Applications of Mathematics, Vol. 68, No. 4, pp. 499-534, 2023

Bifurcation analysis of macroscopic traffic flow model based on the influence of road conditions

Wenhuan Ai, Ting Zhang, Dawei Liu

Received July 19, 2022.   Published online May 18, 2023.

Abstract:  A macroscopic traffic flow model considering the effects of curves, ramps, and adverse weather is proposed, and nonlinear bifurcation theory is used to describe and predict nonlinear traffic phenomena on highways from the perspective of global stability of the traffic system. Firstly, the stability conditions of the model shock wave were investigated using the linear stability analysis method. Then, the long-wave mode at the coarse-grained scale is considered, and the model is analyzed using the reduced perturbation method to obtain the Korteweg-de Vries (KdV) equation of the model in the sub-stable region. In addition, the type of equilibrium points and their stability are discussed by using bifurcation analysis, and a theoretical derivation proves the existence of Hopf bifurcation and saddle-knot bifurcation in the model. Finally, the simulation density spatio-temporal and phase plane diagrams verify that the model can describe traffic phenomena such as traffic congestion and stop-and-go traffic in real traffic, providing a theoretical basis for the prevention of traffic congestion.
Keywords:  macro traffic flow; curves; ramps; bifurcation analysis
Classification MSC:  35A35

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Affiliations:   Wenhuan Ai (corresponding author), Ting Zhang, College of Computer Science and Engineering, Northwest Normal University, 967 Anning East Rd, Lanzhou, Gansu, 730070, P. R. China, e-mail:,; Dawei Liu, College of Electrical Engineering, Lanzhou Institute of Technology, 1 Gongjiaping E Road, Lanzhou, Gansu, 730050, P. R. China, e-mail:

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