Saddle-node bifurcation analysis and control based on a continuum model under the action of new energy vehicles

Traffic flow has long been a focal point of urban research. Drawing upon the theory of traffic flow bifurcation, investigating the propagation characteristics and evolutionary dynamics of traffic flow can provide a theoretical foundation for the design and management of urban road networks. In this paper, we develop a macroscopic traffic flow model that incorporates the influence of new energy vehicles (NEVs). By applying bifurcation theory from nonlinear dynamics, we analyze congestion and stability transitions in the traffic system induced by bifurcations. First, we propose a macroscopic traffic flow model that accounts for both the proportion of NEVs and their driving range. This model is then reformulated into a traffic stability framework suitable for phase-plane analysis, thereby recasting abrupt changes in traffic states as stability problems of the traffic system. This approach enables a macroscopic interpretation of sudden stability shifts—such as those leading to traffic congestion. Next, we derive the types of equilibrium points of the model and establish the conditions under which saddle-node bifurcations occur. We further examine the model through analyses of stop-and-go traffic and shock wave phenomena, which serve to validate the theoretical conditions for the emergence of saddle-node bifurcations. Finally, we incorporate stochastic effects into the traffic flow model and apply bifurcation control strategies based on the observed bifurcation behavior. While few studies have addressed saddle-node bifurcation control in traffic flow models that consider NEV-related factors, this paper investigates such control using a stochastic extension of the improved full velocity difference (FVD) model. Specifically, a feedback controller is designed to either delay or advance the onset of saddle-node bifurcations at equilibrium points. The theoretical findings are subsequently verified through numerical simulations.