On Discretization of Phase Portraits of Dynamical Systems
УДК 517.938
Abstract
In this paper, we consider the dynamical systems that model the functioning of circular gene networks, in which the rate of the concentration change of any substance monotonically depends on the concentration of the substance preceding this one. Previously, the problems of existence, uniqueness and stability of periodic solutions for dynamical systems, whose equations contain monotonically decreasing and monotonically increasing functions, were studied, and a positively invariant domain was constructed to describe the phase portraits of such systems. In the case of large dimensions, it becomes necessary to narrow the constructed domain. For this, the authors developed an algorithm based on the snake-in-the-box problem from graph theory and applied to dynamical systems of various types and different dimensions. The algorithm allows us to construct a transition diagram corresponding to the shifts along the trajectories of dynamical systems. The examples of using the algorithm in the case of a sixdimensional system with step functions and in the case of a ten-dimensional system with smooth functions have been also given. The results of this construction have been applied in the study of the existence of cycles for these systems.
Downloads
Metrics
References
Hastings S., Tyson J.J., Webster D. Existence of periodic solutions for negative feedbacks cellular control systems // Journal of Differential Equations. 1977. V. 25.
Колесов А.Ю., Розов Н.Х., Садовничий В.А. Периодические решения типа бегущих волн в кольцевых генных сетях // Известия РАН, серия математическая. 2016. Т. 80. № 3.
Глызин С.Д., Колесов А.Ю., Розов Н.Х. Периодические решения типа бегущих волн в кольцевых цепочках однонаправленно связанных уравнений // Теоретическая и математическая физика. 2013. Т. 175. № 1.
Glass L., Pasternack J.S. Stable Oscillations in Mathematical Models of Biological Control Systems // J. of Math. Biology. 1978. V. 6.
Murray J.D. Mathematical biology: I. An introduction, third edition — Springer. 2007.
Голубятников В.П., Кириллова Н.Е. О циклах в моделях функционирования кольцевых генных сетей // Сиб. жур. чистой и прикл. математики. 2018. Т. 18. № 1.
Евдокимов А.А. Цепные коды и snake-in-the-box problem // Ученые записки Казанского университета, физико-математические науки. 2014. Т. 156, кн. 3.
Казанцев М.В. О некоторых свойствах графов доменов динамических систем // Сиб. журнал индустриальной математики. 2015. Т. 18. № 4.
Elowitz M.B., Leibler S. A Synthetic Oscillatory Network of Transcriptional Regulators // Nature. 2000. V. 403.
Bukharina T.A., Furman D.P., Golubyatnikov V.P., Kirillova N.E., et al. Mathematical and numerical models of two asymmetric gene networks // SEMR. 2018. V. 15.
Banks H.T., Mahaffy J.M. Stability of cyclic gene models for systems involving repression // J. theor. Biol. 1978. V. 74.
Аюпова Н.Б., Голубятников В.П., Казанцев М.В. О существовании цикла в одной несимметричной модели молекулярного репрессилятора // Сиб. журнал вычислительной математики. 2017. Т. 20. № 2.
Izvestiya of Altai State University is a golden publisher, as we allow self-archiving, but most importantly we are fully transparent about your rights.
Authors may present and discuss their findings ahead of publication: at biological or scientific conferences, on preprint servers, in public databases, and in blogs, wikis, tweets, and other informal communication channels.
Izvestiya of Altai State University allows authors to deposit manuscripts (currently under review or those for intended submission to Izvestiya of Altai State University) in non-commercial, pre-print servers such as ArXiv.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License (CC BY 4.0) that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
