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March  2021, 14(3): 1103-1110. doi: 10.3934/dcdss.2020392

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  January 2019 Revised  February 2020 Published  March 2021 Early access  June 2020

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.
Keywords: Delay differential equations, symmetric periodic solution, elliptic functions, pendulum equation.
Mathematics Subject Classification: 34K13.
Citation: Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
References:
[1]

D. Breda, O. Diekmann, D. Liessi and F. Scarabel, Numerical bifurcation analysis of a class of nonlinear renewal equations, Electron. J. Qual. Theo. Diff. Equ., 2016 (2016), 24 pp. doi: 10.14232/ejqtde.2016.1.65.

[2]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954.

[3]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

P. Dormayer and A. F. Ivanov, Symmetric periodic solutions of a delay differential equation, Dynamical Systems and Differential Equations, Discrete Contin. Dynam. Systems, 1 (1998), 220-230. 

[5]

P. Dormayer and A. F. Ivanov, Stability of symmetric periodic solutions with small amplitude of $x^{\prime}(t) = \alpha f(x(t), x(t-1))$, Discrete Contin. Dynam. Systems, 5 (1999), 61-82.  doi: 10.3934/dcds.1999.5.61.

[6]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[7]

J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl., 48 (1974), 317-324.  doi: 10.1016/0022-247X(74)90162-0.

[8]

B. B. Kennedy, Symmetric periodic solutions for a class of differential delay equations with distributed delay, Electron. J. Qual. Theo. Diff. Equ., 2014 (2014), 18 pp. doi: 10.14232/ejqtde.2014.1.4.

[9]

J. B. Li and X. Z. He, Proof and generalization of Kaplan-Yorke's conjecture under the condition $f^{\prime}(0)>0$ on periodic solution of differential delay equations, Sci. China Ser. A., 42 (1999), 957-964.  doi: 10.1007/BF02880387.

[10]

K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, The American Mathematical Monthly, 108 (2001), 729-737.  doi: 10.1080/00029890.2001.11919804.

[11]

R. E. Mickens, Oscillations in Planar Dynamic Systems, Series on Advances in Mathematics for Applied Sciences, 37. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/2778.

[12]

Y. Nakata, An explicit periodic solution of a delay differential equation, J. Dyn. Diff. Equ., 32 (2020), 163-179.  doi: 10.1007/s10884-018-9681-z.

[13]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4), 101 (1974), 263-306.  doi: 10.1007/BF02417109.

[14] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. 
[15]

H.-O. Walther, Topics in delay differential equations, Jahresbericht Der Deutschen Mathematiker-Vereinigung, 116 (2014), 87-114.  doi: 10.1365/s13291-014-0086-6.

[16]

J. S. Yu, A note on periodic solutions of the delay differential equation $x^{\prime}(t)=f(x(t-1))$, Proc. Amer. Math. Soc., 141 (2012), 1281-1288.  doi: 10.1090/S0002-9939-2012-11386-3.

show all references

References:
[1]

D. Breda, O. Diekmann, D. Liessi and F. Scarabel, Numerical bifurcation analysis of a class of nonlinear renewal equations, Electron. J. Qual. Theo. Diff. Equ., 2016 (2016), 24 pp. doi: 10.14232/ejqtde.2016.1.65.

[2]

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954.

[3]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. O. Walther, Delay Equations: Functional, Complex and Nonlinear Analysis, Applied Mathematical Sciences, 110. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2.

[4]

P. Dormayer and A. F. Ivanov, Symmetric periodic solutions of a delay differential equation, Dynamical Systems and Differential Equations, Discrete Contin. Dynam. Systems, 1 (1998), 220-230. 

[5]

P. Dormayer and A. F. Ivanov, Stability of symmetric periodic solutions with small amplitude of $x^{\prime}(t) = \alpha f(x(t), x(t-1))$, Discrete Contin. Dynam. Systems, 5 (1999), 61-82.  doi: 10.3934/dcds.1999.5.61.

[6]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[7]

J. L. Kaplan and J. A. Yorke, Ordinary differential equations which yield periodic solutions of differential delay equations, J. Math. Anal. Appl., 48 (1974), 317-324.  doi: 10.1016/0022-247X(74)90162-0.

[8]

B. B. Kennedy, Symmetric periodic solutions for a class of differential delay equations with distributed delay, Electron. J. Qual. Theo. Diff. Equ., 2014 (2014), 18 pp. doi: 10.14232/ejqtde.2014.1.4.

[9]

J. B. Li and X. Z. He, Proof and generalization of Kaplan-Yorke's conjecture under the condition $f^{\prime}(0)>0$ on periodic solution of differential delay equations, Sci. China Ser. A., 42 (1999), 957-964.  doi: 10.1007/BF02880387.

[10]

K. R. Meyer, Jacobi elliptic functions from a dynamical systems point of view, The American Mathematical Monthly, 108 (2001), 729-737.  doi: 10.1080/00029890.2001.11919804.

[11]

R. E. Mickens, Oscillations in Planar Dynamic Systems, Series on Advances in Mathematics for Applied Sciences, 37. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/2778.

[12]

Y. Nakata, An explicit periodic solution of a delay differential equation, J. Dyn. Diff. Equ., 32 (2020), 163-179.  doi: 10.1007/s10884-018-9681-z.

[13]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl. (4), 101 (1974), 263-306.  doi: 10.1007/BF02417109.

[14] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Second edition, Westview Press, Boulder, CO, 2015. 
[15]

H.-O. Walther, Topics in delay differential equations, Jahresbericht Der Deutschen Mathematiker-Vereinigung, 116 (2014), 87-114.  doi: 10.1365/s13291-014-0086-6.

[16]

J. S. Yu, A note on periodic solutions of the delay differential equation $x^{\prime}(t)=f(x(t-1))$, Proc. Amer. Math. Soc., 141 (2012), 1281-1288.  doi: 10.1090/S0002-9939-2012-11386-3.

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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