Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation

The singularly perturbed differential-delay equation {Mathematical expression}is studied. Existence of periodic solutions is shown using a global continuation technique based on degree theory. For small e{open} these solutions are proved to have a square-wave shape, and are related to periodic points of the mappingf:R→R. Whenfis not monotone the convergence of x(t) to the square-wave typically is not uniform, and resembles the Gibbs phenomenon of Fourier series.