On exact stationary solutions of equivalent non-linear stochastic systems

Abstract

The response of a dynamical system to Gaussian white-noise excitations may be represented by a Markov vector whose probability density is governed by the well-known Fokker-Planck equation. In this paper a general procedure is developed to obtain the exact solutions for Fokker-Planck equations in the state of statistical stationarity. The dynamical systems considered are generally oscillatory and non-linear, and the random excitations may be additive, or multiplicative, or both. The procedure is based on the idea of splitting each drift coefficient and each diffusion coefficient in a Fokker-Planck equation into two parts, associated with the circulatory and potential probability flows, respectively. In so doing two sets of equations are derived for the probability potential which is the essential ingredient required to construct the probability density of the response. The procedure also provides a natural means to identify equivalent stochastic systems which share the same probability distribution.