DescriptionWe study nonlinear autonomous real-valued differential delay equations with several fixed delays x'(t) = sum_{i=1} D F_i(x(t-d_i)), where the F_i are continuous, have nonzero limits at plus and minus infinity, and are similar (in a sense we make precise) to step functions. Our focus is on periodic solutions, and our approach is to link the above problem to an appropriately related equation y'(t) = sum_{i=1} D h_i(y(t-d_i)), where the h_i are in fact step functions.
Given a periodic solution of the second problem, we describe conditions under which this solution implies the existence of a similar periodic solution of the first problem. We also prove some results on stability of periodic solutions, and make a partial study of the global dynamics of the second problem.