# American Institute of Mathematical Sciences

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

Received  November 2017 Revised  January 2018 Published  April 2018

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.

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
##### References:
 [1] P. Bates and C. K. R. T. Jones, Invariant manifolds for semilinear partial differential equations, Dynamics Reported, 2 (1989), 1-38.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. Lasota and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.   Google Scholar [7] M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289. Google Scholar [8] H.-O. Walther, Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263.   Google Scholar

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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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] A. Lasota and M. Wazewska-Czyzewska, Matematyczne problemy dynamiki ukladu krwinek czerwonych, Matematyka Stosowana, 6 (1976), 23-40.   Google Scholar [7] M. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, New Series, (197) (1977), 286-289. Google Scholar [8] H.-O. Walther, Homoclinic and periodic solutions of scalar differential delay equations, Banach Center Publ., 23 (1989), 243-263.   Google Scholar
Functions from class $\Gamma$
Approximate shape of the solution for $f_{\alpha_{0}}$
Approximate shape of the solution for $f_{\alpha_1}$
Invariant cone
Solution with $\alpha = 0$
Solution with $\alpha = 0.3649$
Solution with $\alpha = 0.340435$
Periodic solution for $\alpha = 0.34182$
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