## 3 CCT-Based Traffic Flow Model  
### Cusp Catastrophe Theory  
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems, which studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances. Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behavior of the system [37]. Catastrophe theory provides a mathematical framework that deals with discontinuous transition between the states of a system, and describes how small, continuous changes in control parameters can have sudden, discontinuous effects on dependent variables. As a powerful mathematical tool in studying the evolution of systematic order,catastrophe theory can better explain and predict sudden phenomena in nature and society [38; 39; 40]. Catastrophic change emphasizes the meaning of discontinuities or mutational changes in the process of changing. In most high-density urban centers, the traffic flow is always within a stable and unstable equilibrium boundary during the peak hours. With a sudden fire disaster, the traffic flow status in high-density urban central area will be catastrophically changing and the traffic flow status will be transiting. The continuous steady traffic flow status will be interrupted and lead to qualitative changes. In the case of sudden fire disasters, the traffic flow in the disaster affected-area can be managed and controlled by intelligent transportation systems (such as adaptive traffic signal control system, advance route guidance system) or human beings with insulated fences. Through controlling the inflow and outflow in the boundary of disaster-affected area and managing the speed within the disaster-affected area, the density or occupancy of regional traffic flow can be estimated and manage by the relationship of traffic flow characteristics with the intelligent transportation systems. From the perspective of catastrophe theory, as the behavior of catastrophe models can become extremely complex when the number of behavioral and control parameters is increased [40]. The cusp catastrophe is the simplest and most often applied catastrophe model to describe the catastrophe characteristics and discontinuous behavior with two control variables and one state variable. When traffic is congested, two control variables already allow for the prediction of quite intricate transitional behavior [11]. Therefore, the relationship of flow-speed-occupancy can correspond to the variables in the cusp catastrophe model. The potential function that goes with the cusp model of catastrophe theory is Equation (1), when one explores what happens to a fold bifurcation if a second parameter \(y\) is added to the control space. Varying the parameters, one finds that there is now a curve of points in \((y,\,z)\) space where stability is lost, where the stable solution will suddenly jump to an alternate outcome [37]. \[U(x,y,z)=x^{A}+yx^{2}+zx \tag{1}\]  
where: \(x\) is the internal state variable of the system; \(y\) and \(z\) are the external control variables. When \(x\) satisfying \(\frac{dII}{dx}=0\), the system is in equilibrium. That is, the states for which the derivative of formula (1) is zero as formula (2). The entire panel of formula (1) is associated with only one, or three points on the cusp surface in Figure 1[40; 41]. \[4x^{3}+2 yx+z=0 \tag{2}\]  
As shown in Figure 1, the critical equilibrium surface is divided into upper, middle and lower parts. The upper and lower parts represent stable regions. In the middle part, the state of the system jumps and becomes unstable. The middle lobe is composed of folding regions, and the folding area can  
Figure 1: Cusp catastrophe equilibrium surface. be projected to the bifurcation set.