Probabilistic analysis of the steady state of weakly perturbed linear oscillators subject to a class of Gaussian inputs

This paper aims to probabilistically study a class of nonlinear oscillator subject to weak perturbations and driven by stationary zero-mean Gaussian stochastic processes. For the sake of generality in the analysis, we assume that the perturbed term is a polynomial of arbitrary degree in the spatial position, that contains, as a particular case, the important case of the Duffing equation. We then take advantage of the so-called stochastic equivalent linearization technique to construct an equivalent linear model so that its behavior consistently approximates, in the mean-square sense, that of the nonlinear oscillator. This approximation allows us to take extensive advantage of the probabilistic properties of the solution of the linear model and its first mean-square derivative to construct reliable approximations of the main statistical moments of the steady state. From this key information, we then apply the principle of maximum entropy to construct approximations of the probability density function of the steady state. We illustrate the superiority of the equivalent linearization technique over the perturbation method through some examples.