Non-autonomous Lattice Dynamical Systems Research Articles

Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems

<p style='text-indent:20px;'>In this paper we study long-time behavior of first-order non-autono-mous lattice dynamical systems in square summable space of double-sided sequences using the cooperation between the discretized diffusion operator and the discretized reaction term. We obtain existence of a pullback global attractor and construct pullback exponential attractor applying the introduced notion of quasi-stability of the corresponding evolution process.</p>

On the forward dynamical behaviour of nonautonomous lattice dynamical systems

In this article, we study the forward dynamical behaviour of nonautonomous lattice systems. We first construct a family of sets in the arbitrary small neighbourhood of a global attractor of the skew-product flow generated by a general nonautonomous lattice system, which is forward invariant and uniformly forward attracts any bounded subset of the phase space. Moreover, under some suitable conditions, we further construct a family of sets such that it uniformly forward exponentially attracts bounded subsets of the phase space. As an application, we study the discrete Gray–Scott model in detail and illustrate how to apply our abstract results to some concrete lattice system.

Second order non-autonomous lattice systems and their uniform attractors

The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators $ \left( f_{1i}\left( \overset{.}{u}_{j}\mid j\in I_{iq_{1}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ and $ \left( f_{2i}\left( u_{j}\mid j\in I_{iq_{2}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions $ f_{1i} $ and $ f_{2i} $ we get the autonomous and non-autonomous second order systems given by [1, 25, 26].

Uniform exponential attractors for second order non-autonomous lattice dynamical systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, α-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.

Non-autonomous lattice systems with switching effects and delayed recovery

The long term behavior of a type of non-autonomous lattice dynamical systems is investigated, where these have a diffusive nearest neighborhood interaction and discontinuous reaction terms with recoverable delays. This problem is of both biological and mathematical interests, due to its application in systems of excitable cells as well as general biological systems involving delayed recovery. The problem is formulated as an evolution inclusion with delays and the existence of weak and strong solutions is established. It is then shown that the solutions generate a set-valued non-autonomous dynamical system and that this non-autonomous dynamical system possesses a non-autonomous global pullback attractor.

Attractors for second order lattice systems with almost periodic symbols in weighted spaces

We study the existence of a uniform global attractor for a family of second order non-autonomous lattice dynamical systems with almost periodic symbols in a suitable weighted space of infinite sequences. The assumptions on the nonlinear part present the main difficulty of this work. Lemmas 2 and 4 are helpful to overcome such a difficulty. A uniform absorbing set O and uniform estimates on the tails of solutions with respect to initial data from O and all translations of the external term involved are introduced. Such uniform estimates are needed to obtain the asymptotic compactness of solution operators which is a major step towards proving the existence of attractors.

Asymptotic Dynamics of Second Order Nonautonomous Systems on Infinite Lattices

We have introduced abstract sufficient conditions for the existence of a uniform exponential attractor for a special family of second order nonautonomous lattice dynamical systems with quasiperiodic symbols in a standard space of infinite sequences. Compared with the lattice dynamical system in [Zhou & Zhao, 2014], here a generalized nonlinear part and weaker assumptions have been presented, kindly see Remark Remark 2.1 for more details.

Pullback and Uniform Exponential Attractors for Nonautonomous Boussinesq Lattice System

We first prove the existence of a pullback attractor and a pullback exponential attractor for a nonautonomous lattice dynamical system of nonlinear Boussinesq equations affected by time-dependent coupled coefficients and forces. Then, we prove the existence of a uniform attractor and a uniform exponential attractor for the system driven by quasi-periodic external forces.

Attractors for non-autonomous retarded lattice dynamical systems

AbstractIn this paperwe study a non-autonomous lattice dynamical system with delay. Under rather general growth and dissipative conditions on the nonlinear term,we define a non-autonomous dynamical system and prove the existence of a pullback attractor for such system as well. Both multivalued and single-valued cases are considered.

Uniform exponential attractors for first order non-autonomous lattice dynamical systems

In Abdallah (2008, 2009) [2,3], we have investigated the existence of exponential attractors for first and second order autonomous lattice dynamical systems. Within this work, in l 2 , we carefully study the existence of a uniform exponential attractor for the family of processes associated with an abstract family of first order non-autonomous lattice dynamical systems with quasiperiodic symbols acting on a closed bounded set.

Uniform global attractors for first order non-autonomous lattice dynamical systems

Recently, many authors investigated the existence of global attractors for different types of autonomous lattice dynamical systems. Within this work, we carefully study the existence of a uniform global attractor for a new class of first order non-autonomous lattice dynamical system in the Hilbert space l 2 .

Compact uniform attractors for dissipative non-autonomous lattice dynamical systems

This paper discusses the long time behavior of solutions for dissipative non-autonomous lattice dynamical systems. We first prove some sufficient and necessary conditions for the existence of a compact uniform attractor for the family of processes defined on a Hilbert space of infinite sequences, and then give an upper bound of the Kolmogorov $\varepsilon$-entropy for the uniform attractor. As an application, we consider the dissipative non-autonomous lattice Zakharov equations.

Attractors for a second order nonautonomous lattice dynamical system with nonlinear damping

In this Letter, the existence of compact pullback attractors is considered for a second order nonautonomous lattice dynamical system with nonlinear damping arising from spatial discretization of wave equations in Rk. And the finite-dimensional approximations of the attractors are studied. Finally, an upper bound of fractal dimension of the attractors is obtained.