Approximate time-dependent solution of Fokker-Planck equation with non-linear drift force

Abstract

In this paper, the unstable state evolution problem of the non-linear dynamical system driven by Gaussian white and colored noise is investigated. Using the eigenvalue and eigenvector theory, the expression of the approximate time-dependent solution (ρ(x, t)) is derived. The effects of parameters on ρ(x, t), mean and normalized variance are also analyzed. Numerical simulations show that 1) ρ(x, t) is a monotonic function of t and x under the certain limits of t, which increases with τ increasing, but decreases with α increasing; it is very remarkable for large τ and large α; 2) the mean of the state variable x is positive, which increases with τ increasing, but decreases with α increasing; the normalized variance of the state variable x is a non-monotonic function of the α and τ. Therefore, a phase transition phenomenon is found in this system.