Analytic-Numerical Construction of Nonstationary Probability Distributions for one Class of Nonlinear Stochastic Systems

Abstract

We propose a method for constructing nonstationary model probability distributions for nonlinear dynamic systems related to the Verhulst stochastic equation. The proposed procedure is based on the numerical solution of relaxation differential equations for the mean and the variance. The set of the moment equations is closed and the probability density is constructed on the basis of rigorous analytical relations for the stationary probability characteristics. As a result, these distributions have correct stationary asymptotics. We show the possibility of numerical control of the accuracy of the proposed procedure. We consider the examples of relaxation of the probability characteristics of the amplitude of a self-oscillator and a parametric oscillator with a noise pump. The evolution of the amplitude probability distribution is found.