Phase Plane Analysis of Traffic Phenomena with Different Input and Output Conditions

  • WenHuan AiEmail author
  • YuHang Su
  • Tao Xing
  • DaWei Liu
  • Huifang Ma
Conference paper
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 581)


In this paper, the traffic flow problem is converted into a system stability problem through variable substitution and a phase plane analysis method is presented for analyzing the complex nonlinear traffic phenomena. This method matches traffic congestion with the unstable system. So these theories and methods of stability can be applied directly to solve the traffic problem. Based on an anisotropic continuum model developed by Gupta and Katiyar (GK model), this paper uses this new method to describe various nonlinear phenomena due to different input and output conditions on ramps which were rarely studied in the past. The results show that the traffic phenomena described by the new method is consistent with that described by traditional methods. Moreover, the phase plane diagram highlights the unstable traffic phenomena we are chiefly concerned about and describes the variation of density or velocity with time or sections more clearly.


Phase plane diagrams Nonlinear traffic phenomena Stability analysis Ramps 



The authors would like to thank of the anonymous referees and the editor for their valuable opinions. This work is partially supported by the National Natural Science Foundation of China under the Grant No. 61863032 and the China Postdoctoral Science Foundation Funded Project (Project No.: 2018M633653XB) and the “Qizhi” Personnel Training Support Project of Lanzhou Institute of Technology (Grant No. 2018QZ-11) and the National Natural Science Foundation of China under the Grant No. 11965019.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Liu, D.W., Shi, Z.K., Ai, W.H.: Enhanced stability of car-following model upon incorporation of short-term driving memory. Commun. Nonlinear Sci. Numer. Simul. 47, 139–150 (2016)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Liu, D.W., Shi, Z.K., Ai, W.H.: A new car-following model accounting for impact of strong wind. Math. Probl. Eng. 1, 1–12 (2017)Google Scholar
  3. 3.
    Liu, D.W., Shi, Z.K., Ai, W.H.: An improved car-following model from the perspective of driver’s forecast behavior. Int. J. Mod. Phys. C 28(4), 1750046 (2017)CrossRefGoogle Scholar
  4. 4.
    Lighthill, M.J., Whitham, G.B.: On kinematic waves: II. a theory of traffic flow on long crowed roads. In: Proceedings of the Royal Society of London, vol. 229, pp. 317–345 (1955)Google Scholar
  5. 5.
    Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ai, W.H., Shi, Z.K., Liu, D.W.: Bifurcation analysis of a speed gradient continuum traffic flow model. Phys. A: Stat. Mech. Appl. 437, 418–429 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ai, W.H., Shi, Z.K., Liu, D.W.: Bifurcation analysis method of nonlinear traffic phenomena. Int. J. Mod. Phys. C 26, 1550111 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ma, C.X., He, R.C.: Green wave traffic control system optimization based on adaptive genetic-artificial fish swarm algorithm. Neural Comput. Appl. 31(7), 2073–2083 (2019)CrossRefGoogle Scholar
  9. 9.
    Yang, B., Kostkova, J., Flusser, J., Suk, T., Bujack, R.: Rotation invariants of vector fields from orthogonal moments. Pattern Recogn. 74, 110–121 (2018)CrossRefGoogle Scholar
  10. 10.
    Gupta, A.K., Katiyar, V.K.: Analyses of shock waves and jams in traffic flow. J. Phys. A. 38, 4069–4083 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Shi, Z.K., Ai, W.H., Liu, D.W.: An improved macro model of traffic flow with the consideration of ramps and numerical tests. Math. Probl. Eng. 2015, 1–13 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2020

Authors and Affiliations

  • WenHuan Ai
    • 1
    Email author
  • YuHang Su
    • 1
  • Tao Xing
    • 1
  • DaWei Liu
    • 2
  • Huifang Ma
    • 1
  1. 1.College of Computer Science and EngineeringNorthwest Normal UniversityLanzhouChina
  2. 2.College of Electrical EngineeringLanzhou Institute of TechnologyLanzhouChina

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