Response of an oscillatory differential delay equation to a single stimulus

TL;DR

This paper analytically investigates how a limit cycle in a differential delay equation responds to a single pulse perturbation, detailing changes in amplitude, period, and phase depending on the pulse timing.

Contribution

It provides a complete analytical characterization of the perturbed response of a limit cycle to a pulse in a delay differential equation, including minima, maxima, and period modifications.

Findings

Pulse alters the limit cycle's amplitude and period.

Timing of the pulse affects the nature of the response.

Analytical expressions for perturbed minima and maxima are derived.

Abstract

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.