Solvable optimal velocity models and asymptotic trajectory

In the optimal velocity model proposed as a 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 with a time delay T. This leads to a global pattern formation in time development of the vehicle's motion, and gives rise to a closed trajectory on Δx-v (headway-velocity) plane connecting congested and free flow points. To obtain the closed trajectory analytically, we propose an 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 ${\mathrm{v}}_{\mathrm{B}}$ (the backward velocity of the pattern), the global parameters associated with the vehicle's collective motion in a congested flow, in terms of parameters, such as the sensitivity a, which appeared in the original coupled equations.