Research On Bifurcation Analysis And Control Based On The Stochastic Traffic Flow Model

The rapid development of the transportation industry has been driven by urban changes,and the growing prosperity of the national economy has made urban motorization a major trend.At the same time,urban traffic congestion is also a problem that needs to be solved.Traffic congestion is an instability phenomenon of traffic flow,which is essentially a bifurcation behavior.In order to fundamentally solve this problem,it is necessary to explore the essential laws that lead to the change of traffic flow state.Starting from the bifurcation phenomenon caused by the stochastic behavior in the process of traffic flow change,and using nonlinear science theory to study its internal mechanism can fundamentally reveal the reasons for the change of traffic system,explain various traffic phenomena,and guide the organization,management,and control of traffic system.In view of this,this paper establishes a stochastic traffic flow model by considering the stochastic change process of traffic flow on the basis of the high-order continuous traffic flow model,analyzes the stochastic nonlinear dynamics of the traffic system,and studies the bifurcation phenomenon of stability mutation in the traffic system based on this model,and conducts the research of related bifurcation control.The following is the paper’s primary contribution:(1)The stochastic behavior of the traffic flow during acceleration or deceleration is considered for modeling,and stability analysis is performed for this model.An optimal speed model considering relative speed is transformed into a high-order continuous traffic flow model by converting micro-macro variables.A stochastic function considering the physical correlation of the stochastic components is added to the higher-order continuous traffic flow model to establish a stochastic traffic flow model that can reflect the uncertain behavior of the traffic flow when it accelerates or decelerates.In this paper,the equilibrium solution types of the new model are discussed by using the qualitative theory of differential equations,and the change of system stability is analyzed from the perspective of phase plane.Numerical simulations of the new model with different initial densities show that the new traffic flow model can describe various complex traffic phenomena caused by stochastic factors in the actual traffic flow,such as the stop-and-go phenomenon,the local clustering phenomenon,etc.(2)Analysis and control research of Hopf bifurcation based on the new stochastic traffic flow model.The bifurcation point reached by the system parameters analyzed when the traffic flow changes from the free-flow state to the stop-and-go phenomenon,and the bifurcation control on the bifurcation point is studied.The stochastic problem of the system is transformed into a bifurcation control problem for its equivalent deterministic system by means of Chebyshev polynomial approximation,and a feedback controller is designed to delay the occurrence of Hopf bifurcation and control the limit cycle amplitude.The complete elimination of Hopf bifurcation without changing the equilibrium point of the system can be achieved by controlling the limit cycle amplitude.The research shows that the controlled system model can realize the movement of the bifurcation point by adjusting the controller parameters,control the amplitude of the limit cycle formed by the Hopf bifurcation,make the unstable bifurcation point disappear,and then control the stability of the system.(3)Analysis and control research of saddle-node bifurcation based on the new stochastic traffic flow model.Firstly,the bifurcation point reached by the system parameters is analyzed when the traffic flow changes from the free-flow state to the local clustering phenomenon.Then,the control of this bifurcation point is performed,that is,the feedback controller is designed to control the occurrence of saddle-node bifurcation,and the changes of saddle-node bifurcation point and system stability in the controlled system are analyzed.The research shows that the bifurcation point can be moved backward and forward by adjusting the control parameters of the controlled system model,so as to prevent or alleviate traffic congestion.(4)The state change in the traffic flow system when the Cusp bifurcation occurs is studied based on the new stochastic traffic flow model.The bifurcation point reached by the system parameter when the equilibrium state of the traffic flow changes abruptly is analyzed.Firstly,the existence condition of Cusp bifurcation is deduced,that is,the sufficient and necessary condition for the Cusp bifurcation to occur in the system under the influence of stochastic variable is performed.Then,the saddle-node bifurcation point is selected as the starting point of the bifurcation calculation,the Cusp bifurcation with codimension 2 is obtained.Furthermore,numerical simulations were performed by substituting the values of relevant parameters when the Cusp bifurcation appeared,and the stability changes and equilibrium state changes of the system after the Cusp bifurcation were analyzed.The study show that the system state changes from one stable equilibrium state to two or more unstable equilibrium states when the uniform traffic flow passes through the Cusp bifurcation,highlighting the instability of the equilibrium state change when the system passes through the Cusp bifurcation.It contributes to a better understanding of the nonlinear traffic flow phenomenon.