# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 1 Introduction  
As the hub of urban transportation, the urban expressway plays an essential role in increasing traffic carrying capacity, improving road service level, and strengthening vehicle operation efficiency in urban transportation systems. With the development of the urban expressway system and the encryption of the road network, the intersection of the urban expressway and ground roads increases, and many complex traffic bottlenecks inevitably appear in the intersection areas and weaving sections, causing traffic congestion. The low efficiency of urban expressways has been criticized for a long time; hence, it is urgent to control and solve the congestion problem. Research on urban expressway traffic congestion formation and the evolution law of dissipation, along with probes into its evolution mechanism from the angle of time and space, can help target the management of expressway traffic to improve the efficiency of urban road network traffic. Research on the evolution law of traffic flow states has a long history. The traditional research method is to establish the fundamental diagram model [1] or LWR (Lighthill-Whitham-Richards) model [2, 3] according to the fundamental relationship among the three parameters of traffic volume, speed, and density, and to elaborate on the traffic flow state by studying the functional relationship. [PERSON] [4] abstracted vehicles into interacting molecules, analyzed the influence of vehicle movement trends on traffic flow using molecular dynamics, and explored its evolution law, in combination with the basic theory of the cellular automata model. [PERSON] [5] analyzed and discussed the correlation relationship between parameters according to the variation tendency of system traffic volume in the phase transformation process and obtained the corresponding parameter correlation function, whichcan better describe the correlation evolution law between traffic flow parameters. On the basis of the LWR model, [PERSON] [6] proposed the concept of density-dependent relaxation time and expounded on the evolution law of traffic flow state by studying the density distribution and speed boundary perturbation. [PERSON] [7] simplified the optimal velocity model using the perturbed modified Korteweg-de Vries (mKdV) equation within the unstable regime; steady traveling wave solutions to this equation were then derived using a multiscale perturbation technique, thereby describing the evolution process of congestion. Later, with the advancement of data acquisition technology, many scholars captured the evolution mechanism of traffic flow state by processing a large amount of traffic flow data. [PERSON] [8] used Internet of vehicles technology to analyze traffic volume, speed, and travel time, and a congestion detection algorithm was proposed to explore the evolution process of congestion with travel time. [PERSON] [9] proposed a practical approach for predicting the congestion boundary due to traffic incidents. Each road type's essential congestion propagation speed was calculated using remote traffic microwave sensors and floating vehicle data systems to capture the congestion propagation characteristics in the existing road network. [PERSON] [10] studied the influence of variable speed limits on the evolution of traffic flow by combining the current emerging complex network theory with the improved traffic flow simulation model. [PERSON] [11] took the standard section of the expressway as an example, simulated the real-time operating state of traffic flow with the help of Vissim traffic simulation software, analyzed its evolution process, and explored the discrimination standards and methods of different traffic flow states. [PERSON] [12] exploited the phenomenon of critical slowing down in dynamic systems near bifurcation, i.e., traffic congestion, to predict the evolution law of traffic congestion after bifurcation by using a few traffic measurements before the tipping point occurs.