Period two solution for a class of distributed delay differential equations

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.