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 and Continuous Dynamical Systems, 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. 

[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 1.  Functions from class $\Gamma$
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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