Transitions Induced by Cross-correlated Gaussian White Noises: An Effective Potential Approach

We consider a first-order one-variable stochastic nonlinear system with a deterministic force and two cross-correlated Gaussian white noises, one of them is additive, and the other is multiplicative. We use an arbitrary interpretation of the corresponding stochastic differential equation that includes in particular the Ito interpretation and the Stratonovich one. We write an exact expression for the effective potential for the system in the form of a quadrature, which is expanded into Maclaurin series. It is shown from general considerations that transitions from unimodal to bimodal probability distribution of the system state induced by cross-correlated Gaussian white noises can be qualitatively described by the normal form of the pitchfork bifurcation perturbed by additive Gaussian white noise. Its explicit expression is obtained for a concrete example of the system with a linear restoring force and multiplicative noise, whose state-dependent amplitude is quadratic for small absolute values of the dynamic variable and constant for large ones. We plot graphs of both single-well and double-well effective potentials of the system using the exact expression and the power series approximation. They are in good agreement with each other in the vicinity of the critical point (origin). Relations between the bifurcation parameter and the noise parameters are investigated. It is determined that the bifurcation parameter changes linearly as the cross-correlation coefficient of the noises or the amplitude of the multiplicative noise are varied, and it changes nonlinearly as the amplitude of the additive noise is varied.