approximation.Key innovation is the efficient solution of a generalized Lyapunovequation using an iterative method involving low-rank approximations. Weapply and illustrate the capabilities of the method using a problem inphysical oceanography, i.e. the occurrence of multiple steady states of theAtlantic Ocean circulation