Analysis and control of saddle-node bifurcation based on continuum model with headway fluctuation

This paper presents a new traffic flow stability model suitable for bifurcation analysis based on headway fluctuation, aiming to reflect unstable characteristics such as traffic congestion to predict traffic congestion or how traffic flow stability changes under specific parameter values. Our method combines bifurcation theory and traffic flow theory to thoroughly examine congestion and stability mutation behavior near equilibrium and bifurcation points by using precise analysis tools, seeking the formation mechanism of traffic congestion and other related phenomena. Based on the stability model, the study proves the existence condition of saddle-node bifurcation and identifies the unstable bifurcation point of the traffic system. To effectively manage the unstable bifurcation point, we have developed a control scheme using Chebyshev polynomial approximation and stochastic feedback control. It enables us to shift the unstable bifurcation point and implement optimal traffic flow control and congestion prevention strategies. The bifurcation analysis method allows for the detailed analysis of the subtle dynamic behavior of a parametric dynamic system as its parameters change. This method offers potential advantages in addressing sudden changes in traffic system stability. The bifurcation control method helps manage the system's dynamic behavior, improving traffic throughput and significantly reducing accident risk. It, in turn, contributes to enhancing road safety and the overall driving experience. Our work advances the development of macroscopic traffic flow bifurcation analysis and control and can potentially have a broader impact on the entire traffic system in specific scenarios.