Branch Analysis of Macro-Traffic Flow Model With Different Driving Characteristics in Its Environment

Research of traffic phenomena is the application basis of intelligent transportation systems and plays an important role in enriching the theoretical system of modern traffic flow. Various nonlinear traffic phenomena in the transportation system often alternately change, and the essence of these changes is theoretically a branching behavior. When the traffic system parameters change to a critical value, the qualitative state of the traffic flow, such as the free-running state, the blocking state, and the stop-and-go state, will undergo abrupt changes. However, previous studies on the bifurcation phenomenon of traffic flow mainly focused on the microscopic car-following model from the perspective of local stability, and the bifurcation analysis of the macroscopic traffic flow model has not been reported. Therefore, this article applies branch theory to study the classic macroscopic traffic flow model. First, the types of equilibrium points and their stable states of the model equations are studied, and the global distribution structure of the equilibrium points in the phase plane is drawn to verify the consistency of the numerical and analytical solutions. Secondly, the theory deduces the existence conditions of the model, and simulations have obtained various systems such as Hopf bifurcation, saddle knot bifurcation, limit cycle bifurcation, cusp bifurcation (CP), Bogdanov-Takens (BT) bifurcation, and so on. Finally, starting from the Hopf branch and the saddle-node branch point, by drawing the density space-time diagram of the system, the stop-and-go phenomenon and local aggregation phenomenon in actual traffic are better explained. The results obtained in this paper show that the branch analysis method can better describe the nonlinear traffic phenomena on urban roads, and can provide a theoretical basis for the implementation of traffic control strategies. At the same time, it also has a very broad application prospect for the development of traffic control software in intelligent transportation systems.