# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 4 Analysis of Evolution Law of Traffic Flow State
### Temporal and Spatial Evolution Characteristics of Traffic Flow State  
When the input traffic volume was 3000 pcu/h, the average speed of the cross-sections at all positions of the outer, middle, and inner lanes was selected as the research object. On the basis of the cusp catastrophe theory, the evolution law of the formation and dissipation of urban expressway traffic congestion and its internal mechanism were studied in terms of space and time. In Figure 10, the green area is free flow, the red area is congested flow, and the blue area is synchronized flow with a speed catastrophe. As shown in Figure 10a, there were 13 speed catastrophes in the outer lane during the temporal simulation, of which eight did not evolve into congested flow but re-evolved into free flow, whereas five evolved into congested flow. Furthermore, the synchronized flow area was narrow, in accordance with the characteristics of the cusp catastrophe. In the spatial simulation, four of the five congestion events occurred at 10 m, near the exit of the weaving section, whereas one occurred at 190 m, near the entrance of the weaving section. Moreover, the speed catastrophe positions of the remaining eight events did not evolve into congestion flow, consistent with their location. As shown in Figure 10b, 13 speed catastrophes also occurred in the middle lane during the temporal simulation, of which four evolved into congested flow and nine re-evolved into free flow. Spatially, all four congestion events occurred near 10 m, i.e., near the exit of the weaving section. For the inner lane, as shown in Figure 10c, there were nine speed catastrophes during the temporal simulation, of which three evolved into the congested flow state, which is a better outcome than the outer and middle lanes. Spatially, the congestion position was also near the exit of the weaving section. The causes of congestion according to the temporal and spatial distribution of traffic flow are analyzed below. 1. For the outer lane, the traffic flow at the entrance of the weaving section evolved into traffic congestion only once. This is because most vehicles could find a gap to freely change lanes during the acceleration process in the weaving lane. These vehicles did not have much impact on the overall traffic flow. Only a few vehicles forcibly changed lanes to enter the mainline when vehicle acceleration was not completed and a gap was not found. This resulted in a traffic flow speed catastrophe, forming a bottleneck in the entrance area. When conservative traffic participants do not find a gap during driving in the weaving lane, they continue to drive in this lane until the exit, before stopping and waiting on the opportunity to change lanes. This results in a traffic flow speed catastrophe, causing congestion at the exit. 2. For the middle lane, since the continuous lane-changing of an entry vehicle into the inner lane rarely occurs after entering the weaving section, there was no congestion near the entrance during the simulation. In the vicinity of the exit, some vehicles in the weaving lane changed lanes into the outer lane, while some vehicles in the outer lane intended to change lanes inward. Thus, some vehicles in the inner lane made a forced lane change near the exit without a suitable gap, thus hindering traffic flow in the middle lane. Therefore, there were more speed catastrophes near the exit in the middle lane than in the outer lane. Moreover, in the congested state, due to the joint obstruction of the outer lane and the inner lane, the congestion in the middle lane lasted longer, and the congestion dissipated more slowly. 3. For the inner lane, similar to the middle lane, no vehicle was forced to change lanes continuously to enter the inner lane without finding a gap during the simulation; hence, there was no speed catastrophe near the entrance.