Prikladnaya Diskretnaya Matematika
General information
Latest issue
Impact factor

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Prikl. Diskr. Mat.:

Personal entry:
Save password
Forgotten password?

Prikl. Diskr. Mat., 2021, Number 52, Pages 114–125 (Mi pdm742)  

Applied Graph Theory

Discrete closed one-particle chain of contours

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

Moscow Automobile and Road Construction State Technical University (MADI), Moscow, Russia

Abstract: 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 $t=0,1,2,…$ 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, $v<1$.

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


Full text: PDF file (975 kB)
References: PDF file   HTML file

Bibliographic databases:

UDC: 519.7

Citation: P. A. Myshkis, A. G. Tatashev, M. V. Yashina, “Discrete closed one-particle chain of contours”, Prikl. Diskr. Mat., 2021, no. 52, 114–125

Citation in format AMSBIB
\by P.~A.~Myshkis, A.~G.~Tatashev, M.~V.~Yashina
\paper Discrete closed one-particle chain of contours
\jour Prikl. Diskr. Mat.
\yr 2021
\issue 52
\pages 114--125

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Прикладная дискретная математика
    Number of views:
    This page:22
    Full text:5

    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021