A Gaussian and non-Gaussian stochastic linearization of a non-linear oscillator governing equation

The basic ideas of the statistical linearization method lies in the replacing a non-linear system by a linear one, that is equivalent to the original system in some probabilistic sense. The question is to know the probability density function (pdf) of the solution, in order to calculate the statistical moments which exist in the coefficient of the new linear differential problem. Aim of this article is the study of a stochastic linear model to predict the dynamic behaviour of a Duffing-van der Pol oscillator. Several cases, corresponding to different values of the differential equation coefficients, are checked and Gaussian (GSL) and non-Gaussian stochastic linearization methods (NGSL) are employed to substitute this nonlinear equation with new equivalent linear relationships.