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Model. Anal. Inform. Sist., 2016, Volume 23, Number 6, Pages 784–803 (Mi mais541)  

Bifurcation of periodic solutions of the Mackey–Glass equation

E. P. Kubyshkin, A. R. Moryakova

P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia

Abstract: We study the bifurcation of the equilibrium states of periodic solutions for the Mackey–Glass equation. This equation is considered as a mathematical model of changes in the density of white blood cells. The equation written in dimensionless variables contains a small parameter at the derivative, which makes it singular. We applied the method of uniform normalization, which allows us to reduce the study of the solutions behavior in the neighborhood of the equilibrium state to the analysis of the countable system of ordinary differential equations. We poot out the equations in “fast” and “slow” variables from this system. Equilibrium states of the “slow” variables equations determine the periodic solutions. The analysis of equilibrium states allows us to study the bifurcation of periodic solutions depending on the parameters of the equation and their stability. The possibility of simultaneous bifurcation of a large number of stable periodic solutions is shown. This situation is called the multistability phenomenon.

Keywords: the Mackey–Glass equation, periodic solutions, multistability bifurcation.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 984
This research was supported by project № 984 within the base part of state assignment on research in YarSU.


DOI: https://doi.org/10.18255/1818-1015-2016-6-784-803

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Bibliographic databases:

UDC: 517.994
Received: 15.03.2016

Citation: E. P. Kubyshkin, A. R. Moryakova, “Bifurcation of periodic solutions of the Mackey–Glass equation”, Model. Anal. Inform. Sist., 23:6 (2016), 784–803

Citation in format AMSBIB
\Bibitem{KubMor16}
\by E.~P.~Kubyshkin, A.~R.~Moryakova
\paper Bifurcation of periodic solutions of the Mackey--Glass equation
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 6
\pages 784--803
\mathnet{http://mi.mathnet.ru/mais541}
\crossref{https://doi.org/10.18255/1818-1015-2016-6-784-803}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3596162}
\elib{http://elibrary.ru/item.asp?id=27517424}


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