Existence of Periodic Solutions in Distribution for Stochastic Newtonian Systems

Abstract

Periodic phenomena such as oscillation have been studied for many years. In this paper, we verify the stochastic version of Levinson’s conjecture, which confirmed the existence of stochastic periodic solutions for second order Newtonian systems with dissipativeness. First, we provide a stochastic Duffing’s equation to display our result. Then, we apply Wong–Zakai approximation method and Lyapunov’s method to stochastic second order Newtonian systems driven by Brownian motions. With the help of Horn’s fixed point theorem, we prove that this kind of systems is stochastic dissipative and admits periodic solutions in distribution.