Period two solution for a class of distributed delay differential equations
Yukihiko Nakata
TL;DR
The paper studies symmetric periodic solutions of a distributed delay differential equation by linking them to periodic orbits of a planar Hamiltonian system. It extends Kaplan–Yorke ideas to distributed delays via the special symmetric periodic solution (SSPS) framework, showing that an SSPS of $x'(t)=-g\left(\int_{0}^{1} f(x(t-s))\,ds\right)$ corresponds to a period-2 orbit of the Hamiltonian system $\begin{cases} x'=-g(y),\\ y'=2f(x) \end{cases}$ with Hamiltonian $H(x,y)=\int_{0}^{y} g(\xi)\,d\xi + 2\int_{0}^{x} f(\xi)\,d\xi$. It provides a period-map analysis giving explicit criteria (in terms of limits $a,b,A,B$ of $f/x$ and $g/y$) for the existence of a period-2 orbit, namely when $ab<\pi^{2}/2<AB$ or its reverse. The paper also supplies concrete Jacobi-elliptic representations of SSPS for two nonlinearities, $f(x)=r\sin x$, $g(x)=x$ and $f(x)=r(e^{x}-1)$, $g(x)=x$, with clear threshold $r>\pi^{2}/2$ and explicit closed-form expressions in terms of $\text{sn},\text{cn},\text{dn}$ and elliptic integrals. These results extend the Kaplan–Yorke connection to distributed delays and yield analytic insight into the structure of periodic solutions in distributed DDEs.
Abstract
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.