Stability analysis of a viscous continuous traffic flow model

. This paper studies the stability of a speed gradient continuous traffic flow model, which is proposed by Ge et al and based on TVDM. The nonlinear and linear systems of traveling wave solutions of the model equation are derived by traveling wave substitution. And the types of equilibrium points and it’s stability are analyzed theoretically. Finally, the phase plane diagram is obtained through simulation, and the global distribution structure of the trajectories is analyzed. The results show that the numerical results are consistent with the theoretical analysis, so some nonlinear traffic phenomena can be analyzed and predicted from the perspective of global stability.


Introduction
In order to alleviate and prevent traffic congestion fundamentally, people have studied the various nonlinear phenomena of traffic flow for decades. However, it is not enough to improve traffic congestion only by increasing traffic infrastructure. We should study and analyze the traffic flow theory and deeply investigate the internal mechanism of traffic phenomenon, which can help us dredge and control the traffic congestion phenomenon fundamentally. In addition to evaluate all kinds of traffic phenomena well by observing experiments, researchers have also reproduced and explained the traffic phenomena through a large number of model equations. Compared with car-following model, macro traffic flow model can better describe the collective behavior of vehicles in traffic flow, so as to alleviate and prevent traffic congestion to a greater extent.
As early as 1955, Lighthill and Whitham [1] first proposed a macroscopic traffic flow model. In 1956, Richards [1] proposed a similar model. These two models are called LWR models. In 1971, Payne [2] proposed the first macroscopic high-order continuous model, which allows the average velocity of the vehicles to deviate from the equilibrium velocity. Later, Kühne [4] added a viscous term in the acceleration equation, it can smooth the discontinuity in the model. But in 1995, Daganzo [5] found that there is always a characteristic velocity greater than the macro flow velocity in these models, which aganist a basic principle of traffic flow. In order to overcome this problem, researchers proposed some anisotropic macroscopic traffic flow models. In 2000, Aw [6] first proposed the anisotropic speed gradient model. After that, some similar macro anisotropic models had been proposed [7][8][9][10] .
At present, the research on macro traffic flow mainly focus on analyzing the stability by using characteristic analysis method, deriving the solution equations of different regions [8,9,10] . There are few studies on the equilibrium point and its stability. Therefore, this paper studies and analyzes the macro model from this point of view. Firstly, we can study the type and stability of equilibrium points in macroscopic traffic flow model, then we can study the most common traffic flow state instability phenomenon from the equilibrium point, so it is particularly necessary to study the equilibrium point and stability of traffic flow.
The rest of this paper is organized as follows. In Section 2, the model is derived. The properties of the equilibrium points are analyzed in Section 3. The numerical simulation is carried out in Section 4, and finally summarized.

Model and it's derivation
In this paper, we study and analyze a new speed gradient continuous traffic flow model, which is based on TVDM and proposed by Ge [10] et al. The expression is as follows: where  is the average density, v is the average velocity, T is the relaxation time, The two sides of Eq. (2) are integral to z, we have From Eq. (4), we can get The second order ordinary differential equation about can be obtained by simplifying. (7) can be transformed into a system of first order ordinary differential equations (or called nonlinear dynamic system) with respect to y .

Equilibrium and stability
If the right term of the system (10) is zero, we can obtain can be determined, i  for short. The right term of the second equation in the system (10) can also be linearized through the Taylor expansion at the equilibrium point, and the higher order term is ignored, then the linearized system of Eq. (10) can be derived as follows: We obtain the Jacobian matrix of the system (11) at the equilibrium points as ' 1 0 (12) and the Jacobian characteristic equation as According to the qualitative theory of differential equation, the types of equilibrium point of linear system (11) can be determined, as shown in table 1. Table 1. Types of equilibrium points for linear dynamical systems (11). ; and stable at the nodal or spiral point for . According to the Hartman-Gorban linearization theorem [11] , when the eigenvalues of nonlinear system have non-zero real parts, that is, the equilibrium point is a non central point, the stability of nonlinear system and linear system at equilibrium point is consistent. It is difficult to appear a central point in the traffic flow model, so we discussed it here.

Numerical simulation
The equilibrium point of linear system (11) can be solved for any given set of c and * q . Then according to table 1, the type of equilibrium point and its stability can be determined and are shown in table 2. Equilibrium velocity function proposed in Ref. [12] is selected for numerical simulation.