Upper semicontinuity of Morse sets of a discretization of a delay-differential equation

In this paper, we consider a discrete delay problem with negative feedback (x)over dot(t) = f(x(t), x(t-1)) along with a certain family of time discretizations with step-size 1/n. In the original problem, the attractor admits a nice Morse decomposition. We prole that the discretized problems have global attractors. It was proved by T. Gedeon and K. Mischaikov (1995, J. Dynamical Differential Equations 7, 141-190) that such attractors also admit Morse decompositions. We then prove certain continuity results about the individual Morse sets, including that if f(x,y) = f(y), then the individual Morse sets are upper semicontinuous at n = infinity. (C) 1999 Academic Press.