PDF Solution of Nonlinear Stochastic Oscillators Excited by Poisson Pulse with EPC Method

Many problems can be considered as nonlinear oscillators excited by non-Gaussian stochastic process in practice. Poisson pulse is a typical stochastic process that is frequently used to model real process in application. Though a lot of researches have been done for analyzing the PDF solutions of nonlinear oscillators excited by Gaussian white noise, few research was done for the PDF solution of nonlinear oscillators excited by Poisson pulse. In this paper, the stationary probability density function (PDF) solution of the responses of nonlinear stochastic oscillators subjected to external Poisson pulses is analyzed. The PDF solutions are obtained with exponential polynomial closure (EPC) method ever proposed for the PDF solutions of nonlinear oscillators excited by Gaussian white noise [1–3]. To assess the effectiveness of the method numerically in the case of Poisson pulse excitation, Duffing oscillator is analyzed. Both weakly and highly nonlinear systems are analyzed. Numerical results show that the PDFs obtained with EPC yield good agreement with those from Monte Carlo simulation. The tail behavior of PDFs is important for reliability analysis and some other statistical analysis, but well estimation of the tail behavior of the PDFs is a challenge problem in the area of analyzing nonlinear stochastic oscillators. The tail behavior of the PDFs obtained with EPC method, stochastic equivalent method and Monte Carlo simulation is also compared. It shows that the tail behavior of the PDFs obtained EPC agrees well with that from Monte Carlo simulation. In the case of polynomial order being 2, the results obtained with EPC is same as those obtained with stochastic equivalent linearization procedure. In the case of polynomial order being 4 or 6, the results can be much improved, especially in the tails of PDFs. It is known that Gaussian PDFs can be obtained with stochastic equivalent linearization procedure. Though it is always stated that stochastic equivalent linearization procedure is suitable for slightly nonlinear stochastic oscillators, but numerical results in this paper show that the tail of the PDFs obtained with stochastic equivalent linearization procedure for slight nonlinear oscillator still deviates from simulated ones by a lot. Numerical results also show that the behavior and the tail behavior of the PDFs obtained with EPC are much improved when polynomial order is 4 or 6 no matter if the system nonlinearity is slight or high.