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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 |
$ x^{\prime}(t) = -f\left(\int_{0}^{1}x(t-s)ds\right), $ |
$ f(x) = r\sin x $ |
$ r>0 $ |
$ 2 $ |
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. |


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