Global stability and bifurcation of macroscopic traffic flow models for upslope and downslope

Abstract

A macro continuum model of the traffic flow is derived from a micro car-following model that considers both the upslope and downslope by using the transformation relationship between macro-and micro variables. The perturbation propagation characteristics and stability conditions of the macroscopic continuum equation are discussed. For uniform flow in the initial equilibrium state, the stability conditions reveal that as the slope angle increased under the action of a small disturbance, the upslope stability increases and downslope stability decreases. Moreover, under a large disturbance, the global stability analysis is carried out by using the wavefront expansion technique for uniform flow in the initial equilibrium state. For the initial nonuniform flow, nonlinear bifurcation analysis is conducted at the equilibrium point. Subcritical Hopf bifurcation exists when the traffic flow state changes，thus the limit cycle formed by the Hopf bifurcation is unstable. Simulation results verify the stability conditions of the model and determine the critical density range. The numerical simulation results show the existence of critical Hopf bifurcation in phase space, and the spiral saddle point varies with the slope angle. Furthermore, the impact of the angle of both the upslope and downslope on the evolution of density waves is investigated.