May/June 2020 Noise-vanishing concentration and limit behaviors of periodic probability solutions
Min Ji, Weiwei Qi, Zhongwei Shen, Yingfei Yi
Differential Integral Equations 33(5/6): 273-322 (May/June 2020). DOI: 10.57262/die/1589594454


The present paper is devoted to the investigation of noisy impacts on the dynamics of periodic ordinary differential equations (ODEs). To do so, we consider a family of stochastic differential equations resulting from a periodic ODE perturbed by small white noises, and study noise-vanishing behaviors of their “steady states” that are characterized by periodic probability solutions of the associated Fokker-Plank equations. By establishing noise-vanishing concentration estimates of periodic probability solutions, we prove that any limit measure of periodic probability solutions must be a periodically invariant measure of the ODE and that the global periodic attractor of a dissipative ODE is stable under general small noise perturbations. For local periodic attractors (resp. local periodic repellers), small noises are constructed to stabilize (resp. de-stabilize) them. Our study provides an elementary step towards the understanding of stochastic stability of periodic ODEs.


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Min Ji. Weiwei Qi. Zhongwei Shen. Yingfei Yi. "Noise-vanishing concentration and limit behaviors of periodic probability solutions." Differential Integral Equations 33 (5/6) 273 - 322, May/June 2020.


Published: May/June 2020
First available in Project Euclid: 16 May 2020

zbMATH: 07217174
MathSciNet: MR4099218
Digital Object Identifier: 10.57262/die/1589594454

Primary: 35Q84 , 37B25 , 60J60 , 93E15

Rights: Copyright © 2020 Khayyam Publishing, Inc.


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Vol.33 • No. 5/6 • May/June 2020
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