Explicit expression of stationary response probability density for nonlinear stochastic systems

Abstract

Identifying the exactly or approximately explicit expression of the stationary response\nprobability density for general nonlinear stochastic dynamical systems is of great\nsignificance in the fields of stochastic dynamics and control. Almost all the existing\nmethods are devoted to determine the exact or approximate solution for specific values of\nsystem and excitation parameters. Herein, aimed at stochastic systems with\npolynomial nonlinearity and excited by Gaussian white noises, a novel method is\nproposed to identify the stationary response probability density which explicitly includes\nsystem and excitation parameters. The stationary probability density is first written as\nan exponential function according to the maximum entropy principle, the power of the\nexponential function is then expressed as a linear combination of prescribed nondimensional\nparameter clusters constituted by system and excitation parameters, and\nstate variables, with the coefficients to be determined. The undetermined coefficients\nare derived by minimizing the residual of the associated Fokker-Planck- Kolmogorov\nequation. The application and efficacy of the proposed method are illustrated by a\ntypical numerical example.