Periodic Solutions in a Simple Delay Differential Equation

TL;DR

This paper investigates the existence and construction of various periodic solutions in a simple delay differential equation with nonlinear feedback, demonstrating their persistence under perturbations and validating findings through numerical simulations.

Contribution

It provides explicit constructions of periodic solutions in a delay differential equation with nonlinear feedback and shows their robustness under smooth perturbations.

Findings

Existence of multiple types of slowly oscillating periodic solutions.

Explicit construction of solutions with same and double periods.

Numerical simulations confirm theoretical results.

Abstract

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.