Bifurcation analysis of a class of ‘car following’ traffic models

We consider a follow-the-leader traffic model describing the dynamics of N cars on a circular road, where each car driver chooses his acceleration according to a certain law. The model is represented by a nonlinear system of ODE’s. This model is known to have a solution with constant velocities and headways which, in a certain parameter regime, is stable. Varying the density of the cars, we prove that the loss of stability is generally due to a Hopf bifurcation. Also we investigate numerically the global bifurcation diagram for periodic solutions and obtain a complete picture of the dynamics of general optimal velocity models. Finally, some analytical results on the stability of solutions in the case of non-equal drivers are given.