Qualitative properties of dynamical system on toroidal chainmail

Journal of Computational and Applied Mathematics

2015

a b s t r a c tA dynamical system is introduced and investigated. The system contains N vertices. The vertices send messages at discrete time instants according to a given rule. A conflict of two vertices takes place if the vertices try to send messages to each other at the same instant. Each vertex sends a message to another vertex at every step if no conflict takes place. In case of a conflict, only one of the two competing vertices sends a message. Deterministic and stochastic conflict resolution rules are considered. We investigate the average number of messages sent by a vertex per a time unit, called the productivity of this vertex, the total productivity of the system and other characteristics. The productivity of vertices depends on the initial state of the system, and the criterion of efficiency is the expected average productivity of vertices provided all possible initial states of the system are equiprobable. An ergodic version of the system is also considered in which any particle moves with approximately equal to 1 probability provided there is no conflict.

Section: Topological Characterization Of the Networkmentioning

confidence: 99%

“…The considered problem is a generalization of the following transport construction, [1][2][3]. Each vertex can be interpreted as a contour, and there are arranged cells on the contour.…”

Section: Transport Analogymentioning

confidence: 99%

A dynamical communication system on a network

Kozlov

Buslaev

Journal of Computational and Applied Mathematics

2015

“…From (2) we have that at each time unit any cluster covers at least one node. As the number of contours is equal to nodes number, then for any system state no cluster can cover two nodes simultaneously.…”

Section: Proofmentioning

confidence: 99%

On Flows Spectrum on Closed Trio of Contours with Uniform Load

Buslaev¹,

Eur. J. Pure Appl. Math.

Yashina

2018

Abstract. Considered dynamical system is a flow of clusters with the same length l on contours of unit length connected in polar-remote points into closed chain. When clusters move through common node, the left-priority rule of competition resolution works. In the paper it is shown that, in the case of chain containing three contours, the dynamical system has a spectrum of velocity and mode periodicity consisted of not more than two components. Spectrum distribution in dependence on load l is developed. Hypotheses on discrete spectrum in the case of arbitrary number of contours are formulated.

“…From (5)- (11), the formula (3) follows. From (5)-(10), (12), the formulas (4), (5) follows. Theorem 1 has been proved.…”

Section: Exact Solution For the Case Of M = 2 And Even Nmentioning

confidence: 99%

“…One of contour network called chainmail was studied in [11], [12], (Buslaev, Tatashev, Yashina, 2013). In this dynamical system, any particles move on a contour.…”

Section: Introductionmentioning

confidence: 99%

A model of particles movement on a discrete contour

Yashina

2019

PEN

We study an exclusive process on a circle. In this paper, we study a discrete closed contour, containing N cells and M particles. Each particle occupies a cell at every time. There is not more than one particle in each cell at every moment. At every time t=0,1, 2, … , each particle tries to move onto a cell forward with probability p, this particle tries to move back with probability q, and the particle does not try to move with probability s, p+q+s=1. Under assumptions that q=0, the system of this type was considered by M. Kanai et. al. As it follows from results of these authors, in the case q=0, the process is time reversible, i.e., in the stationary state, the behavior of process does not change if the direction of time-axis is changed. The ergodic properties of some more general exclusive process were studied by M. Blank but, in the case 00, or N is odd and s=0. We have proved that the process can be nonreversible if M ≥ 3, s=0.