Dynamics of constrained differential delay equations

A class of forced first-order differential delay equations with piecewise-affine right-hand sides is introduced, as a prototype model for the speed of a motor under control. A simple pure delay form is mainly considered. When forcing is zero, an exact stable periodic solution is exhibited. For large amplitude periodic forcing, existence of stable solutions, whose period is equal to that of the forcing function, is discussed, and these solutions are constructed for square wave forcing. Traditional numerical methods are discussed briefly, and a new approach based on piecewise-polynomial structure is introduced. Simulations are then presented showing a wide range of dynamics for intermediate values of forcing amplitude, when the natural period of the homogeneous equation and the period of the forcing function compete.