Wen-Xian( ) a) † , Xu Wei( ) a) , and Cai Li( ) b)

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.

The stochastic resonance is investigated in the generalized Langevin equation with exponential memory kernel subjected to the joint action of internal noise, external noise and external sinusoidal forcing. The system is converted into three-dimensional Markovian Langevin equations. Furthermore, using the Shapiro-Loginov formula and the Laplace transformation technique, the exact expressions of the first moment and the steady response amplitude are obtained. The research results show that with the variations of external sinusoidal force frequency and the parameters of memory kernel and external noise, the system presents bona-fide stochastic resonance, conventional stochastic resonance and stochastic resonance in a broad sense under the condition of Routh-Hurwitz stability. In addition, the stochastic resonance can be weakened as the memory time increases. Moreover, the numerical results of power spectrum of system are in agreement with the analytic results.