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

TL;DR

This paper establishes conditions under which solutions to certain stochastic diffusion oscillators, including Duffing and Van Der Pol types, remain non-explosive, using Lyapunov functions and numerical simulations.

Contribution

It weakens existing conditions on SDE coefficients and provides new criteria for non-explosion in stochastic diffusion oscillators.

Findings

Derived sufficient conditions for non-explosion of solutions.

Constructed Lyapunov functions ensuring solution stability.

Applied Euler-Maruyama method for simulation of oscillators.

Abstract

In this work, we are interested in problems that are related to the physical phenomena of diffusion. We will focus on the theoretical aspect of the study, such as existence, uniqueness and non-explosive solutions. We will weaken the conditions imposed on the coefficients of the stochastic differential equations (SDE) that model some diffusion phenomena of mechanics. The work will be based on a general non-explosion criterion and we will obtain sufficient conditions so that the solution for a certain class of diffusions does not explode. We will construct Lyapunov functions that ensure the non-explosion of the solutions. Two important oscillators, namely the Duffing and the Van Der Pol oscillators, belong to this class. The Euler-Maruyama method is applied to these two oscillators to give us a simulation solution for them.