Upper Semicontinuity of Morse Sets of a Discretization of a Delay-Differential Equation: An Improvement

In this paper, we consider a discrete delay problem with negative feedback x(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 proved in (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36–78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations7, 141–190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations151, 36–78) we proved 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=∞. In this paper we extend this result to the general case; that is, we prove for general f(x, y) with negative feedback that the Morse sets are upper semicontinuous.