Search

In this paper, the existence and the limiting behavior of periodic measures for the periodic stochastic modified Swift-Hohenberg lattice systems with variable delays are analyzed. We first prove the existence and uniqueness of global solution when the nonlinear T -periodic drift and diffusion terms are locally Lipchitz continuous and linearly growing. Then we show the existence of periodic measures of the system under some assumptions. Finally, by strengthening the assumptions, we prove that the set of all periodic measures is weakly compact, and we also show that every limit point of a sequence of periodic measures of the original system must be a periodic measure of the limiting system when the noise intensity tends to zero.

In this paper, two problems related to FitzHugh-Nagumo lattice systems are analyzed. The first one is concerned with the asymptotic behavior of random delayed FitzHugh-Nagumo lattice systems driven by nonlinear Wong-Zakai noise. We obtain a new result ensuring that such a system approximates the corresponding deterministic system when the correlation time of Wong-Zakai noise goes to infinity rather than to zero. We first prove the existence of tempered random attractors for the random delayed lattice systems with a nonlinear drift function and a nonlinear diffusion term. The pullback asymptotic compactness of solutions is proved thanks to the Ascoli-Arzel`a theorem and uniform tailestimates. We then show that the upper semi-continuous of attractors as the correlation time tends to infinity. As for the second problem, we consider the corresponding deterministic version of the previous model, and study the convergence of attractors when the delay approaches zero. Namely, the upper semicontinuity of attractors for the delayed system to the non-delayed one is proved.

We study the locally uniform convergence from a family of pullback random attractors to a deterministic attractor. We establish criteria by using joint convergence of the cocycles, collective local...

In this paper, we study the pullback attractor for a general reaction-diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction-diffusion system is proved to have an invariant (H, V )-pullback attractor A = {A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover ∪ s6τ A(s) is precompact in V, ∀τ ∈ R. A non-autonomous Fitz-Hugh-Nagumo equation is studied as a specific example of the reaction–diffusion system. State Scholarship Fund (China) Junta de Andalucía Brazilian-European partnership in Dynamical Systems European Union National Natural Science Foundation of China

We consider the pullback attractors for non-autonomous dynamical systems generated by stochastic lattice differential equations with non-autonomous deterministic terms. We first establish a sufficient condition for existence of pullback attractors of lattice dynamical systems with both non-autonomous deterministic and random forcing terms. As an application of the abstract theory, we prove the existence of a unique pullback attractor for the first-order lattice dynamical systems with both deterministic non-autonomous forcing terms and multiplicative white noise. Our results recover many existing ones on the existences of pullback attractors for lattice dynamical systems with autonomous terms or white noises.

The paper is devoted to studying the existence of a random attractor for stochastic porous media equations on infinite lattices under some conditions., Comment: 15 pages. Some key details have been added

The present paper is devoted to the existence of a random attractor for stochastic lattice FitzHugh-Nagumo system driven by $\alpha$-stable L\'evy noises under some dissipative conditions.

We consider the synchronization of the solutions to coupled stochastic systems of $N$-stochastic ordinary differential equations (SODEs) driven by Non-Gaussian L��vy noises ($N\in \mathbb{N})$. We discuss the synchronization between two solutions and among different components of solutions under certain dissipative and integrability conditions. Our results generalize the present work obtained in Liu et al (2010) and Shen et al (2010)., arXiv admin note: substantial text overlap with arXiv:1402.1790 by other authors

The present paper is devoted to the existence of a random attractor for stochastic lattice FitzHugh-Nagumo system driven by $��$-stable L��vy noises under some dissipative conditions.