Bifurcation analysis of the full velocity difference model

Bifurcation is investigated with the full velocity difference traffic model. Applying the Hopf theorem, an analytical Hopf bifurcation calculation is performed and the critical road length is determined for arbitrary numbers of vehicles. It is found that the Hopf bifurcation critical points locate on the boundary of the linear instability region. Crossing the boundary, the uniform traffic flow loses linear stability via Hopf bifurcation and the oscillations appear.