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

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 such as Hopf bifurcation and saddle–node bifurcation is carried out 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. And the existence condition of saddle-node bifurcation is obtained. Simulation results verify the stability conditions of the model and determine the critical density range. The numerical simulation results show the existence of Hopf bifurcation in phase space, and the spiral saddle point of saddle-node bifurcation 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.