An Explicit Periodic Solution of a Delay Differential Equation

In this paper we prove that the following delay differential equation ddtx(t)=rx(t)1-∫01x(t-s)ds,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{d}{dt}x(t)=rx(t)\left( 1-\int _{0}^{1}x(t-s)ds\right) , \end{aligned}$$\end{document}has a periodic solution of period two for r>π22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r>\frac{\pi ^{2}}{2}$$\end{document} (when the steady state, x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=1$$\end{document}, 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 (J Math Anal Appl 48:317–324, 1974). The periodic solution is expressed in terms of the Jacobi elliptic functions.