# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 4 Analysis of Evolution Law of Traffic Flow State  
### Formation Law of Traffic Flow Congestion  
The equilibrium surface diagram of cusp catastrophe theory was analyzed. As shown in Figure 7, the system had a dual modality. The traffic flow had two steady states, free flow and congested flow, in the upper and lower lobes of the equilibrium surface, respectively. The region characterized by surface folds was unstable, i.e., a synchronized flow state, which could be either continuous or discontinuous, indicating an unattainable stable state, which could jump to the upper lobe or the lower lobe. Thus, the synchronized flow state can evolve into a congestion or a free-flow state in a traffic flow system. When the traffic flow evolves from free to congestion, the speed evolves from point A to point B and then catastrophically to point B\({}_{1}\). In the process of congestion dissipation, the speed evolves catastrophically from point C to point C\({}_{1}\), indicating that the catastrophe of the system has hysteresis. The point where the surface folds disappear is the coordinate origin of the equilibrium surface. At this time, the traffic volume, occupancy, and speed after data transformation are 0, corresponding to the maximum traffic volume, critical occupancy, and optimal speed of the original data.  
The equilibrium surface plot of cusp catastrophe theory was projected onto three planes for analysis.  
(1) Analysis of the relationship between traffic volume and occupancy  
In the cusp catastrophe model, the bifurcation set equation for traffic volume and occupancy can be obtained by eliminating V. It is crucial to study the two-state potential and situation catastrophe using cusp catastrophe theory. Figure 8a shows that the traffic volume was distributed in the second and third quadrants, with the free flow on the left and the congested flow on the right. There was a prominent bifurcation area in the middle, indicating synchronous flow. In the free flow state, the time occupancy increased with traffic volume, and the traffic volume reached the maximum at the origin. Then, the traffic flow fell into the synchronous flow state. In the synchronous flow state, the traffic flow had a lower occupancy rate than the free flow, but the traffic volume was not as large. Subsequently, with the continuous input of vehicles, the occupancy increased, the traffic volume decreased gradually, and the traffic flow entered the state of congestion.  
Figure 6: Discrimination results of traffic state in the weaving section of Hang’an Expressway.  
Analysis of the relationship between speed and occupancy  
In the process of traffic flow operation, as shown in Figure 8b, the system state was in the upper leaf of the equilibrium surface. That is, the vehicle flow speed was in the free flow state. When it gradually approached the origin, the occupancy increased gradually, and the change process of the state was continuous. Upon reaching the origin, a slight change in the control variable would result in the velocity jumping to the lower leaf of the equilibrium surface at the origin, thereby suddenly changing the speed. Upon increasing the occupancy, the system state was stable in the lower half of the leaf, i.e., the congested flow state.  
Analysis of the relationship between speed and traffic volume  
The cusp catastrophe model could obtain the singular point set equation of traffic volume and speed by deriving the equilibrium surface equation. As shown in Figure 8c, when the traffic flow was in the free flow state, the speed was in the upper leaf of the system. At this time, the vehicles traveled at the desired speed. When the traffic volume increased, the speed decreased smoothly, but the decrease was not significant until reaching the singularity, at which point the speed reached the optimal speed. This resulted in a speed catastrophe, which reached the congestion state via the synchronous flow state, and the speed and traffic volume decreased simultaneously.  
Figure 7: Equilibrium surface of cusp catastrophe theory.