Hopf bifurcation analysis and control of the continuum model considering the new energy vehicles effect

The research of traffic congestion control has been developed rapidly in recent years. Researching the bifurcation characteristics of traffic flow and designing control schemes for unstable bifurcation points can provide some new methods for relieving the traffic congestion. In this paper, we study the bifurcation characteristics and nonlinear control of traffic flow based on the continuum model considering the new energy vehicles effect. First, a macroscopic traffic flow model considering the number of new energy vehicles and their range is proposed. Then, we transform it into a traffic flow stability model suitable for branching analysis using the traveling wave transformation. Therefore, we can transform the traffic flow problem into the stability analysis problem of systems and reflect unstable traffic characteristics such as congestion. Based on the stability model, the existence conditions of the Hopf bifurcation are proved and some bifurcation points of the traffic system are found. Second, we study the congestion and stability abrupt change behaviors near the equilibrium point and branching point to seek for the formation mechanism of traffic congestion. Finally, for the unstable bifurcation point, the control scheme is designed by Chebyshev polynomial approximation, stochastic feedback control, and other methods to delay or disappear the unstable bifurcation point and alleviate traffic congestion. It has important theoretical significance and application value. Graphical abstract