Existence of a period two solution of a delay differential equation

Abstract

<p style='text-indent:20px;'>We consider the existence of a symmetric periodic solution for the following distributed delay differential equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ f(x) = r\sin x $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ r&gt;0 $\end{document}</tex-math></inline-formula>. It is shown that the well studied second order ordinary differential equation, known as the nonlinear pendulum equation, derives a symmetric periodic solution of period <inline-formula><tex-math id="M3">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>, 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.

Topics & Concepts