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Persistence of smooth manifolds for a non-autonomous coupled system under small random perturbations.

Authors :
Zhao, Junyilang
Shen, Jun
Source :
Journal of Differential Equations. Apr2024, Vol. 387, p200-255. 56p.
Publication Year :
2024

Abstract

We consider a non-autonomous coupled system (x , y) , whose x coordinate satisfies a semilinear parabolic equation, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. Our aim is to study the persistence of dynamical behavior for such coupled system under a small random perturbation driven by stationary multipilcative noise. We show that for the perturbed system there exists a C 1 invariant manifold S (t , ω) = { (x , y) ∈ X × Y | x = σ (t , y , ω) } under the condition that the linear part of x equation satisfies the exponential dichotomy. Meanwhile, we observe that typically if the linear part of x equation is uniformly attracting or uniformly expanding, then the corresponding invariant manifold shares the same qualitative properties. In the end, as the perturbation tends to 0, we show that the invariant manifold and its derivative in y are approaching to those of the original system, suggesting that such structure is persistent under the small random perturbation. [ABSTRACT FROM AUTHOR]

Details

Language :
English
ISSN :
00220396
Volume :
387
Database :
Academic Search Index
Journal :
Journal of Differential Equations
Publication Type :
Academic Journal
Accession number :
175296322
Full Text :
https://doi.org/10.1016/j.jde.2023.12.027