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

of a Delay-Di erential Equation Tom a s Gedeon1 Department of Mathematical Sciences, Montana State University, Bozeman,MT 59717 [email protected] Gwendolen Hines2 Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588 [email protected] 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 prove that the discretized problems have global attractors. It was proved in [G,M] 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 =1.