# Extraction of Catastrophe Boundary and Evolution of Expressway Traffic Flow State
## 2 Analysis of Catastrophe Characteristics of Traffic Flow  
### Cusp Catastrophe Theory  
Cusp catastrophe theory explores the intrinsic mechanism of a catastrophe from one steady state to another steady state in a system by establishing a catastrophe model [18]. According to the catastrophe theory, the system can be divided into two modes: steady-state continuity and unsteady discontinuity. The state change of the system is realized through the transformation of the two modes. During the system's evolution, when one or more factors change continuously, there may be a noncontinuous catastrophe in one of the factors, resulting in a change in the overall situation of the system. This change in state mode can be analyzed by studying the potential function of the system. In the catastrophe theory model, there are control variables and state variables. With the continuous changes of the control variables, the state variables change suddenly. By constructing the state space and control space, the boundary points of the system's equilibrium state can be obtained. Catastrophe theory analyzes the catastrophe characteristics of the system according to the mutual transformation between the research boundary points.  
When catastrophe theory is applied to traffic flow systems [19; 20], the traffic volume and occupancy rate are the control variables, and speed is the state variable in traffic flow evolution. Catastrophe theory describes the phenomenon between two steady states of free and congested flow. In addition, to accurately define the catastrophe interval in the model, the concept of synchronized flow is introduced in this paper [21]. With the continuous increase in traffic volume, the following distance of vehicles gradually decreases during operation, the vehicles drive evenly on the road, and the speed gradually reaches synchronization, representing a synchronized flow state. At this time, there is a strong interaction between vehicles, and a slight disturbance may change their running state.  
The fundamental theoretical model of cusp catastrophe is expressed as follows [22]:  
\[F(x)=ax^{4}+byx^{2}+czx, \tag{1}\]  
where \(x\) is the state variable, and \(y\) is the control variable.  
In order to conform to the traffic flow theory, variables are converted into traffic flow parameters.  
\[F(v)=av^{4}+byqv^{2}+cov. \tag{2}\]  
The equilibrium surface equation is obtained by derivation.  
\[4 av^{3}+2 bqv+co=0. \tag{3}\]  
The singularity set equation is obtained by further derivation.  
\[12 av^{2}+2 bq=0 \tag{4}\]  
Equations (3) and (4) can be combined to eliminate \(v\), thereby yielding the bifurcation set.  
\[8 bq^{3}+27 ac^{2}o^{2}=0,\]where \(v\) is the state variable of speed (km\(\cdot\)h\({}^{-1}\)), \(q\) is the control variable of traffic volume (\(pcu\)), \(o\) is the control variable of time occupancy (%), and \(a,b,c\) are parameters.  
In order to satisfy the cusp catastrophe theory, fundamental data transformation is required when traffic flow parameters are applied to the catastrophe theory.  
1. Coordinate translation: \[\left\{\begin{array}{l}v_{1}=v_{0}-v_{q_{m}}\\ q_{1}=q_{0}-q_{m}\\ o_{1}=o_{0}-o_{q_{m}}\end{array}\right.,\] (6) where \(v_{q_{m}}\) is the optimal speed at maximum traffic volume (km\(\cdot\)h\({}^{-1}\)), \(q_{m}\) is the maximum traffic (\(pcu\)), and \(o_{q_{m}}\) is the time occupancy at maximum traffic volume (%).
2. Coordinate rotation: \[\left\{\begin{array}{l}v_{2}=v_{1}\\ q_{2}=q_{1}\cos\theta-mo_{1}\sin\theta\\ o_{2}=q_{1}\sin\theta+mo_{1}\cos\theta\end{array}\right.,\] (7) where \(m\) is the graphic factor, whose value is \(q_{m}/o_{q_{m}}\), and \(\theta\) is the rotation angle (\({}^{\circ}\)). Equilibrium surface equation after transformation: \[v_{2}^{3}+aq_{2}v_{2}+bo_{2}=0.\] (8)