Branch analysis of macro-traffic flow model with different driving characteristics in ITS environment

Research of traffic phenomena is the application basis of intelligent transportation systems and plays an important role in enriching the theoretical system of modern traffic flow. Various nonlinear traffic phenomena in the transportation system often alternately change, and the essence of these changes is theoretically a branching behavior. When the traffic system parameters change to a critical value, the qualitative state of the traffic flow, such as the free-running state, the blocking state, and the stop-and-go state, will undergo abrupt changes. However, previous studies on the bifurcation phenomenon of traffic flow mainly focused on the microscopic car-following model from the perspective of local stability, and the bifurcation analysis of the macroscopic traffic flow model has not been reported. Therefore, this article applies branch theory to study the classic macroscopic traffic flow model. First, the types of equilibrium points and their stable states of the model equations are studied, and the global distribution structure of the equilibrium points in the phase plane is drawn to verify the consistency of the numerical and analytical solutions. Secondly, the theory deduces the existence conditions of the model, and simulations have obtained various systems such as Hopf bifurcation, saddle knot bifurcation, limit cycle bifurcation, cusp bifurcation (CP), Bogdanov-Takens (BT) bifurcation, and so on. Finally, starting from the Hopf branch and the saddle-node branch point, by drawing the density space-time diagram of the system, the stop-and-go phenomenon and local aggregation phenomenon in actual traffic are better explained. The results obtained in this paper show that the branch analysis method can better describe the nonlinear traffic phenomena on urban roads, and can provide a theoretical basis for the implementation of traffic control strategies. At the same time, it also has a very broad application prospect for the development of traffic control software in intelligent transportation systems.


I. INTRODUCTION
In today's rapid economic development, building a powerful country in transportation is a pioneering area for building a modern economic system and important support for building a powerful modern socialist country in an allaround way. However, with the continuous increase in traffic demand, the number of motor vehicles has increased continuously, which has increased the degree of congestion in the city and reduced the efficiency of travel. To solve the problem of traffic congestion, we cannot simply increase the number of traffic roads and restrict vehicle driving. We must rely on theoretical analysis and research of traffic flow, and in-depth consideration of the internal mechanism of traffic phenomena, to dredge and control the traffic, alleviate and prevent traffic congestion.
The study of traffic phenomena is the basis of applications in the fields of intelligent transportation systems, traffic engineering design, traffic management, and control, and plays an important role in enriching the theoretical system of modern traffic flow. Researchers analyze actual traffic phenomena and use traffic flow theory to establish a mathematical model that can describe the general characteristics of the traffic, explore the relationship between traffic flow density and speed, reproduce complex traffic phenomena. The mathematical model describing the actual traffic law is the main tool to reveal the basic law of traffic flow. Researchers have now proposed a large number of traffic flow models, mainly including three types: cellular automata model [1][2][3] , micro car-following model [4][5][6][7][8] , and macro dynamics model [9][10][11][12][13] . However, due to the complexity of the traffic system, various traffic phenomena in actual traffic such as vehicles Stopping at all times, increased vehicle density, increased vehicle speed, etc. often appear alternately, and it is difficult to find a universal model to solve all problems. Secondly, since the previous research on traffic flow theory mainly focused on the microscopic carfollowing model, the study of traffic flow phenomena from the perspective of local stability has limitations. The new generation of intelligent transportation systems puts forward a higher level of research on traffic phenomena. The research on the microscopic car-following model traffic flow phenomenon can no longer meet the needs of current intelligent systems. Therefore, this paper proposes a branch analysis method of a macroscopic traffic model, which can describe and predict the nonlinear traffic phenomena on the road from the perspective of the global stability of the system. The study of traffic flow branching theory can not only effectively explain various traffic flow phenomena, but also contribute to the development of traffic control software in intelligent transportation systems.
The transportation system is a complex integrated system and a time-varying nonlinear dynamic system. There are complex non-linear relationships among the various factors that make up the transportation system, and the various factors that reflect traffic characteristics and affect traffic are also non-linear, which makes a large number of non-linear phenomena in the transportation system. At present, researchers have further analyzed traffic system problems from the perspective of nonlinear scientific theory. For example, in 2001, Y. Igarashi et al. proved that there will be multiple exact solutions in a certain range of automobile density in the optimal speed model with time lag, and confirmed the existence of the subcritical Hopf branch [14]. In 2002, L.A. Safonov et al. studied the phenomena of Hopf branch and limit cycle caused by the delay time of the driver [15] . In the same year, H.J.C. Huijberts et al. studied the synchronous movement of the circular bus route following the model, found their local stability described the branches of the synchronous movement [16] . In 2004, Orosz.G et al. found the Hopf bifurcation point after analyzing the stability of the steadystate solution and local branching in the study of the optimal speed model of the driver's reflection time [17] . In the same year, I. Gasser described the model of N cars on a ring road and found that if the density of the cars is changed, it will prove that the loss of stability is usually caused by the Hopf branch [18] . In 2006, Liu et al. considered branching to be one of the main reasons for various nonlinear behaviors in traffic flow, and reviewed the current research on branching phenomena in traffic flow [19] . In 2008, Zhou Wei et al. analyzed the cycle-doubling bifurcation and chaos phenomena in traffic flow through a new type of traffic flow car-following model [20] . In 2009, Ling Daijian et al. studied a class of nonlinear time-delay car-following models, analyzed the stability of the system and Hopf branch characteristics and obtained the influence of the driver's response time and safety distance on the stability range of the system [21] . In 2010, Jin Yanfei et al. used the Hopf bifurcation theory to calculate the full speed difference vehicle following model using Hopf bifurcation theory. The study found that the Hopf bifurcation critical point is located at the boundary of the linearly unstable regions, the uniform traffic flow will lose linear stability and appear when passing through the Hopf bifurcation point [22] . In 2013, Xu Jian et al. reviewed the research on various branching phenomena in the time-delayed carfollowing model with continuous time and space, proposed to study various nonlinear traffic phenomena in the timedelayed car-following model through the theory of timedelay dynamic system [23] . In 2015, Delgado.J et al. studied the traveling wave solution of the Kerner-Konhauser model and proved the existence of the DTB branch in the model, thus explaining the existence of the Hopf branch and the GH branch in the model, and compared it with the GH branch curve, depicts the stability limit cycle of the system [24] . In 2017, Yasunari Miura et al.'s analysis of the optimal speed traffic flow model revealed a branch structure [25] . In 2018, in a heterogeneous traffic flow model with periodic boundary conditions, Yu-Qing et al. used branch analysis to discuss the branching mode of the heterogeneous system in three cases in detail, explained the branching pattern diagram [26] . In the same year, Wang Wei et al. obtained two balance points by solving the low-dimensional macroscopic traffic flow model, and discussed its stability, and found that the model has a transcritical bifurcation point [27] . In 2020, Ren Weilin et al. carried out a bifurcation analysis on the heterogeneous continuous traffic flow model that took into account the differences in the psychological travel of drivers, and discussed the existence and stability of the Hopf bifurcation from a theoretical perspective [28] . At the same time, Zhu et al. proposed a new short-term traffic flow prediction method based on wavelet transform (WT) and multidimensional Taylor network (MTN). The proposed prediction model provides better prediction performance and time correlation. In addition, when considering data from different dates and locations, the new model shows good robustness and generalization ability [29] . Arzoo Miglani et al. discussed various deep learning models used for self-driving vehicle traffic flow prediction and compared the applicability of these models in modern intelligent transportation systems [30] . Chen et al. used multiple heterogeneous data to establish a random traffic flow model method based on vehicle trajectory information, revealing the internal mechanism of the complex dynamics and random evolution of traffic flow, and providing a theoretical basis for a new generation of intelligent transportation systems [31] . Jia et al. developed an enhanced cooperative micro-traffic model considering V2V and V2I communication, and studied how vehicle communication affects vehicle cooperative driving, especially in the case of traffic interference [32] . D. Ngoduy and others applied coupled Langevin equations to simulate complex human driver behaviors, used stochastic processes to describe time-varying random acceleration, and proposed a new continuous stochastic car-following model to study implies a theoretical significance for the design of CACC or ACC control algorithm to reduce the negative impact of human/machine delays in traffic flow [33] . In addition, D. Ngoduy et al. established a multi-delay car-following model and studied the boundary of the linearly unstable region formed by the Hopf bifurcation point, and found that increasing the delay of the perception space interval will expand the unstable region, thereby destroying the traffic flow stability [34] .
However, the researchers mainly studied the branching phenomenon of traffic flow based on the car-following model. Compared with the car-following model, the macrotraffic flow model can better describe the collective behavior of traffic flow, is more suitable for dealing with the traffic flow problem composed of a large number of vehicles, and to a greater extent alleviate and prevent traffic congestion. At present, there are few studies on the analysis of traffic flow branching phenomena based on macroscopic traffic flow models. Therefore, this paper studies the bifurcation phenomenon of traffic flow based on the macroscopic traffic flow model. First, we solve the balance point of the model, and determine the type and stability of the balance point of the traffic system, and analyze the impact of the balance point on the traffic state from the perspective of the overall stability of the system. Then, using the equilibrium point as the starting point to study the various branches of the nonlinear traffic system, and further study the branch dynamics and branch conditions of the traffic flow and the sudden change of stability caused by the traffic flow bifurcation, through the density space-time diagram of the traffic system and Phase plane diagrams can better explain traffic phenomena such as stop-and-go and sudden changes in stability caused by branch points. Finally, the stop-and-go waves and stability changes in actual traffic are explained through numerical simulations to further verify the correctness of the theory.

II. MODEL EQUATIONS
Serge P. Hoogendoorn et al. proposed a macroscopic traffic flow model considering the different driving characteristics [35] , which consists of the following two equations, one is the equation for the conservation of the number of vehicles, (1) and an equation of motion, Where  is the density, v is the velocity,  is the driver reaction time， k is the speed difference of different driving characteristics, is a sticky item, its role is to help eliminate the impact or discontinuities, without significantly changing the dissection of the original model.
  e V  is the optimal velocity function and has the following form：  

III. TYPES AND STABILITIES OF EQUILIBRIUM POINTS IN THE MODEL
The significance of studying the type and stability of the balance point of the new model is to find the various branch points of the transportation system from the balance point as the starting point on the phase plan and draw the global distribution structure of the traffic trajectory near the branch point. Theoretically analyze the possible branches of the traffic system, branch conditions, and the stability changes of the traffic flow passing these branch points.
If we set the terms on the right-hand side of Eq. (13) to be zero, then we have The Jacobian characteristic equation of (14) reads: The system is always unstable at a saddle point for z   .It is stable at a nodal or a spiral point for . The non-linear system (13) shares the same equilibrium points as the linearized system (14). Moreover, the two systems are uniformly stable or unstable at these equilibrium points if the equilibrium point is not the center point according to the Hartman-Gorban linearization theorem. The equilibrium points    Types and stabilities of some equilibrium points of the linearized system (14) with the given parameters, where Since the analytical solution of the nonlinear system (13) is unlikely to be derived, the numerical solutions nearby the equilibrium points can be given in the phaseplane. We use the four sets of parameters in Table 1 respectively to study the stabilities of system (13) Figure 1 corresponds to the first case in Table 1. We use the equilibrium point as the starting point to draw the trajectory distribution structure diagram of the nonlinear system, and we can find that the various equilibrium points in the phase diagram affect each other, resulting in a relationship of divergence and aggregation. From the perspective of the overall stability of the system, it shows the influence of the equilibrium point on the traffic state. This is the same as what we have observed in actual traffic. It can be seen from Figure 1: When z   , the system is unstable at the equilibrium point   , the trajectory close to this point is also far away from this point. When z   , multiple spiral trajectories near saddle point . When z   , these trajectories move away from the focus and eventually tend to infinity. This shows that: When z   , the system is the trajectory can be regarded as the saddle-focus solution of the system.  Table 1. As the parameters increase, we observe that the equilibrium point gradually changes from an unstable state to an aggregate state over time. This is consistent with the formation of traffic congestion in actual traffic flow over time. It can be seen from Figure  z   , it tends to infinity far. Therefore, a limit cycle is created between the trajectories.  Table 1. With the continuous increase of the parameters, we observe that the equilibrium point changes with time, and the aggregation state of the trajectory also increases significantly, which indicates that the traffic congestion has also reached the maximum. In actual traffic, the vehicle has fallen into a standstill. It can be seen from Figure 3:   Table 1. It can be seen from Figure 4: when z   , the system is unstable at the equilibrium point   , its nearby trajectories are far away from this point. Since the value of vehicle density in the figure continues to show a monotonous increasing relationship, it indicates that the value of vehicle density corresponding to this set of parameters will also continue to increase, and the transportation system is unstable and eventually tends to be congested. It can be seen from the above figure that the type and stability of the equilibrium point on the phase plane are consistent with the theoretical results in Table 1. In addition, they show the influence of the equilibrium point on the traffic state from the perspective of the global stability of the system.

A. Hopf bifurcation conditional derivation
According to the Hopf bifurcation condition theorem given in literature [36] , analysis of system (13), let * q be a variable parameter, which has an equilibrium point 0 for all * q . The derivative matrix L at the equilibrium point is also the Jacobian matrix of the system at the equilibrium point, as follows: , then its characteristic equation is: Suppose that the equation has a pair of complex eigenvalues At this time, the system has Hopf bifurcation at * *0 qq  .

B. THE HOPF TYPE DERIVATION OF THE MODEL
For the Hopf bifurcation problem, because the ndimensional system can be restricted to the 2-dimensional center manifold by the center manifold method, only the Hopf bifurcation of the 2-dimensional system is considered.
Lemma 3.1 [36] : suppose a two-dimensional system with parameters , , The equilibrium point is the origin   Through coordinate transformation, the system (25) can be written as： ,, For the system (13) The system can be linearized by Taylor expansion at the equilibrium point     Where f is a smooth vector function, its constituent element 1,2 f is the Taylor expansion of the least quadratic term of x , which can be expressed as follows: 4  2  2  3  3  2  2  11  22  12  111  222  112  122   0  = ,

 
Jacobian matrix   * Lq can be expressed as follows: Its eigenvalue is the root of the following characteristic equation: According to the Hopf bifurcation condition, it is shown that： For smaller * q variables can be introduced： Thus, the following eigenvalue expression can be obtained： Considering that the real part and imaginary part of both sides of the equal sign are equal, we can get： The finishing equation (42) is as follows: (49) Equation (49) has the same form as equation (26), so according to equation (38), the value in system (13) Therefore, for the system (13), the Hopf bifurcation is supercritical when 0 a  , and subcritical when 0 a  .

C. Derivation of saddle node bifurcation conditions
Lemma 3.2 [36] : consider that system For the system (13), we let * q be a variable parameter and the differential matrix at this equilibrium point is written as Eq. (17).
The equation (56) So the system (13) undergoes a saddle-node bifurcation at

VII. NUMERICAL SIMULATIONS
In order to study the various branching phenomena of nonlinear systems, we need to choose different equilibrium points as the starting point and draw the corresponding branch diagrams. Next, we use the mathematical software package MATCONT that can perform numerical analysis, select the equilibrium point     2 ,0 = 0.0546,0  calculated in the third part as an example, select the parameter * q as a variable parameter with an initial value of 0.2 . The actual calculation ranges of * q is about -30 30 -. Three special points can be found within this range. As shown in Figure  5(a), one is the Hopf branch (H) and the other is the limit point branch (LP). Figure 5(b) is a bifurcation diagram of the appropriate range of the parameter * q . The characteristic values of it are    From Figure 6(a)-6(b), it can be seen that since the parameters and state variable values of the last four limit cycles are not much different, these four limit cycles almost overlap together, and compared with the first limit cycle, they are all small in size and only appear as a single point.
Next, we analyze the stability variation of traffic system when the parameters pass through some bifurcation critical points calculated above. First, to investigate the effects of Hopf bifurcation on traffic flow, we study the system stabilities in the phase-plane when * q passes through 0.64112 .  We can also see the effects of Hopf bifurcation in the density temporal evolution of traffic flow. By selecting the Hopf bifurcation point as the initial average density of the density temporal evolution, it may help to improve our understanding of complex phenomena in congested traffic. The case that a localized perturbation is exerted in an initial homogeneous condition is analyzed by the traffic flow models. The initial density is used as in Ref. 21.
    For computational purpose, the space domain was divided into equal intervals of length 100m and time interval was chosen as 1s . The related parameters of our model were as follows: Corresponding to the above parameters, according to the stability conditions, the critical density of the model is 0.042veh/m and 0.07veh/m, that is, the traffic flow is linearly unstable when the initial density is within the range of  From Figure 8 we can see that when the initial density value is within the unstable range of the model, as time increases, the initial small disturbance will continue to propagate downward, the amplitude of the initial small disturbance will increase with time, and finally formed a constant amplitude periodic oscillation. The traffic flow is in an unstable state.
From the nature of the Hopf branch analyzed above, it can be known that when the variable parameter passes through the Hopf branch point in the system, the transportation system will generate a limit cycle branch from the corresponding equilibrium point. This limit cycle branch can also be called a period solution. This is the characteristic of the limit cycle solution, it also illustrates the existence of the limit cycle solution.
From the results of the density space-time diagram and theoretical analysis, this is consistent with the development of an oscillating and congested traffic pattern under the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3121139, IEEE Access VOLUME XX, 2017 9 uniform road conditions in actual traffic. It further shows that the obtained results are consistent with actual phenomena and numerical calculation results, which verifies the correctness of the theoretical analysis. If the limit point (LP) is selected as the starting point of the bifurcation calculation, the bifurcation calculation starting at this limit point will have a bifurcation type with a codimension of 2. At the same time, the activation parameters * q and 1 c are free parameters, and the codimension can be found to be 2 bifurcations, such as Bogdanov-Takens bifurcation (BT), cusp bifurcation (CP). The corresponding bifurcation diagram is shown in Figure  9.   Fig.  9(a). As the density of vehicles has been increasing monotonically, all the curves on the left of the red line are attracted and approach the point   0.1017,0 . Therefore, the system on the left of the red line is stable, and the system on the right of the red line is unstable. With the change of the variable parameter * q , the two equilibrium points gradually move closer to the middle, but when the parameter * =0.143582  Figure 10(b). Similarly, the traffic system within the red line is stable, and the traffic system outside the red line is unstable. However, the curves within the red line all approach the point   0.1124,0 . At this time, the linear system has a zero eigenvalue. When the variable parameters continue to change, when * 0.143582 q  , the equilibrium point disappears and all the solutions move to the right, as shown in Figure 10(c), the traffic system becomes unstable. We can also see the effects of LP bifurcation in the density temporal evolution of traffic flow. By selecting the LP bifurcation point as the initial average density of the density temporal evolution, it may help to improve our understanding of complex phenomena in congested traffic. The case that a localized perturbation is exerted in an initial homogeneous condition is analyzed by the traffic flow models .The initial density is used as in Ref. 21 For computational purpose, the space domain was divided into equal intervals of length 100m and time interval was chosen as 1s . The related parameters of our model were as follows: Corresponding to the above parameters, according to the stability conditions, the critical density of the model is 0.042veh/m and 0.07veh/m, that is, the traffic flow is linearly unstable when the initial density is within the range of 0 0.042< <0.07  . Firstly, we select corresponding to the LP bifurcation point as the initial average density and the density temporal evolution is shown as the Figure 10  Knowing the nature of the LP branch, when the variable parameters pass through the LP branch point, the traffic system will produce a local clustering phenomenon from the equilibrium point. Since the initial density value at this time is set within the stable range of the model, although the small disturbance on the initial uniform density is magnified, as time increases, the small disturbance eventually diverges, as shown in Figure 11(a). It can be seen from Figure 11(b) that the initial density value at this time is within the unstable range of the model, and the small disturbance on the initial uniform density is amplified, and then it evolves into a local clustering phenomenon, that is, traffic congestion. This is consistent with the characteristics of the LP branch point, which shows that under the initial uniform traffic conditions, when the parameters pass the LP branch point, the small disturbance will change into a local cluster wave, which further shows the obtained results and actual phenomena and numerical calculation results They are consistent, verifying the correctness of the theoretical analysis.  corresponding to the Hopf bifurcation point as the initial average density, and the time evolution of the density is shown in Figure 13.
It can be seen from Figure 13 that when the initial density value is within the instability range of the model, as time increases, the initial small disturbance will continue to propagate downward, and the amplitude of the initial small disturbance will increase with time, eventually forming a denser Large-scale periodic oscillation. The traffic flow is in an unstable state. This is a clear comparison with Figure  8.
As the vehicle speed decreases, the traffic density value gradually increases, the number of vehicles on the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3121139, IEEE Access VOLUME XX, 2017 9 road section increases, the distance between vehicles becomes smaller, and the transportation system becomes unstable. Combining the density space-time diagram and the branch diagram, it can be seen that when the variable parameter * q passes through the Hopf branch point in the system, the transportation system produces a periodic solution from the equilibrium point. The small disturbance in the initial uniform density was amplified and then evolved into a constant amplitude periodic oscillation. Small disturbance changes to stop the wave while walking. It further shows that the obtained results are consistent with actual phenomena and numerical simulation results. and v =20 f ms.By choosing the LP bifurcation point as the initial average density of the density time evolution, it may help to improve our understanding of complex phenomena in congested traffic. The traffic flow model analyzes the local disturbance under the initial uniform condition. Selecting the same numerical simulation conditions as above, at this time we choose 0 =0.61354  corresponding to the LP bifurcation point as the initial average density, and the time evolution of the density is shown in Figure 14.
We can see from the above-mentioned density spacetime diagram that the traffic system will produce a local clustering phenomenon from the equilibrium point. The transportation system will generate local clustering from the equilibrium point. Compared with Fig. 11(a), with the increase of time, the small disturbance is greatly amplified, but eventually diverges, as shown in Fig. 14(a). It can be seen from Figure 14(b) that the small disturbance of the initial uniform density is greatly amplified and then evolved into a strong local clustering phenomenon, that is, severe traffic congestion, which has a significant change from Figure 11(b). This shows that the decrease in vehicle speed, the sharp increase in traffic volume, the smaller the distance between vehicles, and the rapid traffic congestion. This is in obvious contrast with Figure 11.