Periodic Solutions of a Delay Differential Equation with a Periodic Multiplier

TL;DR

This paper investigates the existence and stability of periodic solutions in a delay differential equation with a periodic multiplier, providing explicit solutions in special cases and demonstrating their persistence under perturbations through numerical simulations.

Contribution

It introduces explicit constructions of periodic solutions in a delay differential equation with periodic coefficients and analyzes their stability and persistence under perturbations.

Findings

Existence of stable slowly oscillating periodic solutions with the period of the feedback coefficient.

Explicit solutions constructed for piecewise constant feedback and periodic coefficients.

Periodic solutions persist under small smooth perturbations, confirmed by numerical simulations.

Abstract

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.