Dynamical Systems on Honeycombs

  • Valery V. Kozlov
  • Alexander P. Buslaev
  • Alexander G. Tatashev
  • Marina V. Yashina
Conference paper


Stochastic and deterministic versions of a discrete dynamical system on a necklace network are investigated. This network contains several contours. There are three cells and a particle on each contour. The particle occupies one of the cells and, at each step, it makes an attempt to move to the next cell in the direction of movement. As well as on neighboring contours the particles move in accordance with rules of stochastic or deterministic type. We prove that the behavior of the model with a rule of the first type is stochastic only at the beginning, and after a time interval the behavior becomes purely deterministic. The system with a rule of the first type reaches a stationary mode which depends on the initial state. The average velocity of particles and other characteristics of the dynamical systems are studied.


  1. 1.
    M. Kac, Probability and Related Topics in Physical Sciences (Interscience Publishers, New York/London, 1958)Google Scholar
  2. 2.
    V. Kozlov, Statistical irreversibility of the Kac reversible circular model. Rus. J. Nonlinear Dyn. 7, 101–117 (2011)Google Scholar
  3. 3.
    K. Nagel, M. Schreckenberg, A cellular automaton model for freeway traffic. J. Phys. I Fr. 2, 1221 (1992)CrossRefGoogle Scholar
  4. 4.
    A. Schadschneider, M. Schreckenberg, Cellular automaton models and traffic flow. J. Phys. A26, L679 (1993)Google Scholar
  5. 5.
    M. Schreckenberg, A. Schadschneider, K. Nagel, N. Ito, Discrete stochastic models for traffic flow. Phys. Rev. E 51, 2939 (1995)CrossRefGoogle Scholar
  6. 6.
    M. Rickert, K. Nagel, M. Schreckenberg, A. Latour, Two lane traffic simulations using cellular automata. Physica A 231, 534 (1996)CrossRefGoogle Scholar
  7. 7.
    K. Nagel, D. Wolf, P. Wagner, P. Simon, Two-lane traffic rules for cellular automata: a systematic approach. Phys. Rev. E 58, 1425 (1998)CrossRefGoogle Scholar
  8. 8.
    P. Simon, K. Nagel, A simplified cellular automation model for city traffic. Phys. Rev. E 58, 1286 (1998)CrossRefGoogle Scholar
  9. 9.
    M.V. Evans, N. Rajewsky, E.R. Speer, Exact solution of a cellular automation for traffic. J. Stat. Phys. 95, 45 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    L. Gray, D. Griffeath, The ergodic theory of traffic jams. J. Stat. Phys. 105, 413 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    M. Blank, Exact analysis of dynamical systems arising in models of traffic flow. Russ. Math. Surv. 55(3), 333 (2000)Google Scholar
  12. 12.
    M. Blank, Dynamics of traffic jams: order and chaos. Mosc. Math. J. 1(1), 1–26 (2001)zbMATHMathSciNetGoogle Scholar
  13. 13.
    A.P. Buslaev, A.G. Tatashev, Particles flow on the regular polygon. J. Concr. Appl. Math. 9, 290 (2011)zbMATHMathSciNetGoogle Scholar
  14. 14.
    A.P. Buslaev, A.G. Tatashev, Monotonic random walk on a one-dimensional lattice. J. Concr. Appl. Math. 10, 130 (2012)zbMATHMathSciNetGoogle Scholar
  15. 15.
    A.P. Buslaev, A.G. Tatashev, On exact values of monotonic random walks characteristics on lattices. J. Concr. Appl. Math. 11, 17 (2013)zbMATHMathSciNetGoogle Scholar
  16. 16.
    A.P. Buslaev, A.G. Tatashev, M.V. Yashina, Cluster flow models and properties of appropriate dynamic systems. J. Appl. Funct. Anal. 8, 54–76 (2013)zbMATHMathSciNetGoogle Scholar
  17. 17.
    A.S. Bugaev, A.P. Buslaev, V.V. Kozlov, A.G. Tatashev, M.V. Yashina, Traffic modelling: monotonic random walks on a network. Matematicheskoe modelirovanie 25, 3 (2013)MathSciNetGoogle Scholar
  18. 18.
    V.V. Kozlov, A.P. Buslaev, A.G. Tatashev, Monotonic Random Walks and Clusters Flows on Networks. Models and Applications (Lambert Academician Publishing, Saarbrücken, 2013). No. 78724, ISBN:978-3-659-33987-5Google Scholar
  19. 19.
    H. Daduna, Queuing Networks with Discrete Time Scale: Explicit Expression for the Steady State Behavior of Discrete Time Stochastic Networks (Springer, Berlin, 2001)CrossRefGoogle Scholar
  20. 20.
    A. Feller, An Introduction to Probability Theory and Its Applications (Wiley, New York, 1968)zbMATHGoogle Scholar
  21. 21.
    K.K. Glukharev, N.M. Ulyukov, A.M. Valuev, I.N. Kalinin, On traffic flow on the arterial network model, in Traffic and Granular Flow ’11, ed. by V.V. Kozlov, A.P. Buslaev, A.S. Bugaev, M.V. Yashina, A. Schadschneider, M. Schreckenberg (Springer, Berlin/Heidelberg, 2013), pp. 399–411. ISBN: 978-3-642-39668-7 (Print), 978-3-642-39669-4 (Online),
  22. 22.
    V.V. Kozlov, A.P. Buslaev, A.G. Tatashev, On synergy of total connected flows on chainmails, in The 13th International Conferences of Computational and Mathematical Methods in Science and Engineering, CMMSE 2013. Proceedings, Cabo de Gata, Almeria, Spain, vol. III, 24–27 June 2013, pp. 861–874. ISBN: 978-84-616-2723-3,
  23. 23.
    A.P. Buslaev, A.G. Tatashev, A system of pendulums on a regular polygon, in The Fifth International Conference on Advances in System Simulation, 2013, Proceedings, Venice, p. 36. ISBN:978-1-61208-308-7Google Scholar
  24. 24.
    A.P. Buslaev, A.G. Tatashev, M.V. Yashina, Qualitative properties of dynamical system on toroidal chainmail. AIP Conf. Proc. 1558, 1144 (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Valery V. Kozlov
    • 1
  • Alexander P. Buslaev
    • 2
  • Alexander G. Tatashev
    • 2
  • Marina V. Yashina
    • 2
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.MADIMTUCIMoscowRussia

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