A periodic solution of period two of a delay differential equation

TL;DR

This paper proves the existence of a period-two solution for a specific delay differential equation when a parameter exceeds a critical value, using elliptic functions and an integrable ODE system.

Contribution

It establishes the existence of a periodic solution of period two for a delay differential equation using elliptic functions, extending previous methods.

Findings

Periodic solution exists for r > π²/2

Solution expressed via Jacobi elliptic functions

Steady state x=1 is unstable in this regime

Abstract

In this paper we prove that the following delay differential equation \[ \frac{d}{dt}x(t)=rx(t)\left(1-\int_{0}^{1}x(t-s)ds\right), \] has a periodic solution of period two for $r>\frac{\pi^{2}}{2}$ (when the steady state, $x=1$, is unstable). In order to find the periodic solution, we study an integrable system of ordinary differential equations, following the idea by Kaplan and Yorke \cite{Kaplan=000026Yorke:1974}. The periodic solution is expressed in terms of the Jacobi elliptic functions.