Global stability and bifurcation of macroscopic traffic flow models for upslope and downslope

A macro-continuum model of the traffic flow is derived from a micro-car-following model that considers both the upslope and downslope by using the transformation relationship between macro- and micro-variables. The perturbation propagation characteristics and stability conditions of the macroscopic continuum equation are discussed. For uniform flow in the initial equilibrium state, the stability conditions reveal that as the slope angle increased under the action of a small disturbance, the upslope stability increases and downslope stability decreases. Moreover, under a large disturbance, the global stability analysis is carried out by using the wavefront expansion technique for uniform flow in the initial equilibrium state. For the initial nonuniform flow, nonlinear bifurcation analysis such as Hopf bifurcation and saddle–node bifurcation is carried out at the equilibrium point. Subcritical Hopf bifurcation exists when the traffic flow state changes; thus, the limit cycle formed by the Hopf bifurcation is unstable. And the existence condition of saddle-node bifurcation is obtained. Simulation results verify the stability conditions of the model and determine the critical density range. The numerical simulation results show the existence of Hopf bifurcation in phase space, and the spiral saddle point of saddle-node bifurcation varies with the slope angle. Furthermore, the impact of the angle of both the upslope and downslope on the evolution of density waves is investigated.

Empirical observations over many years indicate that various congested traffic patterns are induced by traffic bottlenecks on roads [28,29]. Helbing and Lee et al. [15,29,30] observed many congested traffic patterns caused by traffic bottlenecks, such as pinned localized clusters (PLCs), triggered stop-and-go waves (TSGs), oscillating congested traffic (OCT), homogeneous synchronized traffic (HST), and homogeneous congested traffic (HCT), via numerical simulations. Notably, uphill and downhill roads represent a type of traffic bottleneck, and the traffic congestion caused by these entities has attracted the attention of traffic engineers and researchers. In 2009, Komada studied vehicles in the context of the gravitational force on sloped roads to extend the optimal velocity model by considering upslope and downslope gradients and clarified the traffic states and jamming transitions induced by the slopes of sags and hills [31]. Zhu et al. focused on the impact of the slope angle on traffic stability and investigated the stability condition and density waves of the traffic flow on a single lane gradient (upslope or downslope) [32]. Wu et al. examined the steady-state traffic flow on a ring road with upslopes and downslopes by using a semidiscrete model [33]. Gupta et al. studied a unidirectional single lane gradient highway by using the optimal flow differential method for a lattice hydrodynamic model [34]. Kaur and Sharma proposed a lattice hydrodynamics model of traffic flow and conducted a simulation study by considering the anticipation effect of drivers in a two-lane traffic system on a slope curve [35]. Moreover, many extension models have been proposed. For example, Li et al. considered the driver habit, slope grade and slope length [36]. Yu et al. focused on the effect of the variable slope [37]. Chen investigated the slope of two-lanes [38], Tan et al. considered the effect of lowvisibility foggy weather on a highway with slopes [39], and other researchers considered several relevant factors [40][41][42][43][44]. With the development of transportation theory and technology, the research of intelligent transportation system has been carried out in recent years [45][46][47]. Nevertheless, the traffic congestion caused by uphill and downhill slopes has not been simultaneously considered. Due to the different uphill and downhill movement states, the traffic flow may be unstable, and complex traffic congestion may occur.
To examine the traffic congestion caused by uphill and downhill slopes, we attempt to establish a macroscopic fluid dynamics model of uphill and downhill slopes by transforming the microscopic model to a macroscopic model. In 2000, Berg et al. established a continuum approach for car-following models by deriving the relation of the headway of successive vehicles with the density [48]. The car-following model and continuum model can be integrated. Helbing derived nonlocal macroscopic traffic equations from microscopic carfollowing models by using gradient expansion, linear interpolation and the smooth particle hydrodynamics approach [49]. Gupta et al. expressed the headway as a perturbation series, obtained an anisotropic higherorder continuum model, and studied the formation of shock and rarefaction waves, the local cluster, and the generation of stop-and-go traffic [50,51].
In this study, we focus on not only the stability of the small disturbance range near the equilibrium state but also attempt to analyze the global stability and state bifurcation of traffic flow. Yi et al. [52], Ou et al. [53] and Gupta et al. [54] studied the nonlinear stability criterion of macroscopic traffic flow models by using a wavefront expansion technique under large traffic disturbances. To examine the traffic flow bifurcation, Carrillo et al. [55] demonstrated the occurrence of Bogdanov-Takens bifurcations in a two-parameter dynamic system for Kerner-Konhauser's model. In 2015, Delgado et al. [56] proved the existence of degenerate Bogdanov-Takens bifurcations for Kerner-Konhauser's model, thereby explaining the presence of Hopf bifurcation. In 2004, Gasser et al. [57] analyzed the bifurcation characteristics of car-following models. They demonstrated that Hopf bifurcation generally leads to instability of quasistationary solutions in the optimal velocity model, and obtained a criterion pertaining to the subor supercriticality of the Hopf bifurcation. In 2004 and 2006, Orosz et al. [58][59][60] investigated Hopf bifurcation for a single fixed time-delay differential equations like a car-following model considering reaction time delay. Similarly, their results show that Hopf bifurcation reduces the stability of the system and subcritical Hopf bifurcations may cause bistability. In 2020, Ngoduy et al. [61] established a general bifurcation structure of a car-following model with multiple time delays. The impact of multiple time delays and other model parameters on the Hopf bifurcation in various existing car-following models with delays was discussed. In 2015, Ai et al. [62] considered that bifurcation corresponds to traffic jams by the bifurcation analysis for a speed gradient macro-continuum traffic flow model. More recently, Miura et al. [63] observed macroscopic collective phenomenon that occur only in multi-body systems while studying bifurcations in optimal velocity models. The authors noted the traffic jams occur as a moving cluster in the bifurcation structure. Ren et al. [64] studied the conditions for occurrence and stability of Hopf bifurcation for heterogeneous continuum traffic flow and explained that stop-and-go traffic corresponds to Hopf bifurcation. The study of traffic flow stability helps to reveal the condition of traffic change, and the bifurcation of traffic state indicates the instability of the equilibrium state of the traffic system. Nevertheless, the research on the bifurcation behaviors of traffic state caused by road conditions such as upslope and downslope, road curves etc. is still limited from the perspective of macroscopic traffic flow model. The traffic stability caused by these different traffic conditions such as uphill, downhill, road reduction and curve type of traffic bottleneck is different. Traffic stability in these road conditions is related to traffic vehicle factors and road conditions. Therefore, it is worth exploring the relationship between traffic instability and traffic bifurcation. Different from the microscopic model, the macroscopic continuum model is applied to study the bifurcation, chaos and other nonlinear behaviors of traffic congestion formation and dissipation caused by road constraints.
In this paper, we derive the macroscopic continuum model of uphill and downhill slopes by transforming the microscopic model to a macroscopic model, as described in Sect. 2, and deduce the stability condition through a linear analysis. As described in Sect. 3, a global stability analysis is conducted using the wavefront expansion method. Section 4 discusses the conditions for the presence of Hopf bifurcation and saddle-node bifurcation. Section 5 describes the numerical simulation performed to explore the evolution of traffic density waves. Finally, the concluding remarks are presented.
2 Proposed model Figure 1 shows the schematic of a vehicle on a slope subjected to gravitational force. The angles of the slope are represented by h, the gravitation acceleration is g, and the vehicle has a mass m. According to Newton's second law, where Dx i ð¼ x iþ1 À x i Þ is the headway, FðDx i Þ is the driving force contributed by the vehicle engine, l denotes the friction coefficient, and Sðx i Þ represents the slope-control function, with Sðx i Þ ¼ 1 and Sðx i Þ ¼ À1 for the upslope and downslope, respectively. An extended optimal velocity model including the effect of the slope in comparison with the optimal velocity model can be defined as follows [8]: where a ¼ l=m ð Þ is the sensitivity, the inverse of which corresponds to the delay time s, with The optimal velocity function is specified with reference to the work of Gupta et al. [34]: For an upslope gradient highway, and for a downslope gradient highway, Sðx i Þ ¼ À1, where v up;max and v down;max denote the maximal reduced and enhanced velocities on the upslope and downslope, respectively, and v up; max ¼ v down;max ¼ mg sin h=l [34]. As the slope angle increases, the safe distance on the upslope (downslope) decreases Fig. 1 Schematic of the gravitational force upon a vehicle on the slope gradient (increases). In particular,x c;up ¼ x c ð1 À a sin hÞ; x c;down ¼ x c 1 þ b sin h ð Þ , and for simplicity, a ¼ b ¼ 1, which leads to x c;up ¼ x c ð1 À sin hÞ; x c;down ¼ x c ð1 þ sin hÞ [32]. The following approximate equation is defined: For simplicity,mg=lv max ¼ 1, and V 0 ðDx i ; x i Þ¼ cVðDx i ; x i Þ. Here the slope Angle parameter Substituting the above expressions, Eq. (2) can be simplified as According to continuum approach to car-following models, transformation relating headway Dx i to density q enables predictions of the global effect and characteristics of microscopic model. The microscopic transformation formulas are converted to macroscopic formulas, defined as [48] x i ðtÞ ! xðx; tÞ ð 7Þ where q denotes the local density. By the relation from the microscopic to macroscopic transformation, the function Sðx i Þ becomes SðxÞ ¼ 1 for upslope, SðxÞ ¼ À1 for a downslope. In this manner, the governing equation of the macroscopic traffic model considering the upslope and downslope can be obtained: The full differentials of density and velocity: [54], and Eqs. (10) and (11) are rewritten in the matrix form: The coefficient matrix of partial derivatives must be singular. Thus, the characteristic velocity can be derived . Two characteristic velocities exist, which are smaller and larger than the local velocity v. This aspect indicates that high-order models exhibit an isotropic traffic flow because the characteristic velocities are greater than the local velocities [16]. However, for most second-order traffic models, Yi et al. [52] demonstrated that the disturbance wave propagating forward, which travels faster than traffic, promptly disappears. Moreover, most of the secondorder models have been used in practice and known to effectively capture several important characteristics of traffic. Therefore, research on the dynamic behavior induced by uphill and downhill roads may have implications for practical applications.
The following stability conditions can be obtained by the linear analysis of Eqs. (10) and (11).
For the upslope, For the downslope, where q 0 is the initial density. When the stability condition is violated, the stability of the uniform traffic flow in equilibrium is lost. According to stability condition (14), in the upslope case, the increase in the slope angle can enhance the stability of traffic flow. In contrast, in the case of a downhill slope, the increase in the slope angle, as indicated in Eq. (15), deteriorates the stability of the traffic flow. This conclusion is consistent with the theoretical analysis of Zhu et al. [32] and Gupta et al. [34].

Global stability analysis
Yi et al. [52], Ou et al. [53] and Gupta et al. [54] studied the nonlinear stability criterion of a macroscopic traffic flow model by using a wavefront expansion technique under large traffic disturbances.
In particular, Yi et al. [52] discussed the propagation stability conditions for second-order traffic flow. Ou et al. [53] and Gupta et al. [54] performed global stability analyses for an anisotropic macroscopic traffic flow model. As described in this section, we conduct a global stability analysis for the governing equations, Eqs. (10) and (11), under large traffic disturbances. By the following coordinate transformation, the solution of the traffic system is expended around the wavefront in powers of n.
where X(t) indicates the location of the wavefront at time t. The characteristic velocity of the wavefront can be obtained at the equilibrium state: The local density q and local velocity v behind the wavefront can be expressed in the power series of n as follows: where q i ðtÞ¼ o i q ox i j ðXðtÞ À ;tÞ ; v i ðtÞ¼ o i v ox i j ðXðtÞ À ;tÞ ; i ¼ 1; 2; 3; :::. By applying Eqs. (18) and (19), the spatiotemporal partial derivative of the density q and velocity v can be calculated as follows: þ ::: ð20Þ Accordingly, the equilibrium velocity Vðq; vÞ can be obtained. (20)-(24) into Eqs. (10) and (11), it can be derived that the coefficients of the first two terms n 0 and n 1 satisfy the following equations: (26) and (28), we can derive the following determinant det Àac It indicates the coefficients of q 2 and v 2 are linearly dependent. Therefore, by eliminating q 2 and v 2 in Eqs. (26) and (28), the Bernoulli equation is obtained: The slope of the points is along the wavefront trace. This equation specifies the slope evolution at the wavefront. The propagation stability defined by Eq. (30) can be analyzed in terms of the initial condition v 1 (0) and parameters C and D. The value of D is greater than zero for the two characteristic velocities. If C = 0, the solutions of the Bernoulli equation, Eq. (30), are Then, the monotonicity of function v 1 ðtÞ is determined by the first derivative with respect to t: If C = 0, the general solutions of Eq. (30) can be specified as follows: where v 1 0 ð Þ is determined by the initial condition for v 1 t ð Þ at t = 0. Similar to the monotonicity of Eq. (31), the monotonicity of Eq. (33) is given by the first derivative with respect to t.
Therefore, the trend of v 1 (t) can be determined, as shown in Table 1. The stability conditions are listed in Table 1. When the density disturbance increases, the velocity perturbation decreases, specifically, Þ is monotonic decreasing function, and tends to zero with time.
Plugging it into Eq. (33), we can deduce v 1 0 ð Þ 2 ÀC D ; 0 À Á . The stability criterion for the traffic flow is: Thus, we can obtain the following expression: According to the comparison of the stability condition specified in Eq. (13), the stability criterion under large traffic disturbances is considerably different from the result of the linear analysis, which is followed by a term À8ðc À 1ÞV When h ¼ 0 and c=1, Eq. (35b) can be expressed as a [ À 2q 2 0 V 0 q , which is the stability condition of the traffic flow on the flat road without considering the slope [52]. The model is stable against any initial condition as long as this stability criterion is satisfied.
In order to compare local stability (13) and global stability (35b), Fig. 2(a) and (b) shows the phase diagrams corresponding to different slope angles on the upslope and downslope, respectively, where the  [52,53] Parameter C

Stable region
Unstable region equilibrium speed-density relationship VðqÞ proposed by Kerner et al. [65] is utilized. Local and global stability are represented by dashed and solid lines, respectively. The region under these curves is the region of instability. Under the action of small disturbance, with the increase of slope Angle, the local stability of upslope increases and the local stability of downslope decreases. Under large disturbance, the global stability also shows a similar insignificant effect, but the global instability region uphill is larger, while the global instability region downhill is smaller than the local instability region.

Hopf bifurcation
Considering the following nonlinear system: where f is a smooth function, and v is a variable parameter. A linear equation around the origin is represented as LðvÞ ¼ D x f ðx; vÞj ðx 0 Þ . It is assumed that the system has a pair of complex eigenvalues k 1;2 ¼ aðvÞ AE iwðvÞ .
Lemma 1 Cao et al. [66]. If the succeeding conditions are satisfied in the equilibrium state Lemma 2 Cao et al. [66]. A dynamical system (36) with a smooth function f, for which v is a variable parameter, exhibits equilibrium x = 0 with eigenvalues The following equation can be derived through a coordinate shift: Subsequently, the first Lyapunov exponent of system (36) can be computed [66] as follows: If u \ 0 and c [ 0, the Hopf bifurcation is supercritical, and if u [ 0 and c [ 0, the Hopf bifurcation is subcritical [66]. For the dynamical system (36), we perform the following coordinate shift to change the reference system.
where c is the constant velocity of the moving coordinate system with respect to the stationary coordinate system. Substituting Eq. (39) into Eq. (10) yields where q Ã is a constant. Substituting Eqs. (39) and (40) into Eq. (11) yields the following equation: Equation (41) can be reformulated as follows: where Gðq; q Ã Þ ¼ À Let ðq i ; 0Þ represent the equilibrium point of Eq. (43). The equilibrium point can be determined by assuming that the right-hand term of Eq. (43) is zero, that is, y ¼ 0 and F q; c; q Ã ð Þ¼0. The Taylor expansion of the right-hand term of the second equation of equation group (43) at the equilibrium point can be obtained: Assuming the matrix L has eigenvalues k ¼ aðq Ã Þ AE iwðq Ã Þ, the eigenvalue solutions to the above characteristic equation can be specified as The following variable transformation is performed, and Lemma 1 is considered to obtain where q Ã ¼ q Ã0 at the equilibrium point ðq 0 ; 0Þ ¼ ðq i ; 0 ð Þ Þ . Thus, the Hopf bifurcation exists according to Lemma 1 when the following equation is satisfied: Specifically, Next, we examine the type of Hopf bifurcation of the system. The eigenvector of Lðq Ã Þ is assumed to be w re þ iw im , and according to Eq. (49), the value can be obtained as follows: The real and imaginary parts of Eq. (51) are equal.
The matrix form is The transpose of the matrix can be obtained as we can obtaiñ For Eq. (43), the coordinate shift is performed, assumingq ¼ q À q 0 . In particular, the equilibrium point translates to the origin, and we obtaiñ This equation is linearized at the original point by the Taylor expansion ðq; yÞ ¼ ð0; 0Þ. In other words, Substituting Eqs. (55) and (58) into Eq. (56) yields The eigenvector of Lðq Ã Þ can be represented as Notably, When u\0, the Hopf bifurcation of the model is a supercritical bifurcation, and the limit cycle formed by the supercritical bifurcation is stable. If u [ 0, the Hopf bifurcation is a subcritical bifurcation [66].
Lemma 3 [66]. Assumed L has a single zero eigenvalue and the other eigenvalues are non-zero real parts, when For system (43), when The vectors where Thus, a saddle-node bifurcation exists in system (43) at q Ã ¼ q Ã0 ¼ Àcq 2 0 V 0 e ðq 0 Þ.

Numerical simulations
We study the spatiotemporal evolution of traffic density waves under no-flux boundary conditions. The traffic flow uphill, downhill and continuous uphill and downhill is simulated. To study the local cluster effect induced by a localized perturbation in an initial homogeneous condition, we use the finite difference method to discretize Eqs. (10) and (11). To discretize the conservation equation, Eq. (10), and motion equation, Eq. (11), we use the first-order upwind scheme to discretize the macroscopic equation that can be adapted to the physical meaning of the traffic flow to obtain: The related parameters are as follows: where L denotes the length of all roads, Dx and Dt are the space interval and time interval, respectively, and q m represents the maximum density or jam density. The initial density is expressed as the following density function, proposed by Herrmann and Kerner [67]: where Dq 0 is the density fluctuation, with The equilibrium speed-density relationship [65] is selected as follows: According to the global stability analysis, the critical density is obtained by substituting the above parameter into Eq. (35b): where q 0 2 ½0; q m , the global instability region lies between the upper critical density and the lower critical density [14,54].

Evolution of the traffic flow on the upslope or downslope
First, we perform a simulation for the spatiotemporal pattern of the traffic density on the upslope or downslope. Figures 3 and 4 display the spatiotemporal evolution patterns of the density waves on the upslope and downslope when the initial density is q 0 ¼ 0:053veh=m, respectively. Figure 3a and b shows that the density wave is attenuated as the slope angle increases from h 1 ¼ p/60 to h 1 ¼ p/15, consistent with the stable condition specified in Eq. (14) for the upslope. Figure 4a and b shows that the density wave intensifies as the slope angle increases from h 2 ¼ p/60 to h 2 ¼ p/15. In the case of downhill slopes, the traffic stability decreases with increasing slope angle, which agrees with the criterion specified in Eq. (15) for the downhill stability condition.

Evolution of traffic flow on the upslope and downslope
Figure 5a-c shows the spatiotemporal density patterns of traffic flow on the upslope and downslope for different initial densityq 0 . Under the condition of slope Angle h 1 ¼ p/90 and h 2 ¼ p= 90, no matter the class of road is upslope or downslope, the traffic system is in a global stable state when the initial density is less than 0.01 and greater than 0.12. When the initial density is smaller than the lower critical densityq c1 , the disturbance decays rapidly over time.
When the value of the initial density lies between the lower and upper critical densities, two or more clusters of dipole-like structures are formed. When the initial density exceeds the upper critical densityq c2 , the disturbance decays and a stable region is generated. This finding is consistent with Gupta's results [54]. Two types of disturbance waves can be observed propagating forward and backward. Notably, the forward-propagating waves decay rapidly, in agreement with the observations of Yi et al. [52].  Figure 6a-c shows the evolution pattern of the density wave when the initial condition q 0 ¼ 0:053 veh=m and the upslope and downslope angles change. Consider a situation in which the length of the upslope and downslope is L/2. When both the upslope and downslope are considered, increasing the upslope or downslope angle reduces the density fluctuation propagating downstream. Increasing only the slope angle can help enhance the stability. Figure 7a and b shows the spatiotemporal density patterns when the initial condition q 0 ¼ 0:053veh=m, and the length of the upslope and downslope changes. The upslope length is L 1 , and the downslope length is L 2 . Comparison of Fig. 7a and b with Fig. 6a shows that when the uphill length is smaller than the downhill length at the same angle, the cluster size decreases, and the clusters become smoother downstream of the traffic flow. The density waves of traffic are more stable downstream (upslope situation) of the traffic flow, reflecting the influence of the slope length on the traffic flow stability.

Saddle-node bifurcation and Hopf bifurcation
As described in this section, a numerical simulation is performed to study the bifurcation characteristics of uphill and downhill traffic flows. The parameters ðc; q Ã Þ ¼ ðÀ1:371; 0:2Þ [62] selected to derive the equilibrium point specified in Sect. 5 are summarized in Table 2. According to the nonlinear stability theory, the stability of the equilibrium point is confirmed. As shown in Fig. 8a, when the disturbance is around the equilibrium point, only the equilibrium point q 2 remains stable, in accordance with the theoretical results. Figure 8b shows that the right shift of the equilibrium point q 2 with decreasing Angle of upslope. Conversely, as the Angle of downslope increases, the equilibrium point q 2 shifts to the right. Taking the equilibrium point ðq 2 ; 0Þ as the initial point, and the variable parameter q Ã is selected as 0.2 at the initial moment.
Using the package MATCONT of MATLAB, the location of the Hopf bifurcation point can be obtained in phase space as shown in Table 3. Substituting the density of Hopf bifurcation point into the Eq. (50), it is found that the existence condition of Hopf bifurcation is satisfied, which is consistence with the theoretical analysis. Figure 9a displays the bifurcation diagram of q À q Ã of the Angle parameter c ¼ 1 þ sinðp=60Þ.  Table 3 lists all Hopf  bifurcation points and the first Lyapunov coefficient u for different slope angles on the upslope and downslope. The phase diagram shown as Fig.10a and b is obtained near the Hopf bifurcation point. According to the trajectory, the system has a stable focus and an unstable limit cycle.
As can be seen from Table 3, the first Lyapunov coefficient u is greater than zero. This indicates that the Hopf bifurcation is a subcritical bifurcation. Many studies have shown that the bifurcation structure is related to traffic congestion through the transformation of traveling waves. Igarashi believes the bifurcation point determines the boundary of different traffic state in the fundamental diagram [68]. The uniform flow equilibrium of the traffic system loses its stability because of Hopf bifurcations [60]. Table 3 and (72) indicate that the direction of the Hopf bifurcation moving with the slope Angle is consistent with the critical density of global stability, and the density at Hopf bifurcation point is between the upper critical density and the lower critical density. It means that the bifurcation causes instability in the traffic system.

Conclusion
We derive a macro-continuum model of traffic flow from a microscale car-following model considering both upslope and downslope by using the transformation relationship between macro-and micro-variables. The perturbation propagation characteristics and stability conditions of the macroscopic continuum equation are discussed. Although the continuum equation has isotropic characteristics, the forward propagation characteristic velocity promptly decays and does not affect the vehicle in front. For the uniform flow in the Table 2 Types and stabilities of several equilibrium points determined using the model parameters initial equilibrium state, the stability conditions show that the upslope stability increases with increasing slope angle, and the downslope stability decreases with increasing slope angle under the action of a small disturbance. Under a large disturbance, the global stability criterion is derived using the wave front expansion technique for uniform flow in the initial equilibrium state. The upslope global stability condition is different from that in the downslope condition.
For the nonuniform flow in the initial non-equilibrium state, we perform a bifurcation analysis of the traffic flow at the equilibrium point. The results illustrate that a subcritical Hopf bifurcation exists when the traffic flow state changes. The limit cycle formed by Hopf bifurcation is unstable. Meanwhile, the existence condition of saddle node bifurcation is deduced by theoretical analysis. Simulation results verify the stability conditions of the model. The critical density range is determined. The forward propagation disturbance decreases rapidly when the characteristic velocity is larger than the traffic flow velocity. The Hopf bifurcation and saddle-node bifurcation with different slope Angle in phase space is verified via numerical simulations. The corresponding bifurcation diagram is displayed. Furthermore, the impact of the angle of both the upslope and downslope on the evolution of density waves is studied. The amplitude of the density wave decreases with increasing ascending slope angle or decreasing descending slope angle, and the length of the slope exerts a certain influence on the traffic density wave. For a given slope angle, the clusters become smaller and smoother downstream of the traffic flow when the upslope length is smaller than the downslope length.
It is of practical significance to study the nonlinear behavior of traffic congestion formation and dissipation caused by road constraints. Road traffic bottlenecks such as road reduction, on-ramp, etc., are prone to traffic congestion. And the bifurcation structure is related to traffic congestion through the change of traveling waves. The research method of this paper can be widely extended to the bifurcation phenomenon induced by various traffic bottlenecks, and the overall stability of traffic at road bottlenecks and the formation mechanism of bifurcation phenomenon can be  Fig. 9 The bifurcation diagram of q À q Ã discussed. This will be the subject of our further research in the future.