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We investigate the limiting behavior of dynamics of non-autonomous stochastic FitzHugh–Nagumo equations driven by a nonlinear multiplicative colored noise on unbounded thin domains. We first establish the existence and uniqueness of random attractors for the equations on the thin domains and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Wong–Zakai approximations and random attractors for stochastic discrete complex Ginzburg–Landau equations driven by nonlinear multiplicative noise. We establish the existence of the random attractor for the approximate system. For linear multiplicative noise or additive noise, we study the convergence of the solutions and the upper semicontinuity of random attractors for the approximate equation when the perturbation parameters tend to zero. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wong-Zakai approximations given by a stationary process via Euler approximation of Brownian motion and the associated long term behavior of the stochastic wave equation driven by an additive white noise on unbounded domains. We first prove the existence and uniqueness of tempered pullback attractors for stochastic wave equation and its Wong-Zakai approximation. Then, we show that the attractor of the Wong-Zakai approximate equation converges to the one of the stochastic wave equation driven by additive noise as the correlation time of noise approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, we study the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic lattice systems. We first prove that the approximate system has a unique tempered pullback attractor under much weaker conditions than the original system. When the stochastic system is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions and the upper semicontinuity of random attractors for Wong-Zakai approximations as the step-length of the Wiener shift approaches zero. [ABSTRACT FROM AUTHOR]

We investigate the asymptotic behavior of a class of non-autonomous stochastic FitzHugh–Nagumo systems driven by additive white noise on unbounded thin domains. For this aim, we first show the existence and uniqueness of random attractors for the considered equations and their limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse into a lower-dimensional unbounded domain. [ABSTRACT FROM AUTHOR]

In this paper, we study the long term behavior of non-autonomous fractional FitzHugh-Nagumo systems with random forcing given by an approximation of white noise, called Wong-Zakai approximation. We first prove the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximation fractional FitzHugh-Nagumo systems, and then establish the upper semicontinuity of attractors of system driven by a linear multiplicative Wong-Zakai approximations as random forcing approaches white noise in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the long term behavior of the solutions to a class of stochastic discrete complex Ginzburg-Landau equations with time-varying delays and driven by multiplicative white noise. We first prove the existence and uniqueness of random attractor in a weighted space for these equations. Then, we analyze the upper semicontinuity of the random attractors as the time delay approaches to zero. [ABSTRACT FROM AUTHOR]

A system of stochastic retarded reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing on thin domains is considered. Relations between the asymptotic behavior for the stochastic retarded equations defined on thin domains in Rn+1 and an equation on a domain in Rn are investigated. We first show the existence and uniqueness of tempered random attractors for these equations. Then, we analyze convergence properties of the solutions as well as the attractors. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic behavior of the solutions of the nonautonomous fractional stochastic reaction-diffusion equations on R n with continuous (but not necessarily differentiable) nonlinear drift terms, which leads to the nonuniqueness of solutions and hence a multivalued random dynamical system with the solution operators of the equation. We first show the existence and uniqueness of random attractors for such a dynamical system and then establish the upper semicontinuity of these attractors as the intensity of noise approaches zero. The measurability of the random attractors is proved by the method of weak upper semicontinuity of multivalued functions, and the pullback asymptotic compactness of the system is derived by the idea of uniform estimates on the tails of the solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of stochastic discrete complex non-autonomous Ginzburg-Landau equations with multiplicative noise. We prove the existence and uniqueness of the random attractor. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided. [Copyright &y& Elsevier]

Abstract: In this paper, a new nonlinear integro-differential inequality is established. Using the properties of -cone and a generalization of Barbalat’s lemma, the boundedness and asymptotic behavior for the solution of the inequality are obtained. Applying this nonlinear integro-differential inequality, the invariant and attracting sets for Cohen–Grossberg neural networks with mixed delays are obtained. The results extend and improve the earlier publications. An example is given to illustrate the efficiency of the obtained results. [Copyright &y& Elsevier]