Attractors For Lattice Dynamical Systems Research Articles

Discretisation of Global Attractors for Lattice Dynamical Systems

The existence of numerical attractors for lattice dynamical systems is established, where the implicit Euler scheme is used for time discretisation. Infinite dimensional discrete lattice systems as well as their finite dimensional truncations are considered. It is shown that the finite dimensional numerical attractors converge upper semicontinuously to the global attractor of the original lattice model as the discretisation step size tends to zero.

Exponential Attractors for Lattice Dynamical Systems in Weighted Spaces

The aim of this paper is to investigate the existence of exponential attractors for lattice reaction-diffusion systems in weighted spaces $l_{\sigma}^{2}$ and for partly dissipative lattice reaction-diffusion systems in weighted spaces $l_{\mu}^{2}\times l_{\mu}^{2}$ , respectively. In contrast to the previous work by Abdallah in J. Math. Anal. Appl. 339, 217---224 (2008) and Commun. Pure Appl. Anal. 8, 803---818 (2009), we get the existence of exponential attractors for lattice dynamical systems in the weak topology spaces.

Exponential attractors for lattice dynamical systems in weighted spaces

We first present some sufficient conditions for the existence of exponential attractors for locally coupled lattice dynamical systems in weighted spaces of infinite sequences. Then we apply this result to discuss the existence of exponential attractors for first order lattice systems, partly dissipative lattice systems, and second order lattice systems in weighted spaces of infinite sequences.

Exponential attractors for first-order lattice dynamical systems

In l 2 , we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l 2 is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.

Upper semicontinuity of the attractor for lattice dynamical systems of partly dissipative reaction diffusion systems

We investigate the existence of a global attractor and its upper semicontinuity for the infinite‐dimensional lattice dynamical system of a partly dissipative reaction diffusion system in the Hilbert space l2 × l2. Such a system is similar to the discretized FitzHugh‐Nagumo system in neurobiology, which is an adequate justification for its study.