Solutions of path integration for nonlinear dynamical system under stochastic parametric and external excitations

The numerical path integration based on Gauss-Legendre scheme is extended to the case of nonlinear dynamical system under stochastic parametric and external excitations. For the purpose of comparison between the numerical solutions and the analytic solution (if the system has) or Monte-Carlo simulation, we discuss the system under parametric and external Gaussian white noise excitations. The numerical method is shown to give accurate results. Via the numerical solutions of path integration, we have studied the P bifurcation of the stochastic system.