Dynamics of systems on infinite lattices

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction–diffusion equation has a global attractor in a weighted l2 space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction–diffusion equation is approached by finite-dimensional systems.