Analysis of stochastic bifurcations with phase portraits

TL;DR

This paper introduces a method to analyze stochastic bifurcations using phase portraits derived from the Fokker-Planck equation, linking fixed points of the convective field to extrema of the stationary distribution.

Contribution

It presents a novel approach to visualize and identify stochastic bifurcations by constructing phase portraits from the convective component of the Fokker-Planck dynamics.

Findings

Stochastic phase portraits reveal extrema of the stationary distribution.

Identification of a new class of stochastic bifurcations involving maxima moving to system edges.

Limit cycles in phase portraits indicate ridges in probability distributions.

Abstract

We propose a method to obtain phase portraits for stochastic systems. Starting from the Fokker-Planck equation, we separate the dynamics into a convective and a diffusive part. We show that stable and unstable fixed points of the convective field correspond to maxima and minima of the stationary probability distribution if the probability current vanishes at these points. Stochastic phase portraits, which are vector plots of the convective field, therefore indicate the extrema of the stationary distribution and can be used to identify stochastic bifurcations that change the number and stability of these extrema. We show that limit cycles in stochastic phase portraits can indicate ridges of the probability distribution, and we identify a novel type of stochastic bifurcations, where the probability maximum moves to the edge of the system through a gap between the two nullclines of the convective field.