On discrete flow networks model spectra: statements, simulation, hypotheses

2019

PEN

Traffic flow models on networks are relevant because of exponential growth of road transport in megalopolises worldwide. But there are not enough adequate approaches to describe such processes. Buslaev A.P. with co-authors had introduced networks of contours with common nodes, local particles movement on each contour in given direction and competition resolution rules in common nodes. The problem is to study limit system behavior in dependence on initial conditions. In mathematical models of communication systems and computer networks, particles correspond to messages or information blocks (message packages). Behavior of particles on a contour chain with two cells on each contour is similar to behavior of Ising model used in quantum mechanics. In particular, Ising model is used for modeling of behavior of experimental computers based on principles of quantum mechanics. Contour network models can be used for study of spectral quantization. We study a behavior contour networks of Buslaev type for regular cases depending on rules of competition resolutions in common nodes. Exact results were obtained for closed chain with opposite and oddeven resolution rules, and stochastic version of left-priority rule with non-zero probability of indecisive movement.

Section: Spectrummentioning

confidence: 99%

Analysis of flow dynamics on Buslaev contour networks

2019

PEN

“…In [11] (Buslaev, Fomina, Tatashev, Yashina, 2018) the concept of spectrum of a contour system has been introduced for a deterministic dynamical system with a finite set of states. In such system, a sequence of states repeated periodically from some moment.…”

Section: Introductionmentioning

confidence: 99%

“…This sequence of states is called a spectral cycle. The system, considered in [11], is a closed chain of contours. Particles move on each contour in accordance with the rule of the cellular automaton 240 (CA 240).…”

Section: Introductionmentioning

confidence: 99%

Spectrum of continuous two-contours system

2019

ITM Web Conf.

A deterministic continuous dynamical system is considered. This system contains two contours. The length of the ith contour equals ci, i = 1, 2. There is a moving segment (cluster) on each contour. The length of the cluster, located on the ith contour, equals li , i = 1, 2. If a cluster moves without delays, then the velocity of the cluster is equal to 1. There is a common point (node) of the contours. Clusters cannot cross the node simultaneously, and therefore delays of clusters occur. A set of repeating system states is called a spectral cycle. Spectral cycles and values of average velocities of clusters have been found. The system belongs to a class of contour systems. This class of dynamical systems has been introduced and studied by A.P. Buslaev.

“…20,23 Contours networks with regular periodic structures were studied in Buslaev et al 19,21,22 These contours networks are the closed chain of contours 19,21 and the open chain of contours. 22 It was assumed in Buslaev et al 19,21,22 that the nodes divide each contour into parts of the same length. This paper considers a pair of discrete contours with two nodes, which divide each contour into nonequal parts.…”

mentioning

confidence: 99%

“…In Buslaev et al, 21 a concept of the spectrum has been introduced for a dynamical system. A periodic sequence of states is repeated from some moment.…”

mentioning

confidence: 99%

Spectral cycles and average velocity of clusters in discrete two‐contours system with two nodes

2020

Math Meth Appl Sci

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 class of dynamical systems introduced by A. P. Buslaev. In these systems, the movement of particles (clusters) can be interpreted as the mass transfer on regular networks. A cyclic trajectory in the system state space is called a spectral cycle. The system state space is divided into sets such that any of this set contains the states of the spectral cycle and the nonrecurrent states from which the system results in the set of the spectral cycle states. We have found that, in the general case, what spectral cycle is realized and with what average velocity the clusters move depends on the initial state of the system. We have found the set of spectral cycles and the values of the clusters average velocities taking into account the delays due to that the clusters cannot pass through the same node simultaneously. In one of two considered versions of the system, the clusters move counterclockwise (one‐directional movement). In the other version, one of the clusters moves clockwise and the other cluster moves counterclockwise (codirectional movement). It is turned out that the behavior of system is significantly different in these versions. In particular, in the case of the codirectional movement, the average clusters velocity does not depend on the initial state.