Period two solution for a class of distributed delay differential equations

TL;DR

This paper investigates the existence of period two solutions in a class of distributed delay differential equations, linking them to Hamiltonian systems and illustrating with elliptic function solutions.

Contribution

It introduces a novel approach to find period two solutions for distributed delay differential equations using Hamiltonian systems and extends known methods to this class.

Findings

Existence of symmetric period 2 solutions for certain distributed delay equations.

Connection between distributed delay solutions and Hamiltonian systems.

Explicit solutions expressed via Jacobi elliptic functions.

Abstract

We study the existence of a periodic solution for a differential equation with distributed delay. It is shown that, for a class of distributed delay diferential quations, a symmetric period 2 solution, where the period is twice the maximum delay, is given as a periodic solution of a Hamiltonian system of ordinary differential equations. Proof of the idea is based on (Kaplan & Yorke, 1974, J. Math. Anal. Appl.) for a discrete delay differential equation with an odd nonlinear function. To illustrate the results, we present distributed delay differential equations that have periodic solutions expressed in terms of the Jacobi elliptic functions.