Existence of a period two solution of a delay differential equation

Yukihiko Nakata

doi:10.3934/dcdss.2020392

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2021, Volume 14, Issue 3: 1103-1110. Doi: 10.3934/dcdss.2020392

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Existence of a period two solution of a delay differential equation

Yukihiko Nakata

Department of Physics and Mathematics, Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara-shi, Kanagawa 252-5258, Japan

Received: December 31, 2018

Revised: January 31, 2020

Early access: June 2020

Published: March 2021

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Abstract

We consider the existence of a symmetric periodic solution for the following distributed delay differential equation

$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $

where $ f(x) = r\sin x $ with $ r>0 $. 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 $ 2 $, 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.

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Mathematics Subject Classification: 34K13.

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Figure 2.1. Symmetric periodic solutions of (2.1) and (1.3)

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Figure 4.1. An SSPS attracts two solutions with different initial conditions ($ r = 8 $)

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