A new single lane car following model of traffic flow is presented. The model is inertial and free of collisions. It demonstrates experimentally observed features of traffic flow such as the existence of three regimes: free, fluctuative (synchronized) and congested (jammed) flow; bistability of free and fluctuative states in a certain range of densities, which causes the hysteresis in transitions between these states; jumps in the density-flux plane in the fluctuative regime and gradual spatial transition from synchronized to free flow. Our model suggests that in the fluctuative regime there exist many stable states with different wavelengths, and that the velocity fluctuations in the congested flow regime decay approximately according to a power law in time.In the last years, growing effort has been made in understanding traffic flow dynamics. Recent experiments [1,2] show that traffic flow demonstrates complex physical phenomena, among which are:(i) The existence of three states: free flow, "synchronized" (or "fluctuative") flow and traffic jams (for low, intermediate and high densities correspondingly). The second state has two essential features: synchronization of flow in different lanes (for the multilane traffic) and fluctuation performed by the system in density-flux plane. Since our model is single-lane we will refer to this state as "fluctuative".(ii) Hysteresis which is observed in transitions between the free and the fluctuative flow.
We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.
We study the presence of chaos in a car-following traffic model based on a system of delay-differential equations. We find that for low and high values of cars density the system has a stable steady-state solution. Our results show that above a certain time delay and for intermediate density values the system passes to chaos following the Ruelle-Takens-Newhouse scenario (fixed point-limit cycles-two-tori-three-tori-chaos). Exponential decay of the power spectrum and non-integer correlation dimension suggest the existence of chaos. We find that the chaotic attractors are multifractal. (1)Similarly to classical Mackey-Glass [5] and Ikeda [6] equations.
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