Abstract
Spontaneous transitions of traffic flows, from the laminar state to the turbulent state, may occur. In 2008, the experimental results of traffic jam formation on a ring was published by Y. Sugiyama et al. Our paper studies a dynamical system which contains two closed contours with two common points (nodes). The nodes divide each contour into two parts with lengths d and \(1-d.\) In each contour, there is a moving segment called a cluster. The clusters move in accordance with given rules. The system can be intepreted as a traffic model and belongs to the class of dynamical systems introduced by A.P. Buslaev. We study the system spectrum connected with the set of cyclic trajectories in the system state space. Some theorems regarding the spectrum have been proved.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
K. Nagel, M. Schreckenberg, Cellular automation models for freeway traffic. J. Phys. I. 2, Nr. 12, S. 2221–2229 (1992). http://dx.doi.org/10.1051/jp1.1992277
Y. Sugiyama, M. Kikuchi, A.K. Husebe, M.K. Makayama, K. Nishinary, S.T. Minory, Y. Satochi, F. Minory, Traffic jam without bottlenecks—experimental evidence for physical mechanism of the formation of a jam. New J. Phys. 10, S. 2221–2229 (2002). http://dx.doi.org/10.1088/1367-2630/10/303301
V.V. Kozlov, A.P. Buslaev, A.S. Bugaev, M.V. Yashina, A. Schadschneider, M. Schreckenberg, in Traffic and granular flow’11 (Springer, 2013)
M. Shreckenberg, K. Nagel, A. Schadshneide, Discrete stochastic models for traffic flow. In: In Phys. Rev. 51, Nr. 4, S. 2939 (1995). https://doi.org/10.1103/PhysRevE.51.2939
M.L. Blank, Exact analysis of dynamical systems arising in models of traffic flow. Russ. Math. Surv. 55, Nr. 3, S. 562–563 (2000). http://dx.doi.org/10.4213/rm95
V. Belitsky, P.A. Ferrari, Invariant measures and convergence properties for cellular automation 184 and related processes. J. Stat. Phys. 118, Nr. 3, S. 589–623 (2005). http://dx.doi.org/10.1007/s10955-044-8822-4
L. Gray, D. Griffeath, The ergodic theory of traffic jams. J. Stat. Phys. 105, Nr. 3/4, S. 413–452 (2001). http://dx.doi.org/10.1023/A:1012202706850
A.S. Bugaev, A.P. Buslaev, V.V. Kozlov, M.V. Yashina, Distributed problems of monitoring and modern approaches to traffic modeling. In: 14th international IEEE conference on intelligent transactions systems (ITSC 2011), Washington, USA, S. 477–481 (2011). http://dx.doi.org/10.1109/ITSC.2011.6082805
V.V. Kozlov, A.P. Buslaev, A.G. Tatashev, On synergy of totally connected flow on chainmails, in: Proceedings of the 13 International Conference on Computational and Mathematical Methods in Science and Engineering, Almeria, Spain, vol. 3, S. 861–874 (2013)
A.P. Buslaev, M.Y. Fomina, A.G. Tatashev, M.V. Yashina, On discrete flow networks model spectra: statement, simulation, hypotheses. J. Phys.: Conf. Ser. 1053, Nr. 1, S. 012034 (2018). http://dx.doi.org/10.1088/1742/6596/1053/1/012034
O. Biham, A.A. Middleton, D. Levine, Self-organization and a dynamical transition in traffic-flow models. Phys. Rev. A. 71, Nr. 6, S. R6124–R6127 (1992). http://dx.doi.org/10.1003/PhysRevA.46.R6124
T. Austin, I. Benjamini, 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
A.P. Buslaev, A.G. Tatashev, M.V. Yashina, Flows spectrum on closed trio of contours. Eur. J. Pure Appl. Math. 11, Nr. 3, S. 893–897 (2018). http://dx.doi.org/10.29020/nybg.ejpam.v11i1.3201
A.G. Tatashev, M.V. Yashina, Behavior of continuous two-contours system. WSEAS Trans. Math. 18, Nr. 5, S. 28–36 (2019)
M. Yashina, A. Tatashev, Spectral cycles and average velocity of clusters in discrete two-contours system with two nodes, in Mathematical Methods in the Applied Sciences First published 04 February 2020, S. 1–28 (2020). http://dx.doi.org/10.1088/1742-6596/1324/1/012001
Acknowledgements
This work has been supported by the RFBR, Grant No. 20-01-00222.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Yashina, M.V., Tatashev, A.G. (2020). Uniform Cluster Traffic Model on Closed Two-Contours System with Two Non-symmetrical Common Nodes. In: Zuriguel, I., Garcimartin, A., Cruz, R. (eds) Traffic and Granular Flow 2019. Springer Proceedings in Physics, vol 252. Springer, Cham. https://doi.org/10.1007/978-3-030-55973-1_71
Download citation
DOI: https://doi.org/10.1007/978-3-030-55973-1_71
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-55972-4
Online ISBN: 978-3-030-55973-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)