Asymptotic Behavior of Solutions of SDE for Relaxation Oscillations

Abstract

A stochastic Lienard equation with a small parameter $\varepsilon > 0$ multiplying the highest derivative is formulated by a two-dimensional stochastic differential equation (SDE). Here fast and slow variables appear. In order to investigate the asymptotic behavior of the fast variable in such a system as $\varepsilon \to 0$, a stochastic process $X^\varepsilon (t)$ as a good approximation for all $t \geqq 0$ is derived by the methods of matched and composite expansions for relaxation oscillations. Then the limit of $X^\varepsilon (t)$ is identified, so that $X^\varepsilon (t)$ converges weakly as $\varepsilon \to 0$ to a solution of a one-dimensional stochastic differential equation. This yields the weak convergence of the slow variable as $\varepsilon \to 0$.