Periodic Solutions of a Delay Differential Equation with a Periodic Multiplier

A simple non-autonomous scalar differential equation with delay, exponential decay, nonlinear negative feedback and a periodic multiplicative coefficient is considered. It is shown that stable slowly oscillating periodic solutions with the period of the feedback coefficient, and also with the double period of the feedback coefficient exist. The periodic solutions are built explicitly in the case of piecewise constant feedback function and the periodic coefficient. The periodic dynamics are shown to persist under small perturbations of the equation which make it smooth. The results are confirmed and illustrated by numerical simulations.