Bifurcation of a class of stochastic delay differential equations

Abstract

In this paper, we consider the bifurcation of a class of two-dimensional stochastic delay differential equations. Firstly, we translate the original system into an It\^{o} limiting diffusion system by applying stochastic Taylor expansion, small time delay expansion, polar coordinate transformation, and stochastic averaging procedure. Then we discuss the dynamical bifurcation by analyzing the qualitative changes of invariant measures, and investigate the phenomenological bifurcation by utilizing Fokker-Planck equation. The obtained conclusions are completely new, which generalize and improve some existing results.