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July
2018, 38(7): 3637-3661.
doi: 10.3934/dcds.2018157
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.
Keywords:
Differential delay equation,
homoclinic solution,
stable periodic solution,
global behavior,
Mackey-Glass type equation.
Mathematics Subject Classification:
Primary: 34K18, 34K13; Secondary: 37N25.
Citation:
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
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References:
| [1] |
P. Bates and C. K. R. T. Jones,
Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38.
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| [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.
|
Figure 2.
Approximate shape of the solution for $f_{\alpha_{0}}$
Figure 3.
Approximate shape of the solution for $f_{\alpha_1}$
Figure 4.
Invariant cone
Figure 5.
Solution with $\alpha = 0$
Figure 6.
Solution with $\alpha = 0.3649$
Figure 7.
Solution with $\alpha = 0.340435$
Figure 8.
Periodic solution for $\alpha = 0.34182$
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