Numerical solutions for a class of Nonlinear Systems and Application to Stochastic Resonance

Abstract: We study a nonlinear system under regime switching and subject to an environmental noise. The stochas-tic differential equations with Markovian switching (SDEwMSs), one of the important classes of hybrid systems, have been used to model many physical systems that are subject to frequent unpredictable structural changes. The research in this area has been both theoretical and applied. Most of SDEwMSs do not have explicit solutions so it is important to have numerical solutions. It is surprising that there are not any numerical methods established for SDEwMSs yet, although the numerical methods for stochastic differential equations (SDEs) have been well studied. We will considered some more general conditions for the coefficient functions and prove a result on the existence using the Schauder’s fixed point theorem extended some similarly results on linear systems. The most important results in this paper is to develop a numerical scheme for SDEwMSs and estimate the error between the numerical and exact solutions. Also, we study the application of these system to control the electronic circuits using the benefit of stochastic resonance Key–Words:stochastic differential equation with Markovian switching, pathwise uniqueness, fixed point technique, numerical solution, Euler-Maruyama method, stopping times, stochastic resonance