Bifurcation analysis and control of macro traffic flow models based on real-time traffic state influence

From the perspective of system stability, the essence of traffic congestion is the bifurcation behavior induced by the instability of the traffic system. In this paper, relying on the real-time traffic flow model, we conduct an in-depth exploration of the bifurcation dynamic behavior of traffic flow and its bifurcation conditions. A stability analysis model for real-time traffic flow is constructed by means of traveling-wave transformation. This model theoretically elucidates the transformation processes among different traffic states, providing a crucial basis for subsequent research. Leveraging the established stability analysis model, we have systematically deduced multiple bifurcation conditions within the system. These conditions trigger abrupt transitions in traffic dynamics, typically including Hopf bifurcation and saddle-node bifurcation. Through an in-depth analysis of these bifurcation conditions and the stability of the system’s equilibrium points, we can locate the critical points at which the traffic operation state mutates. We can clarify the causes of the mutations. And we can comprehensively describe the characteristics of the behavioral mutations at the bifurcation critical points of the traffic system from the perspective of the system’s global stability. When key parameters such as vehicle density and traffic flow on the road change to specific critical values, the traffic system state is highly likely to plunge into chaos or even collapse. Specifically, when passing through the Hopf bifurcation point, a limit-cycle phenomenon will occur, causing the vehicle density to exhibit a remarkable stop-and-go equal-amplitude oscillation state. When passing through the saddle-node bifurcation point, a series of highly destructive dynamic behaviors such as amplitude jumps and lags will be induced. At the same time, the stability of the traffic flow will also undergo a sudden change. In this study, an inquiry into plausible control strategies for the prevention or mitigation of traffic mutation phenomena is conducted. Specifically, a feedback controller tailored to unstable bifurcation points is formulated, with careful consideration given to its design parameters. By reasonably adjusting the control parameters of the feedback controller, we can effectively delay or even eliminate the occurrence of Hopf bifurcation, thus effectively achieving the goal of alleviating or preventing traffic congestion and providing new theoretical and practical bases for improving traffic conditions.