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