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

Abstract 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 Equations 151 , 36–78) that the discretized problems have global attractors. It was proved in (T. Gedeon and K. Mischaikow, 1995, J. Dynam. Differential Equations 7 , 141–190) that such attractors also admit Morse decompositions. In (T. Gedeon and G. Hines, 1999, J. Differential Equations 151 , 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.