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Chaos in a dynamic model of urban transportation network flow based on user equilibrium states

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  • Xu, Meng
  • Gao, Ziyou

Abstract

In this study, we investigate the dynamical behavior of network traffic flow. We first build a two-stage mathematical model to analyze the complex behavior of network flow, a dynamical model, which is based on the dynamical gravity model proposed by Dendrinos and Sonis [Dendrinos DS, Sonis M. Chaos and social-spatial dynamic. Berlin: Springer-Verlag; 1990] is used to estimate the number of trips. Considering the fact that the Origin–Destination (O–D) trip cost in the traffic network is hard to express as a functional form, in the second stage, the user equilibrium network assignment model was used to estimate the trip cost, which is the minimum cost of used path when user equilibrium (UE) conditions are satisfied. It is important to use UE to estimate the O–D cost, since a connection is built among link flow, path flow, and O–D flow. The dynamical model describes the variations of O–D flows over discrete time periods, such as each day and each week. It is shown that even in a system with dimensions equal to two, chaos phenomenon still exists. A “Chaos Propagation” phenomenon is found in the given model.

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  • Xu, Meng & Gao, Ziyou, 2009. "Chaos in a dynamic model of urban transportation network flow based on user equilibrium states," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 586-598.
    • Handle: RePEc:eee:chsofr:v:39:y:2009:i:2:p:586-598
    DOI: 10.1016/j.chaos.2007.01.077
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    References listed on IDEAS

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    1. Michael J. Smith, 1984. "The Stability of a Dynamic Model of Traffic Assignment---An Application of a Method of Lyapunov," Transportation Science, INFORMS, vol. 18(3), pages 245-252, August.
    2. Anna Nagurney & Ding Zhang, 1997. "Projected Dynamical Systems in the Formulation, Stability Analysis, and Computation of Fixed-Demand Traffic Network Equilibria," Transportation Science, INFORMS, vol. 31(2), pages 147-158, May.
    3. D. Zhang & A. Nagurney, 1997. "Formulation, Stability, and Computation of Traffic Network Equilibria as Projected Dynamical Systems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 417-444, May.
    4. Horowitz, Joel L., 1984. "The stability of stochastic equilibrium in a two-link transportation network," Transportation Research Part B: Methodological, Elsevier, vol. 18(1), pages 13-28, February.
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    1. Guo, Ren-Yong & Huang, Hai-Jun, 2009. "Chaos and bifurcation in dynamical evolution process of traffic assignment with flow “mutation”," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1150-1157.

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