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 l 2 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 .