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Existence of a period two solution of a delay differential equation

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  • 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.

    Mathematics Subject Classification: 34K13.


    \begin{equation} \\ \end{equation}
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  • Figure 2.1.  Symmetric periodic solutions of (2.1) and (1.3)

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

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