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The stability of the transportation system refers to the structural stability of the system. When the system structure is unstable, local or global bifurcation phenomena will occur, which is one of the main reasons for nonlinear traffic phenomena such as congestion. To truly understand the internal mechanism of the formation of these phenomena, it is necessary to analyze the bifurcation of traffic flow. In this paper, the Hopf bifurcation control of a modified viscous macroscopic traffic flow model is studied by using the linear state feedback method, which changes the characteristics of the bifurcation phenomenon of the dynamic system and obtains the required dynamic behavior of the system. First, we can convert the original traffic model into the nonlinear ordinary differential form suitable for bifurcation analysis, solve the equilibrium point of the system, and carry out phase plane analysis. Then, the linear state feedback term is added and the corresponding controlled system is generated, the existence and type of Hopf bifurcation and the existence of saddle node bifurcation are proved. Numerical simulation results show that the analysis and control of Hopf bifurcation in the traffic model are well realized in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a novel nonuniform continuous traffic flow model, which takes into account the differences in drivers' expected headway and further improves the model by incorporating the influences of relative distance and relative velocity of vehicles through linear weighting. Additionally, the model also considers the dynamic response time of drivers for further refinement. With this model, bifurcation theory can also be applied in the stability analysis of traffic systems to study the stability mutation behavior of traffic systems at bifurcation points. Through linear and nonlinear analysis methods, the stability conditions of the model and the KdV–Burgers equation can be derived, the type of equilibrium point of the model can be judged, the conditions for the existence of Hopf bifurcations and the type of bifurcations can be proved, and the traffic flow problem can be transformed into the stability analysis problem of the system. Through numerical simulations in both high-density and low-density scenarios, it is shown that the model can well describe the actual traffic phenomena and can describe the stability abrupt behavior of the traffic system at Hopf bifurcation points and saddle-node bifurcation points through density-time and phase plane diagrams. [ABSTRACT FROM AUTHOR]

In this paper, the improved Aw-Rascle model with viscosity term is developed to a new nonlinear dynamical system composed of ordinary differential equations, by using traveling wave substitution and a kind of variable substitution. The stability of the system is discussed by the method of phase plane analysis after getting the travelling-wave solutions, then the final data is simulated by Matlab to verify the analysis conclusion.

At present, researchers have proposed many traffic flow models and research methods of traffic phenomena, but few of them are analyzed from the perspective of system stability. Therefore, this paper proposes a phase plane analysis method from the perspective of traffic flow stability. This method can describe the nonlinear traffic flow phenomenon on the road from the perspective of system global stability. The nonlinear system of the model is obtained by traveling wave substitution and Taylor expansion, and the equilibrium point of the model is solved by specifying the model parameters. According to the qualitative theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, the simulation diagram is used for numerical verification.

In this paper, the traveling wave solution of Kerner konhauser model is obtained by phase plane analysis method, and a new model suitable for stability analysis is obtained by variable transformation. The new model can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. In this paper, the nonlinear system and linear system are obtained respectively by traveling wave substitution and Taylor expansion at the equilibrium point. According to the qualitative theory of differential equations, the equilibrium point type and stability of the linear system are determined. Finally, the simulation results show the consistency between the numerical results and the theoretical analysis.

In this paper, the traveling wave solution of the macro continuum model considering the automobile tail lamp effect is obtained by using the phase plane analysis method, the type of equilibrium point of the model is determined, and the stability of the system at the equilibrium point is analyzed. This method can describe and predict the nonlinear traffic phenomena on Expressway from the perspective of system global stability. According to the theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, the simulation diagram is used for numerical simulation. Through numerical verification, it is concluded that the numerical results are consistent with the theoretical analysis.

In this paper, the phase plane analysis method is used, which can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. The nonlinear system of the model is obtained by the traveling wave substitution and Taylor expansion of the model, and the equilibrium point of the model is solved by specifying the model parameters. According to the definite theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, numerical verification is carried out according to the simulation diagram. Through numerical verification, the simulation results of the model are consistent with the conclusions of theoretical analysis. It can more clearly describe the change of density or speed with time or road section.

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.

The stochastic function describing the stochastic behavior of traffic flow in the process of acceleration or deceleration can better capture the stochastic characteristics of traffic flow. Based on this, we introduce the stochastic function into a high-order viscous continuous traffic flow model and propose a stochastic traffic flow model. Furthermore, we performed the bifurcation analysis of traffic flow system based on the model. Accordingly, the traffic flow problem is transformed into the stability analysis problem of the system, highlighting the unstable traffic characteristics such as congestion. The model can be used to study the nonlinear dynamic behavior of traffic flow. Based on this model, the existence of Hopf bifurcation and the saddle-node bifurcation is theoretically proved. And the type of the Hopf bifurcation is theoretically derived. The model can also be used to study the mutation behavior of system stability at bifurcation point. From the density space-time diagram of the system, we find that the system undergoes a stability mutation when it passes through the bifurcation point, which is consistent with the theoretical analysis results. [ABSTRACT FROM AUTHOR]

The essence of traffic congestion is a kind of bifurcation behavior incurred by the loss of stability in the traffic flow from the perspective of system stability. Therefore, researching the bifurcation characteristics of traffic flow can provide some new methods for relieving the traffic congestion. In this paper, we study the bifurcation dynamic behavior and the bifurcation conditions of traffic flow based on Kühne's continuum traffic flow model. We discussed the types and stabilities of the equilibrium solutions and proved the existence conditions of some bifurcation of the model. Then various bifurcations of the system are found by numerical simulation and the stability catastrophic behaviors initiated by some bifurcation in traffic flow are investigated. Combined with the measured data, we study the actual traffic flow bifurcation phenomena and analyze the internal reasons for the stability catastrophic behavior of traffic flow passed by the bifurcation point. The nonlinear characteristics and formation mechanism of traffic flow phenomena are revealed from a new perspective. It may help to design the corresponding control schemes for preventing and alleviating traffic congestion. [ABSTRACT FROM AUTHOR]

A modified continuum traffic flow model is established in this paper based on an extended car-following model considering driver's reaction time and distance. The linear stability of the model and the Korteweg–de Vries (KdV) equation describing the density wave of traffic flow in the metastable region are obtained. In the new model, the relaxation term and the dissipation term exist at the same time, thus the type and stability of equilibrium solution of the model can be analyzed on the phase plane. Based on the equilibrium point, the bifurcation analysis of the model is carried out, and the existence of Hopf bifurcation and saddle-node bifurcation is proved. Numerical simulations show that the model can describe the complex nonlinear dynamic phenomena observed in freeway traffic, such as local cluster effect. Various bifurcations of the model, such as Hopf bifurcation, saddle-node bifurcation, Limit Point bifurcation of cycles, Cusp bifurcation and Bogdanov–Takens bifurcation, are also obtained by numerical simulations, and the traffic behaviors of some bifurcations are studied. The results show that the numerical solution is consistent with the analytical solution. Consequently, some nonlinear traffic phenomena can be analyzed and predicted from the perspective of global stability. [ABSTRACT FROM AUTHOR]