doi: 10.3934/dcdss.2020392

Existence of a period two solution of a delay differential equation

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.
Citation: Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 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 4.1.  An SSPS attracts two solutions with different initial conditions ($ r = 8 $)
[1]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[2]

Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057

[3]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[4]

H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301

[5]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369

[6]

Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39

[7]

Changrong Zhu, Bin Long. The periodic solutions bifurcated from a homoclinic solution for parabolic differential equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3793-3808. doi: 10.3934/dcdsb.2016121

[8]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031

[9]

Miguel V. S. Frasson, Patricia H. Tacuri. Asymptotic behaviour of solutions to linear neutral delay differential equations with periodic coefficients. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1105-1117. doi: 10.3934/cpaa.2014.13.1105

[10]

Joan Gimeno, Àngel Jorba. Using automatic differentiation to compute periodic orbits of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020130

[11]

Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529

[12]

Vera Ignatenko. Homoclinic and stable periodic solutions for differential delay equations from physiology. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3637-3661. doi: 10.3934/dcds.2018157

[13]

Feng Wang, Jifeng Chu, Zaitao Liang. Prevalence of stable periodic solutions in the forced relativistic pendulum equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4579-4594. doi: 10.3934/dcdsb.2018177

[14]

Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263

[15]

Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2271-2292. doi: 10.3934/dcdsb.2019227

[16]

Benjamin B. Kennedy. A periodic solution with non-simple oscillation for an equation with state-dependent delay and strictly monotonic negative feedback. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : 47-66. doi: 10.3934/dcdss.2020003

[17]

A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373

[18]

Jan Sieber. Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2607-2651. doi: 10.3934/dcds.2012.32.2607

[19]

Jehad O. Alzabut. A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay. Conference Publications, 2007, 2007 (Special) : 35-43. doi: 10.3934/proc.2007.2007.35

[20]

Teresa Faria, José J. Oliveira. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2451-2472. doi: 10.3934/dcdsb.2016055

2019 Impact Factor: 1.233

Article outline

Figures and Tables

[Back to Top]