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Homoclinic and stable periodic solutions for differential delay equations from physiology
1. | Justus Liebig University, 35392, Arndtstrasse 2, Giessen, Germany |
2. | National Research University Higher School of Economics, St. Petersburg, Russia |
A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified numerically. Numerically computed solutions are shown.
References:
[1] |
P. Bates and C. K. R. T. Jones,
Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38.
|
[2] |
O. Diekmann, S. M. Verduyn Lunel, S. A. van Gils and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[3] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[4] |
T. Krisztin, H. -O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, American Mathematical Society, Providence, Rhode Island, 1999. |
[5] |
B. Lani-Wayda, Wandering solutions of delay equations with sine-like feedback, Mem. Amer. Math. Soc., 151 (2001), ⅹ+121 pp.
doi: 10.1090/memo/0718. |
[6] |
A. Lasota and M. Wazewska-Czyzewska,
Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.
|
[7] |
M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289. |
[8] |
H.-O. Walther,
Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263.
|
show all references
References:
[1] |
P. Bates and C. K. R. T. Jones,
Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38.
|
[2] |
O. Diekmann, S. M. Verduyn Lunel, S. A. van Gils and H. -O. Walther, Delay Equations Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995.
doi: 10.1007/978-1-4612-4206-2. |
[3] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[4] |
T. Krisztin, H. -O. Walther and J. Wu, Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback, American Mathematical Society, Providence, Rhode Island, 1999. |
[5] |
B. Lani-Wayda, Wandering solutions of delay equations with sine-like feedback, Mem. Amer. Math. Soc., 151 (2001), ⅹ+121 pp.
doi: 10.1090/memo/0718. |
[6] |
A. Lasota and M. Wazewska-Czyzewska,
Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.
|
[7] |
M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289. |
[8] |
H.-O. Walther,
Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263.
|








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