Homoclinic and stable periodic solutions for differential delay equations from physiology

Vera Ignatenko

doi:10.3934/dcds.2018157

Article Contents

2018, Volume 38, Issue 7: 3637-3661. Doi: 10.3934/dcds.2018157

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Homoclinic and stable periodic solutions for differential delay equations from physiology

Vera Ignatenko^1,2,

1.
Justus Liebig University, 35392, Arndtstrasse 2, Giessen, Germany
2.
National Research University Higher School of Economics, St. Petersburg, Russia

Received: October 31, 2017

Revised: December 31, 2017

Published: June 2018

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K18, 34K13; Secondary: 37N25.

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Figure 1. Functions from class $\Gamma$

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Figure 2. Approximate shape of the solution for $f_{\alpha_{0}}$

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Figure 3. Approximate shape of the solution for $f_{\alpha_1}$

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Figure 4. Invariant cone

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Figure 5. Solution with $\alpha = 0$

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Figure 6. Solution with $\alpha = 0.3649$

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Figure 7. Solution with $\alpha = 0.340435$

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Figure 8. Periodic solution for $\alpha = 0.34182$

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References

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