Bifurcation analysis of a speed gradient continuum traffic flow model

• A bifurcation analysis approach is presented on a macroscopic traffic flow model.
• The types and stabilities of the equilibrium solutions of the model are discussed.
• The conditions of the model’s Hopf bifurcation and saddle–node bifurcation are deduced.
• Various bifurcations such as Hopf, saddle–node, limit cycle, Cusp and Bogdanov–Takens are found.
• The Hopf bifurcation can help to explain the stop-and-go traffic phenomena.

A bifurcation analysis approach is presented based on the macroscopic traffic flow model. This method can be used to describe and predict the nonlinear traffic phenomena on the highway from a system global stability perspective. Based on a recently proposed speed gradient continuum traffic flow model, the types and stabilities of the equilibrium solutions are discussed and the existence of Hopf bifurcation and saddle–node bifurcation is proved. Then various bifurcations such as Hopf bifurcation, saddle–node bifurcation, Limit Point bifurcation of cycles, Cusp bifurcation and Bogdanov–Takens bifurcation are found and the traffic flow behaviors at some of them are analyzed. When the Hopf bifurcation is selected as the starting point of density temporal evolution, it may help to explain the stop-and-go traffic phenomena.