Attractors for lattice FitzHugh-Nagumo systems

The FitzHugh-Nagumo system on infinite lattices is studied. By means of “tail ends” estimates on solutions, it is proved that the system is asymptotically compact in a weighted l2 space and has a compact global attractor containing traveling wave solutions. The singular limiting behavior of global attractors is also investigated as a singular parameter ϵ→0. It is shown that the limiting system for ϵ=0 has no global attractor, but all the global attractors for perturbed systems are contained in a common compact subset of the phase space when ϵ is positive but small. Further, a compact local attractor for the limiting system is constructed, and the upper semicontinuity of global attractors is established when ϵ→0.