Solvable Optimal Velocity Models and Asymptotic Trajectory

Details Solvable Optimal Velocity Models and Asymptotic Trajectory Solvable Optimal Velocity Models and Asymptotic Trajectory

1996-10-23

arxiv.org/abs/patt-sol/9610009v1

Nonlinear Sciences

Pattern Formation and Solitons

25 pages, 15 eps figures, LaTeX

Scientific paper

10.1103/PhysRevE.55.6519

In the Optimal Velocity Model proposed as a new version of Car Following Model, it has been found that a congested flow is generated spontaneously from a homogeneous flow for a certain range of the traffic density. A well-established congested flow obtained in a numerical simulation shows a remarkable repetitive property such that the velocity of a vehicle evolves exactly in the same way as that of its preceding one except a time delay $T$. This leads to a global pattern formation in time development of vehicles' motion, and gives rise to a closed trajectory on $\Delta x$-$v$ (headway-velocity) plane connecting congested and free flow points. To obtain the closed trajectory analytically, we propose a new approach to the pattern formation, which makes it possible to reduce the coupled car following equations to a single difference-differential equation (Rondo equation). To demonstrate our approach, we employ a class of linear models which are exactly solvable. We also introduce the concept of ``asymptotic trajectory'' to determine $T$ and $v_B$ (the backward velocity of the pattern), the global parameters associated with vehicles' collective motion in a congested flow, in terms of parameters such as the sensitivity $a$, which appeared in the original coupled equations.

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