Geometry and Topology of Phase Portraits of Integrable Block-Linear Dynamical Systems

  • В.П. Голубятников Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk, Russia); Novosibirsk Military Institute of National Guard Troops (Novosibirsk, Russia) Email: golubyatn@yandex.ru
  • В.С. Градов Novosibirsk State University (Novosibirsk, Russia) Email: gnets2008@outlook.com
DOI_disc_logo https://doi.org/10.14258/izvasu(2019)1-11
Keywords: block-linear dynamical systems, feedbacks, phase portraits, cycles, invariant domains, integral manifolds, gene networks, Poincare map

Abstract

We consider 3-dimensional block-linear dynamical systems as models of two classes of circular gene networks. The gene networks of the first class are regulated by negative feedbacks only. Networks of the second class are regulated by positive feedbacks. Investigation of principles of functioning of these gene networks allows to control biochemical processes in living organisms. Combinatorial structure of the considered gene networks is a circular one, and we study the case when the rate of change of concentration of any substance in the gene network depends on the concentration of the «previous» substance monotonically. Sufficient conditions of existence and uniqueness of a cycle are elaborated for the first class of these dynamical systems. These systems are integrable, and we construct for each of them an integral manifold containing a cycle. In the case of the systems symmetric with respect to cyclic permutations of the variables, we find a solution of an inverse problem of parameter identification when some information about a period of this cycle is known. For another class of similar blocklinear dynamical systems, we show the non-existence of a cycle which is symmetric to that permutation of the variables.

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Author Biographies

В.П. Голубятников, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences (Novosibirsk, Russia); Novosibirsk Military Institute of National Guard Troops (Novosibirsk, Russia)

доктор физико-математических наук, профессор, главный научный сотрудник

В.С. Градов, Novosibirsk State University (Novosibirsk, Russia)

магистрант механико-математического факультета

References

Elowitz M.B., Leibler S. A Synthetic Oscillatory Network of Transcriptional Regulators Nature. 2000. V. 403.

Glass L., Pasternack J.S. Stable oscillations in mathematical models of biological control systems // Journal of Mathematical Biology. 1978. V. 6.

Hastings S., Tyson J.J., Webster D. Existence of periodic solutions for negative feedbacks cellular control systems // Journal of Differential Equations. 1977. V. 25.

Акиньшин K.K., Голубятников В.П., Голубятников И.В. О некоторых многомерных моделях функционирования генных сетей // Сибирский журнал индустриальной математики. 2013. Т. 16, № 1.

Аюпова Н.Б., Голубятников В.П. О двух классах нелинейных динамических систем. Четырехмерный случай // Сибирский математический журнал. 2015. Т. 56, № 2.

Голубятников В.П., Калёных А.Э. О строении фазовых портретов некоторых нелинейных динамических систем // Сибирский журнал чистой и прикладной математики. 2015. Т. 15, № 1.

Акиньшин А.А., Голубятников В.П. Циклы в симметричных динамических системах // Сибирский журнал чистой и прикладной математики. 2012. Т. 12, № 2.

Gaidov Yu.A., Golubyatnikov V.P., Kleshchev A.G., Lashina E.A., Likhoshvai V.F. Regular and chaotic dynamics in the gene network modeling // Proceedings of 8-th International conference “Human&Computers”, Japan, Aizu-Wakamatsu, 2005.

Гайдов Ю.А., Голубятников В.П. О некоторых нелинейных динамических системах, моделирующих несимметричные генные сети //Сибирский журнал чистой и прикладной математики. 2007. Т. 7, № 2.

Gaidov Yu.A., Golubyatnikov V.P., Kleshchev A.G., Volokitin E.P. Modeling of asymmetric gene networks functioning with different types of regulation. Biophysics. 2006. V. 51, suppl. 1.

Табачников С. Геометрия и биллиарды. Москва ; Ижевск, 2011.
Published
2019-03-06
How to Cite
Голубятников В., Градов В. Geometry and Topology of Phase Portraits of Integrable Block-Linear Dynamical Systems // Izvestiya of Altai State University, 2019, № 1(105). P. 69-74 DOI: 10.14258/izvasu(2019)1-11. URL: http://izvestiya.asu.ru/article/view/%282019%291-11.
Issue
No 1(105) (2019): Izvestiya of Altai State University
Section
Математика и механика

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