  Discrete Ginzburg-Landau (DGL) equations with non-local nonlinearities have
been established as significant inherently discrete models in numerous physical
contexts, similar to their counterparts with local nonlinear terms. We study
two prototypical examples of non-local and local DGLs on the one-dimensional
infinite lattice. For the non-local DGL, we identify distinct scenarios for the
asymptotic behavior of the globally existing in time solutions depending on
certain parametric regimes. One of these scenarios is associated with a
restricted compact attractor according to J. K. Hale's definition. We also
prove the closeness of the solutions of the two models in the sense of a
"continuous dependence on their initial data" in the $l^2$ metric under general
conditions on the intrinsic linear gain or loss incorporated in the model. As a
consequence of the closeness results, in the dissipative regime we establish
the congruence of the attractors possessed by the semiflows of the non-local
and of the local model respectively, for initial conditions in a suitable
domain of attraction defined by the non-local system.
