Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation

In this paper, we consider a discrete delay problem with negative feedbackx(t)=f(x(t), x(t−1)) along with a certain family of time discretizations with stepsize 1/n. In the original problem, the attractor admits a nice Morse decomposition. We prove that the discretized problems have global attractors. It was proved by T. Gedeon and K. Mischaikov (1995,J. Dynamical Differential Equations7, 141–190) that such attractors also admit Morse decompositions. We then prove certain continuity results about the individual Morse sets, including that iff(x, y)=f(y), then the individual Morse sets are upper semicontinuous atn=∞.