Study On The Bifurcation Analysis Of Nonlinear Traffic Phenomena

It is really necessary to study various nonlinear traffic phenomena for ultimately alleviating and preventing traffic jams.Only when the formation mechanism and inherent law of the complex traffic phenomena were investigated comprehensively,can we organize the traffic scientifically and obtain the desired effect eventually.This paper describes various complex nonlinear traffic phenomena and analyzes the nonlinear dynamical behaviors of the traffic flow from a system stability perspective.Particularly,previous research on traffic flow bifurcation phenomena has focused on car-following models from a system local stability perspective.Few bifurcation analyses about the macroscopic traffic flow models are known.In this paper,a bifurcation analysis approach for analyzing the nonlinear traffic phenomena on the highway from a system global stability perspective is presented based on some viscid macroscopic traffic flow models.Using the bifurcation analysis approach we study the bifurcation phenomena incurred by the loss of stability in the traffic flow to seek the inherent mechanism of the nonlinear traffic phenomena and to present some suggestions to deal with the traffic congestion.The main contents of the paper are as follows:1.A phase plane analysis method for analyzing the complex nonlinear traffic phenomena is presented.This method makes use of variable substitution to transform a traditional traffic flow model into a new model which is suitable for the stability analysis in phase plane.According to the new model,various traffic phenomena,such as the well-known shock waves,rarefaction waves,and stop-and-go waves,are analyzed in the phase plane.From the phase plane diagrams,we can see the relationship between traffic jams and system instability.So the problem of traffic flow could be converted into that of system stability.The results show that the traffic phenomena described by the new method is consistent with that described by traditional methods.Moreover,the phase plane analysis highlights the unstable traffic phenomena we are chiefly concerned about and describes the variation of density or velocity with time or sections more clearly.2.Some traffic phenomena on a highway with ramps raised by different input and output flow are described by the phase plane analysis method.Using the phase plane diagrams,various complex phenomena of fixed vehicle generation rate but increasing initial homogeneous density with a single ramp and the situation of morning peak are described.According to the actual road sections of Xi'an-Baoji highways,the situations of morning peak with several ramps and the traffic jam are also analyzed.Moreover,the phase plane diagrams highlight the instability of the system.As long as the traffic becomes congested,the curves will be divergent and approach infinity in phase plane.3.A bifurcation analysis approach is presented based on the macroscopic traffic flow model.This method can be used to describe and predict the nonlinear traffic phenomena on the highway from a system global stability perspective.Based on a recently proposed speed gradient continuum traffic flow model,the types and stabilities of the equilibrium solutions are discussed and the existence of Hopf bifurcation and saddle-node bifurcation is proved.Then various bifurcations such as Hopf bifurcation,saddle-node bifurcation,Limit Point bifurcation of cycles,Cusp bifurcation and Bogdanov-Takens bifurcation are found and the traffic flow behaviors at some of them are analyzed.When the Hopf bifurcation is selected as the starting point of density temporal evolution,it may help to explain the stop-and-go traffic phenomena.4.Using the variable substitution,a density gradient continuum traffic flow model is transformed into a new model which is suitable for the stability analysis.Various bifurcation phenomena of traffic flow are described by the new model.First,the types and stabilities of the equilibrium solutions of the traffic flow system are identified based on the new model and the overall distribution structure of the nearby equilibrium solutions is given in the phase plane.Then we deduce the existence conditions of Hopf bifurcation and saddle-node bifurcation of the new model and find some bifurcations such as Hopf bifurcation,saddle-node bifurcation and Limit Point bifurcation of cycles.Furthermore,the nonlinear dynamical behaviors of the traffic flow at the Hopf bifurcation and saddle-node bifurcation are analyzed in the density temporal evolution and in the phase plane.The losses of stability in the traffic flow incurred by the bifurcation are also evaluated by using real data.It will be helpful for improving our understanding of stop-and-go wave and the sudden change of stability in real traffic flow.