Abstract
A follow-the-leader model of traffic flow on a closed loop is considered in the framework of the extended optimal velocity (OV) model where the driver reacts to both the following and the preceding car. Periodic wave train solutions that describe the formation of traffic congestion patterns are found analytically. Their velocity and amplitude are determined from a perturbation approach based on collective coordinates with the discrete modified Korteweg–de Vries equation as the zero order equation. This contains the standard OV model as a special case. The analytical results are in excellent agreement with numerical solutions.
Highlights
To model real traffic situations with a high density of cars in the flow, the optimal velocity (OV) models are usually used on an infinite line or on a ring
A follow-the-leader model of traffic flow on a closed loop is considered in the framework of the extended optimal velocity (OV) model where the driver reacts to both the following and the preceding car
In [22], a bifurcation analysis was carried out for a rather general class of OV functions V (u), and it was proven that the loss of stability of the free-flow solution is generally due to a Hopf bifurcation
Summary
We will assume that the driver reacts to a decreasing distance as well as an increasing distance between his car and the car that is in front of him and take the forward looking OV in the form. H is the sum of the car length and safety distance between cars. The parameter f 0 denotes the forward sensitivity of the model. We will assume that the driver reacts to a change of the distance between his car and the car that is behind it and take the backward looking OV in the form. The parameter b 0 gives the backward sensitivity and we use the same safety distance h as in the forward OV expression in (4)
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