We study the relationship between the dynamics of delay differential equations and their discrete versions obtained via Euler discretizations. It is shown that the dynamics of the two can be essentially different for some important examples of the delay equations coming from applications. They include the well-known singularly perturbed differential delay equation a[x'(t) + bx'(t-1)] = -x(t) + f(x(t-1)), 0 < a (small) which dynamics is far from being completely understood. On the other hand, its discrete version, when x'(t) is replaced by the Euler difference, is shown to possess stable periodic solutions that correspond to hyperbolic attracting cycles of the map f. The discrepancies between the dynamics can potentially have important computational implications.

MINISIMPOSIUM SESSION: 3. Numerical methods for delay equations (2 sessions)