Boundary Layer Phenomena for Di erential-Delay Equations with State Dependent Time Lags: II

We study the limiting shape of solutions of the singularly perturbed di erential-delay equation " _ x(t) = f(x(t); x(t r)); r = r(x(t)) (1) as " ! 0: More precisely, we take a sequence xn(t) of solutions of (1) for " = "n ! 0; and consider the set IR2 de ned as the limit, in the Hausdor sense, of the corresponding sequence of graphs n IR2: Using the geometry of ; we make precise the sense in which points ( k; k) 2 satisfy the di erence equation 0 = f( k ; k 1); k 1 = k r( k); (2) thereby providing a means for determining : In the particular case that xn(t) is a sequence of slowly oscillating periodic solutions and f and r satisfy appropriate hypotheses, we use this theory with subtle scaling arguments to show that 6= IR f0g; or equivalently, for the sup norm, lim inf n!1 kxnk > 0: (3) Despite its pedestrian appearance, (3) is the crucial rst step in determining exactly. We also prove generally that if the delay function r(x) is not constant on any x-interval, and if f(x; y) is piecewise monotone in its rst argument, the set is almost a graph. More precisely, for all but a countable set of t 2 IR the vertical slices t = f( ; ) 2 j = tg are single points. This is in marked contrast to the case of a constant delay, where it is possible for t to be a nontrivial interval for all real t: