Discrete closed one-particle chain of contours

P. A. Myshkis ; A. G. Tatashev ; M. V. Yashina

Prikladnaâ diskretnaâ matematika, no. 2 (2021), pp. 114-125.

Résumé

A discrete dynamical system called a closed chain of contours is considered. This system belongs to the class of the contour networks introduced by A. P. Buslaev. The closed chain contains $N$ contours. There are $2m$ cells and a particle at each contour. There are two points on any contour called a node such that each of these points is common for this contour and one of two adjacent contours located on the left and right. The nodes divide each contour into equal parts. At any time $t=0,1,2,\dots$ any particle moves onto a cell forward in the prescribed direction. If two particles simultaneously try to cross the same node, then only the particle of the left contour moves. The time function is introduced, that is equal to $0$ or $1$. This function is called the potential delay of the particle. For $t\ge m$, the equality of this function to $1$ implies that the time before the delay of the particle is not greater than $m$. The sum of all particles potential delays is called the potential of delays. From a certain moment, the states of the system are periodically repeated (limit cycles). Suppose the number of transitions of a particle on the limit cycle is equal to $S(T)$ and the period is equal to $T$. The ratio $S(T)$ to $T$ is called the average velocity of the particle. The following theorem have been proved. 1) The delay potential is a non-increasing function of time, and the delay potential does not change in any limit cycle, and the value of the delay potential is equal to a non-negative integer and does not exceed $ 2N/3$. 2) If the average velocity of particles is less than 1 for a limit cycle, then the period of the cycle (this period may not be minimal) is equal to $(m+1)N$. 3) The average velocity of particles is equal to $v=1-{H}/({(m+1)N})$, where $H$ is the potential of delays on the limit cycle. 4) For any $m$, there exists a value $N$ such that there exists a limit cycle with $H>0$ and, therefore, $v1$.

Détail

Keywords: dynamical system, contour network, limit cycle, potential of delays.

Bibliographie

[1] Kozlov V. V., Buslaev A. P., Tatashev A. G., “On synergy of totally connected flows on chainmails”, Proc. Intern. Conf. CMMSE, v. 3, 2013, 861–874

[2] Buslaev A. P., Tatashev A. G., “Spectra of local cluster flows on open chain of contours”, Europ. J. Pure Appl. Math., 11:3 (2018), 628–641

[3] Blank M. L., “Exact analysis of dynamical systems arising in models of traffic flow”, Russian Math. Surveys, 55:3 (2000), 562–563

[4] Belitsky V., Ferrari P. A., “Invariant measures and convergence properties for cellular automation 184 and related processes”, J. Stat. Phys., 118:3/4 (2005), 589–623

[5] Biham O., Middleton A. A., Levine D., “Self-organization and a dynamic transition in traffic-flow models”, Phys. Rev. A, 46:10 (1992), R6124–R6127

[6] D'Souza R. M., “Coexisting pases and lattice dependence of a cellular automaton model for traffic flow”, Phys. Rev. E, 71:6 (2005), 066112

[7] Angel O., Horloyd A. E., Martin J. B., “The jammed phase of the Biham — Middelton — Levine traffic model”, Elec. Commun. Probability, 10 (2005)

[8] Austin D., Benjamini I., For what number of cars must self organization occur in the Biham — Middleton — Levine traffic model from any possible starting configuration?, 2006, arXiv: math/0607759

[9] Bugaev A. S., Buslaev A. P., Kozlov V. V., Yashina M. V., “Distributed problems of monitoring and modern approaches to traffic modeling”, 14th Intern. IEEE Conf. ITSC, 2011, 477–481

[10] Buslaev A. P., Tatashev A. G., “Spectra of local cluster flows on open chain of contours”, 7th Intern. Conf. ICCMA, 2019, 283–288

[11] Kozlov V. V., Buslaev A. P., Tatashev A. G., “Monotonic walks on a necklace and a coloured dynamic vector”, Int. J. Comput. Math., 92:9 (2015), 1910–1920

[12] Buslaev A. P., Tatashev A. G., Yashina M. V., “Flows spectrum on closed trio of contours”, Europ. J. Pure Appl. Math., 11:1 (2018), 260–283

[13] Buslaev A. P., Fomina M. Yu., Tatashev A. G., Yashina M. V., “On discrete flow networks model spectra: statement, simulation, hypotheses”, J. Physics: Conf. Ser., 1053 (2018), 012034, 1–7

[14] Tatashev A. G., Yashina M. V., “Spectrum of elementary cellular automata and closed chains of contours”, Machines, 7:2 (2019), 28

[15] Zharkova A. V., “On number of inaccessible states in finite dynamic systems of complete graphs orientations”, Prikladnaya Diskretnaya Matematika. Prilozhenie, 2020, no. 13, 100–103 (in Russian)