# American Institute of Mathematical Sciences

November  2016, 21(9): 3163-3174. doi: 10.3934/dcdsb.2016091

## Approximation for random stable manifolds under multiplicative correlated noises

 1 School of Mathematics and Statistics, Huazhong University of Sciences and Technology, Wuhan 430074, China, China 2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616

Received  December 2015 Revised  February 2016 Published  October 2016

The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.
Citation: Tao Jiang, Xianming Liu, Jinqiao Duan. Approximation for random stable manifolds under multiplicative correlated noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3163-3174. doi: 10.3934/dcdsb.2016091
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