Weak forms of the locally transversal linearization (LTL) technique for stochastically driven nonlinear oscillators

Abstract

We explore several weak forms of the locally transversal linearization (LTL) method for stochastically driven nonlinear oscillators. Owing to their computational expediency, weak forms are typically suited to cases wherein it suffices to compute the statistical moments of the response of such oscillators. We first consider a variation of stochastic LTL (SLTL) method [D. Roy, M.K. Dash, A novel stochastic locally transversal linearization (LTL) technique for engineering dynamical systems: strong solutions, Appl. Math. Mod. 29(10) (2005) 913–937, doi:10.1016/j.apm.2005.02.001 ] through weak replacements of the Gaussian stochastic integrals, appearing in the linearized solutions, by random variables with considerably simpler and discrete probability distributions. We also formalize another weak version wherein the linearized equations corresponding to a higher order SLTL schemes are arrived at by conditionally replacing nonlinear drift and multiplicative diffusion fields using backward Euler–Maruyama expansions [Nilanjan Saha, D. Roy, Higher order weak linearizations of stochastically driven nonlinear oscillators, Proc. Roy. Soc. A 463 (2083) (2007) 1827–1856, doi:10.1098/rspa.2007.1852 ]. Error estimates for this weak form of SLTL are also briefly reported. Following this, we suggest a novel procedure to weakly correct the SLTL-based strong solutions, which capture the analyticity and continuity of flow of a dynamical system, using Girsanov transformation of measures. Here error in the replacement of the nonlinear drift field by a linearized one is corrected through the Radon-Nikodym derivative following a Girsanov transformation of probability measures. Since the Radon-Nikodym derivative is computable in terms of a stochastic exponential of an SLTL solution, a remarkably high numerical accuracy is potentially achievable. Numerical illustrations are provided for a few nonlinear oscillators driven by additive and multiplicative noises.