Existence of a period two solution of a delay differential equation

Abstract

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation \begin{document}$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $\end{document} where \begin{document}$ f(x) = r\sin x $\end{document} with \begin{document}$ r>0 $\end{document} . It is shown that the well studied second order ordinary differential equation, known as the nonlinear pendulum equation, derives a symmetric periodic solution of period \begin{document}$ 2 $\end{document} , expressed in terms of the Jacobi elliptic functions, for the delay differential equation. We here apply the approach in Kaplan and Yorke (1974) for a differential equation with discrete delay to the distributed delay differential equation.