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The paper considers a discrete dynamical system containing two contours. There are n cells and m particles in each contour. At any time, the particles of each contour form a cluster. There are two common points of the contours. These common points are called nodes. The nodes divide the contours into two nonequal parts. The system belongs to the cla...
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Citations
... In 19-34 , the limit cycles and values of average velocities, depending of the initial state of the system, are obtained for some deterministic systems with two contours and contour networks with regular structures. For the contour networks that were considered in [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] , both the state space and time scale are discrete, both the state space and time scale are continuous. States of a deterministic discrete contour network are repeated from a moment. ...
A dynamical system is studied. This system belongs to the class of contour networks introduced by A.P. Buslaev. Earlier, contour networks were considered such that the system state space and time were continuous, or the system space and time were discrete. This paper considers a contour network with continuous state space and discrete time. The system is a version of the system called an open chain of contours. The version of open chain of contours introduced in this paper is a generalization of a dynamical system called circumference shifts, which was considered by A. Katok and B. Hasselblatt. We have obtained the limit distribution of the system states.
... The main problems include finding the average velocity of motion of the clusters (particles), finding conditions for the system to enter a state of free-flow traffic (from some time instant on, all clusters move without delays at the current instant and in the future) or a state of traffic breakdown (the motion of the particles ceases completely), and choosing a rule of competition settlement that is best-suited to maximize the average velocity of motion. Analytical results were obtained for two-contour systems with one [12][13][14][15] or two [16][17][18][19] nodes, for systems with several contours and one common node [20] and for contour networks with regular periodic one-dimensional [21][22][23][24] or two-dimensional [25][26][27] contour systems. ...
... In [16] we investigated a contour network containing two contours on each of which there is one cluster. The length of the contour and the length of the cluster located on it do not depend on the contour. ...
... In this version, the system is called a system with one-directional motion (Fig. 2), since in its geometric interpretation either both clusters move counterclockwise or both clusters move clockwise. In [16], we considered a system with the left-priority rule of competition settlement, according to which, during competition, priority is always given to the same cluster, which is located on the left. The main characteristic under study is the average velocity of motion, which is equal to the average distance traveled by the cluster in unit time, with delays taken into account. ...
A system belonging to the class of dynamical systems such as Buslaev contour networks is investigated. On each of the two closed contours of the system there is a segment, called a cluster, which moves with constant velocity if there are no delays. The contours have two common points called nodes. Delays in the motion of the clusters are due to the fact that two clusters cannot pass through a node simultaneously. The main characteristic we focus on is the average velocity of the clusters with delays taken into account. The contours have the same length, taken to be unity. The nodes divide each contour into parts one of which has length d, and the other, length 1-d. Previously, this system was investigated under the assumption that the clusters have the same length. It turned out that the behavior of the system depends qualitatively on how the directions of motion of the clusters correlate with each other. In this paper we explore the behavior of the system in the case where the clusters differ in length.
... To obtain analytical results, networks with regular structures periodic structure were considered (in particular, closed and open chains of contours, [13] - [15]), two contour systems, [16]- [19], systems with one common nodethe flower, [20]. ...
... In [19], it was shown how the spectrum of the system varies depending on the direction of motion. This article explores the effect of competition resolution rules on the spectrum of a system. ...
... In the case of the left-priority rule, the statement 3 is proved in [19]. If no competition occurs, a spectral cycle is realized such that this spectral cycle does not depend on the competition resolution rule. ...
p>In computer networks based on the principle of packet switching, the important transmitting function is to maintain packet queues and suppress congestion. Therefore, the problems of optimal control of the communication networks are relevant. For example, there are users, and no more than a demand of one user can be served simultaneously. This paper considers a discrete dynamical system with two contours and two common points of the contours called the nodes . There are n cells and particles, located in the cells. At any discrete moment the particles of each contour occupy neighboring cells and form a cluster. The nodes divide each contour into two parts of length and (non-symmetrical system). The particles move in accordance with rule of the elementary cellular automaton 240 in the Wolfram classification. Delays in the particle movement are due to that more than one particle cannot move through the node simultaneously. A competition (conflict) occurs when two clusters come to the same node simultaneously. We have proved that the spectrum of velocities contains no more than two values for any fixed n , d and l . We have found an optimal rule which minimizes the average velocity of clusters. One of the competition clusters passes through the node first in accordance with a given competition rule. Two competition resolutions rules are introduced. The rules are called input priority and output priority resolution rules. These rules are Markovian, i.e., they takes into account only the present state of the system. For each set of parameters n, d and l , one of these two rules is optimal, i.e., this rule maximizes the average velocity of clusters. These rules are compared with the left-priority resolution rule, which was considered earlier. We have proved that the spectrum of velocities contains no more two values for any fixed n, l, and d . We have proved that the input priority rule is optimal if , and the output priority rule is optimal if .</p
Исследуется динамическая система типа бинарной цепочки Буслаева. Система содержит $N$ контуров. На каждом контуре имеются две ячейки и одна частица. Для каждого контура имеется по одной общей точке, называмой узлом, с каждым из двух соседних контуров. В детерминированном варианте системы в любой дискретный момент времени каждая частица перемещается в другую ячейку, если нет задержки. Задержки обусловлены тем, что две частицы не могут проходить через узел одновременно. Если две частицы стремятся пересечь один и тот же узел, то перемещается только одна частица в соответствии с заданным правилом разрешения конкуренции. В стохастическом варианте частица стремится переместиться, если система находится в состоянии, соответствующем состоянию детерминированной системы, в котором частица перемещается. Эта попытка реализуется в соответствующей системе с вероятностью $1-\varepsilon,$ где $\varepsilon$~--- малая величина. Получено правило разрешения конкуренции, называемое правилом длинного кластера. Это правило переводит систему в такое состояние, что все частицы перемещаются без задержек в настоящий момент и в будущем (состояние свободного движения), причем система попадает в состояние движения за минимальное возможное время. Среднее число $v_i$ перемещений частицы $i$-го контура в единицу времени называется средней скоростью этой частицы, $i=1,\dots,N.$ В предположении, что $N=3,$ для стохастического варианта системы получены следующие результаты. Для правила длинного кластера получена следующая формула для средней скорости частиц: $v_1=v_2=v_3=1-2\varepsilon+o(\varepsilon)$ $(\varepsilon\to 0).$ Для левоприоритетного правила, в соответствии с которым при конкуренции приоритет имеет частица контура с меньшим номером, для средней скорости частиц получена следующая формула: $v_1=v_2=v_3=\frac{6}{7}+o(\sqrt{\varepsilon}).
Исследуется динамическая система с непрерывным временем и непрерывным пространством состояний. Система относится к классу контурных сетей Буслаева. Контурные сети могут использоваться при моделировании трафика на сложных сетях, а также иметь другие приложения, в частности, применяться при моделировании коммуникационных систем. Рассматриваемая система содержит замкнутую последовательность контуров, каждый из которых имеет по две симметрично расположенные общие точки, называемые узлами, с соседними контурами. На каждом контуре движется с постоянной скоростью отрезок, называемый кластером. Это название объясняется тем, что в дискретном варианте транспортной модели такому отрезку соответствует группа частиц, располагающихся в соседних ячейках и перемещающихся одновременно, причем каждая частица соответствует автотранспортному средству. Задержки в перемещении кластеров обусловлены невозможностью одновременного прохождения двух кластеров через общий узел. Динамика системы такова, что с некоторого момента времени периодически повторяются состояния, принадлежащие некоторому множеству (предельный цикл). Каждому предельному циклу соответствуют значение средней скорости кластеров. Это значение в общем случае зависит от начального состояния. Исследуется поведение системы на предельных циклах. Получены результаты о характере поведения рассматриваемой системы на предельном цикле, о значении периода цикла, о поведении функции от состояния, называемой потенциалом задержек. Найдены возможные значения средней скорости кластеров при известных значениях числа контуров и длины кластера. Получены достаточные условия существования предельных циклов при малых длинах кластеров с наличием задержек в движении. Доказана теорема о непрерывной замкнутой цепочке контуров с длиной, равной половине длины контуров. Доказательство этой теоремы основано на сравнении поведения данной системы с поведением рассматривавшейся ранее дискретной системы, называемой бинарной замкнутой цепочкой контуров.
