Transient solution of stochastic oscillators with both odd and even nonlinearity under modulated Gaussian white noise.

Abstract

This study presents an improved solution procedure for determining the transient probability density function (PDF) solutions of the nonlinear oscillators with both odd and even nonlinearity under modulated random stimulation, which is an extension of the exponential-polynomial-closure approach. An evolutionary exponential-polynomial function with time-varying undetermined variables is considered as the transient probabilistic solution. By selecting a set of independent evolutionary base functions spanning a R^{n} space as weight functions, a set of ordinary differential equationscan be formulated by integrating the weighted residual error. The undetermined variables can be determined numerically by solving those ordinary differential equations. Three numerical examples illustrate that the improved solution procedure can acquire the transient probabilistic responses of the stochastic dynamic systems effectively and efficiently even in the PDF solution tails when compared with Monte Carlo simulation. Moreover, the results indicate that the PDF solutions of the oscillators are asymmetrical at their nonzero means due to the influence of even nonlinearity. The nonstationary behaviors of the system responses are also investigated along with the behavior of modulated Gaussian white noise.