Existence of Periodic Solutions in Distribution for Stochastic Newtonian Systems

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 Du...

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Bibliographic Details
Published inJournal of statistical physics Vol. 181; no. 2; pp. 329 - 363
Main Authors Jiang, Xiaomeng, Li, Yong, Yang, Xue
Format Journal Article
LanguageEnglish
Published New York Springer US 01.10.2020
Springer
Springer Nature B.V
Subjects
Approximation
Differential equations
Fixed points (mathematics)
Mathematical and Computational Physics
Newton's laws of motion
Physical Chemistry
Physics
Physics and Astronomy
Quantum Physics
Statistical Physics and Dynamical Systems
Theoretical
Levinson’s conjecture
Periodic solutions in distribution
Wong–Zakai approximations
Lyapunov’s method
Stochastic Newtonian systems
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Summary: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.
ISSN:0022-4715
1572-9613
DOI:10.1007/s10955-020-02583-3