  Simple form scalar differential equation with delay and nonlinear negative
periodic feedback is considered. The existence of several types of slowly
oscillating periodic solutions is shown with the same and double periods of the
feedback coefficient. The periodic solutions are built explicitly in the case
of piecewise constant nonlinearities involved. The periodic dynamics are shown
to persist under small perturbations of the equation which make it smooth. The
theoretical results are verified by extensive numerical simulations.
