Non-explosion solutions for a class of stochastic physical diffusion oscillators

Abstract

This study investigates the theoretical aspects of diffusion phenomena, focusing on the existence, uniqueness, and non-explosive behavior of solutions to stochastic differential equations (SDEs). We relax the conditions imposed on the coefficients of these SDEs, broadening their applicability to various diffusion phenomena in mechanics. By establishing a general non-explosion criterion, we derive sufficient conditions to guarantee non-explosive solutions for a specific class of diffusions. This is achieved through the construction of Lyapunov functions. Notably, the Duffing and Van der Pol oscillators fall within this class. To illustrate the practical application of our findings, we employ the Euler-Maruyama method to simulate solutions for these two oscillators.