Traveling Waves, Catastrophes and Bifurcations in a Generic Second Order Traffic Flow Model

Carrillo, Francisco A.; Delgado, Joaquín; Saavedra, Patricia; Velasco, R. M.; Verduzco, Fernando

doi:10.1142/s0218127413501915

Cited by 10 publications

(9 citation statements)

References 14 publications

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“…The stability of the critical points is given in the following proposition [2] . • If v e (v c ) = 1 then c = 0 and one eigenvalue becomes zero.…”

Section: The Dynamical System and The Surface Of Critical Pointsmentioning

confidence: 99%

“…for some positive integer m, where T is the period of the limit cycle, give rise to traveling wave solutions, see [2]. If T is the minimal period, then we call L 0 = T /ρ max the minimal road length.…”

Section: Periodic Boundary Conditionsmentioning

confidence: 99%

“…Here in analogy with compressible fluids, the rate of change in momentum in (2) is due to a decreasing gradient in "pressure" P . The bulk forces are modeled as a tendency to acquire a safe velocity V e (ρ).…”

Section: Introductionmentioning

confidence: 99%

“…We have shown in a previous work [2], that under general properties of the fundamental diagram, a one parameter curve of Takens-Bogdanov (BT) bifurcations exists, associated to a folding projection of the surface of critical points into the two-dimensional space of parameters Q g -V g . The family of BT points can be parametrized by the value of Θ 0 .…”

Section: Introductionmentioning

confidence: 99%

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Global Bifurcation Diagram for the Kerner–Konhäuser Traffic Flow Model

Delgado¹,

Saavedra²

2015

Int. J. Bifurcation Chaos

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We study traveling wave solutions of the Kerner-Konhäuser PDE for traffic flow. By a standard change of variables, the problem is reduced to a dynamical system in the plane with three parameters. In a previous paper [2] it was shown that under general hypotheses on the fundamental diagram, the dynamical system has a surface of critical points showing either a fold or cusp catastrophe when projected under a two dimensional plane of parameters named q g -v g . In any case a one parameter family of Bogdanov-Takens (BT) bifurcation takes place, and therefore local families of Hopf and homoclinic bifurcation arising from each BT point exist.Here we prove the existence of a degenerate Bogdanov-Takens bifurcation (DBT) which in turn implies the existence of Generalized Hopf or Bautin bifurcations (GH). We describe numerically the global lines of bifurcations continued from the local ones, inside a cuspidal region of the parameter space. In particular, we compute the first Lyapunov exponent, and compare with the GH bifurcation curve. We present some families of stable limit cycles which are taken as initial conditions in the PDE leading to stable traveling waves.

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“…The stability of the critical points is given in the following proposition [2] . • If v e (v c ) = 1 then c = 0 and one eigenvalue becomes zero.…”

Section: The Dynamical System and The Surface Of Critical Pointsmentioning

confidence: 99%

Section: Periodic Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Global Bifurcation Diagram for the Kerner–Konhäuser Traffic Flow Model

Delgado¹,

Saavedra²

2015

Int. J. Bifurcation Chaos

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“…Luo et al established a three-dimensional traffic flow model to describe the change characteristics of the three parameters of traffic flow, verified that the three basic parameters of traffic flow had similar characteristics and change rules with cusp catastrophe theory, and further proved the feasibility of using cusp catastrophe theory to describe the relationship characteristics of the three basic parameters of traffic flow [16]. Carrillo et al proved the mutation and bifurcation of traffic wave in the second order traffic flow model [17]. Aiming at the frequent congestion phenomenon of urban expressway, Xu L. et al analyzed the evolution process of traffic congestion through the cusp catastrophe model [18].…”

Section: Introductionmentioning

confidence: 99%

An Analysis of the Catastrophe Model and Catastrophe Characteristics of Traffic Flow Based on Cusp Catastrophe Theory

Huang

Zhang

Liu

et al. 2022

Journal of Advanced Transportation

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—The state of urban road traffic flow shows discontinuity and jumping phenomenon in the process of running. There was a data gap in the collected traffic flow data. Through the data analysis, it was found that the traffic flow state had the characteristics of multimode, mutation, inaccessibility, divergence and hysteresis, which were similar to the mutation characteristics of the basic model of catastrophe theory when the system state changed. The cusp catastrophe model of traffic flow based on traffic wave theory was established by analyzing the movement process of traffic flow. In this model, the traffic density was taken as the state variable, and traffic flow and wave speed were taken as the control variable. Referring to the basic idea of catastrophe theory, the solution method of the model was given, and the structural stability of the traffic flow state was analyzed. Through the critical equilibrium surface equation, the stability of the extreme value of the system potential function can be analyzed, and the bifurcation set equation when the traffic flow state changed can be obtained, which can be used to determine the critical range of the structural stability of the system. This paper discussed and analyzed the changing trend and constraint relationship among the wave speed, traffic density and traffic flow when the traffic flow state changed suddenly in different running environments. The analysis results were consistent with the actual road traffic flow state. A case was given, and the results showed that the cusp catastrophe model could describe the relationship among the three parameters of traffic flow from three-dimensional space, and could effectively analyze the internal relationship of the parameters when the traffic flow state changed. The validity of the model and analysis method was verified. The goal of this paper is to provide an analysis method for the judgment of urban road traffic state.

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Game-Theoretic Context and Interpretation of Kerner’s Three-Phase Traffic Theory

Hausken

Rehborn

2014

Springer Series in Reliability Engineering

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Traveling Waves, Catastrophes and Bifurcations in a Generic Second Order Traffic Flow Model

Cited by 10 publications

References 14 publications

Global Bifurcation Diagram for the Kerner–Konhäuser Traffic Flow Model

Global Bifurcation Diagram for the Kerner–Konhäuser Traffic Flow Model

An Analysis of the Catastrophe Model and Catastrophe Characteristics of Traffic Flow Based on Cusp Catastrophe Theory

Game-Theoretic Context and Interpretation of Kerner’s Three-Phase Traffic Theory

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