Higher order weak linearizations of stochastically driven nonlinear oscillators

Abstract

We present derivative-free weak and strong solutions of stochastically driven nonlinear oscillators of engineering interest using higher order forms of the locally transversal linearization (LTL) method. Unlike strong solutions, weak stochastic solutions attempt to predict only mathematical expectations of functions of the true solution and are obtainable with much less computational effort. The linearized equations corresponding to the higher order implicit LTL schemes are arrived at using backward Euler–Maruyama and Newmark expansions in order to conditionally replace nonlinear drift and multiplicative diffusion vector fields. We also briefly describe alterations through which explicit forms of such higher order linearizations are obtained. In the weak forms, the Gaussian stochastic integrals, appearing in the linearized solutions, are replaced by random variables with simpler and discrete probability distributions. The resulting local approximations to the true solution are of significantly higher formal order of accuracy, as reflected through local error estimates. Numerical illustrations are presently provided for the Duffing and Van der Pol oscillators driven by additive and multiplicative noises, which are indicative of the numerical accuracy, computational speed and algorithmic simplicity.