Abstract:
A dynamical system with continuous time and continuous state space is studied. The system belongs to the class of Buslaev contour networks. Contour networks can be used to
simulate of traffic on complex networks, as well as have other applications, in particular,
to be used in modeling communication systems. Considered system contains a closed sequence of contours, each of which has two symmetrically located common points, called
nodes, with adjacent contours. There is a segment on each contour. It is called a cluster
and moves at a constant speed. This title is explained by the fact that in the discrete version of the transport model, such a segment corresponds to a group of particles located in
adjacent cells and moving simultaneously, and each particle corresponds to a vehicle.
Delays of clusters moving are caused by the impossibility of simultaneous passage of
two clusters through a common node. The dynamics of the system is such that, from a
certain moment in time, the states are belonging to a certain set (limit cycle) are periodically repeated. Every limit cycle corresponds to the value of the average cluster velocity.
A value depends on the initial state in general case. System behavior on limit cycles is
developed in dependence on initial conditions. Results are obtained on the nature of the
behavior of the system under consideration at the limit cycle, on the value of the cycle
period, on the behavior of the function of the state, called the delay potential. The possible values of the average velocity of the clusters are obtained for the prescribed values of
the number of contours and the cluster length. Sufficient conditions for the existence of
limit cycles for small cluster lengths with delays in motion are obtained.
Citation:
A. S. Bugaev, A. G. Tatashev, M. V. Yashina, “Spectrum of a continuous closed symmetric chain with an arbitrary number of contours”, Mat. Model., 33:4 (2021), 21–44; Math. Models Comput. Simul., 13:6 (2021), 1014–1027
\Bibitem{BugTatYas21}
\by A.~S.~Bugaev, A.~G.~Tatashev, M.~V.~Yashina
\paper Spectrum of a continuous closed symmetric chain with an arbitrary number of contours
\jour Mat. Model.
\yr 2021
\vol 33
\issue 4
\pages 21--44
\mathnet{http://mi.mathnet.ru/mm4277}
\crossref{https://doi.org/10.20948/mm-2021-04-02}
\transl
\jour Math. Models Comput. Simul.
\yr 2021
\vol 13
\issue 6
\pages 1014--1027
\crossref{https://doi.org/10.1134/S207004822106003X}
Linking options:
https://www.mathnet.ru/eng/mm4277
https://www.mathnet.ru/eng/mm/v33/i4/p21
This publication is cited in the following 6 articles:
M. V. Yashina, A. G. Tatashev, “Dvukhkonturnaya sistema s razlichnymi po dline klasterami i neodinakovym raspolozheniem dvukh uzlov na konturakh”, Kompyuternye issledovaniya i modelirovanie, 16:1 (2024), 217–240
A. G. Tatashev, M. V. Yashina, “Optimalnoe pravilo razresheniya konkurentsii dlya upravlyaemoi binarnoi tsepochki”, Vladikavk. matem. zhurn., 26:1 (2024), 142–153
Marina A. Trapeznikova, Marina V. Yashina, Alla G. Garibyan, 2024 Intelligent Technologies and Electronic Devices in Vehicle and Road Transport Complex (TIRVED), 2024, 1
A. S. Bugaev, M. V. Yashina, A. G. Tatashev, “O skorosti potoka na regulyarnoi neodnorodnoi otkrytoi odnomernoi seti s nesimmetrichnym raspolozheniem uzlov”, Avtomat. i telemekh., 2023, no. 9, 106–119
A. G. Garibyan, 2023 Intelligent Technologies and Electronic Devices in Vehicle and Road Transport Complex (TIRVED), 2023, 1
A. S. Bugaev, M. V. Yashina, A. G. Tatashev, “Velocity of Flow on Regular Non-Homogeneous Open One-Dimensional Net with Non-Symmetrical Arrangement of Nodes”, Autom Remote Control, 84:9 (2023), 974
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