# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 1 Introduction
[PERSON] [13] studied the evolution law of traffic flow state by determining the critical change nodes of urban road traffic flow state using the profile of the equilibrium model of cusp catastrophe theory. In recent years, machine learning algorithms have gradually emerged. When applied to transportation, they can significantly improve the accuracy and reliability of traffic flow data processing. [PERSON] [14] combined data mining historical trajectory data to detect and predict traffic congestion and used the track clustering algorithm to study the evolution law of road bottleneck formation and proposed a solution to reduce congestion. [PERSON] [15] used a k-means clustering algorithm to put forward a quantitative method that can effectively identify traffic state offline to analyze its evolution law. [PERSON] [16] proposed a neural network to calculate vehicles' relative position and speed on the expressway to monitor the traffic volume and analyze the traffic flow state. [PERSON] [17] studied a vehicle wireless positioning fusion algorithm suitable for the actual vehicle-road collaborative environment, which can help identify real-time traffic congestion status and other practical scenarios all day. As reviewed in this section, it can be seen that studies on the evolution law of traffic flow state are relatively mature. However, most of them were based on the continuous traffic flow model using a two-dimensional surface, which struggles to explain the discontinuous catastrophe phenomenon in complex three-dimensional traffic data. Moreover, in the limited studies on the catastrophe phenomenon of traffic flow state, the length of catastrophe interval was chiefly set artificially. Then, the interval corresponding to the minor data within this length was found by traversing the sample set as the range of the catastrophe interval, easily leading to insufficient accuracy and reliability. This paper introduces the cusp catastrophe theory based on the traffic flow theory. It applies a new spectral clustering algorithm to improve the catastrophe boundary extraction method to establish a calculation model and criteria that are more in line with the actual traffic flow state of the expressway. This method can achieve the goal of distinguishing the traffic flow state of the expressway and accurately describing its evolution law to accurately evaluate and predict the traffic flow state of the road network, analyze and mine the evolution mechanism of traffic congestion, and make a reasonable judgment on the development of traffic state. The remainder of the paper is organized as follows: Section 2 introduces the cusp catastrophe theory and analyzes the catastrophe characteristics of the traffic flow state on the basis of simulation data. Section 3 uses the spectral clustering analysis algorithm to accurately extract the boundary of catastrophe interval of traffic flow state. Then, a sensitivity analysis of the critical factors affecting the velocity catastrophe boundary is carried out. Furthermore, the accuracy of the method is verified using an example. In Section 4, on the basis of the cusp catastrophe theory, we conduct a comprehensive and detailed analysis of the evolution law of urban expressway traffic flow state from the aspects of traffic volume, speed, occupancy rate, and different spatial and temporal distribution. Lastly, we outline the conclusions in Section 5.