American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs
  • DCDS-S Home
  • This Issue
  • Next Article
    An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow
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  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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321
[2]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044
[3]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042
[4]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[5]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468
[6]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[7]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[8]

Neil S. Trudinger, Xu-Jia Wang. Quasilinear elliptic equations with signed measure. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 477-494. doi: 10.3934/dcds.2009.23.477
[9]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[10]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[11]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[12]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354
[13]

Jiangtao Yang. Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020371
[14]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264
[15]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050
[16]

Daniele Bartolucci, Changfeng Gui, Yeyao Hu, Aleks Jevnikar, Wen Yang. Mean field equations on tori: Existence and uniqueness of evenly symmetric blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3093-3116. doi: 10.3934/dcds.2020039
[17]

Thazin Aye, Guanyu Shang, Ying Su. On a stage-structured population model in discrete periodic habitat: III. unimodal growth and delay effect. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021005
[18]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037
[19]

Taige Wang, Bing-Yu Zhang. Forced oscillation of viscous Burgers' equation with a time-periodic force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1205-1221. doi: 10.3934/dcdsb.2020160
[20]

Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020052

2019 Impact Factor: 1.233

Tools

Article outline

Figures and Tables

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences