Stability analysis in Greenberg’s traffic model

Abstract

In this document, Greenberg’s classical traffic model is analyzed by varying the critical velocity [Formula: see text] that appears in it as a parameter. This model is transformed as a discrete expression and manipulated by means of isolating variables such as vehicular density [Formula: see text], flow [Formula: see text] and velocity [Formula: see text] that appear in the model. So-called fundamental diagrams are obtained by pairing two out of these three variables, and then they are used to support trajectories of iterated processes depicting distinct behaviors of traffic. As it is shown in this paper, those behaviors have monotonous and stable patterns for some range of values of the parameter [Formula: see text], but they can change to cyclic and stable trajectories when this parameter is modified. If the modifications of that parameter are taken to a higher range of values, then cycles of different periods appear, reaching chaotic trajectories. These behaviors are analyzed from a dynamical approach, proving their stability conditions, and then they are illustrated with simulations in each case.