(Student Paper) Girsanov Linearization of Stochastically Driven Nonlinear Oscillators

Abstract

A weak linearization technique, based on Girsanov transformation of probability measures, is proposed to accurately estimate statistical moments of the response of non-linear oscillators driven by (filtered) white noise processes. Although there are exact solutions for the stationary probability density functions for a rather restricted class of non-linear oscillators, neither are these solutions valid in nonstationary regimes nor are they generally available for higher dimensional non-linear oscillators of engineering interest. The technique most generally employed to solve such problems is Monte Carlo simulations through direct numerical integration. However, compared with numerical integration of deterministic dynamical systems, orders of accuracy for most of the available numerical techniques for stochastically driven oscillators are prohibitively low. Generally engineers are interested in weak solutions to obtain the statistical moments of response functions. In this study, we use a measure transformation method for studying the response of non-linear dynamical systems under additive white noise excitation. In the proposed linearization, the nonlinear part of the drift vector is removed, resulting in an exactly solvable linear system under additive noises. Following this, a Girsanov transformation of measures, which appears as an exponential function (called the Radon-Nikodym derivative), is effected to weakly account for the nonlinear parts of the drift vector. Since the Radon-Nikodym derivative is expressible purely in terms of the linearized solution and can be computed with a high accuracy, one can readily achieve a correspondingly high level of accuracy through the proposed method. The proposed method is numerically illustrated through applications to a non-linear Duffing oscillator.