On pattern formation in a class of traffic models

We consider a simple microscopic follow-the-leader model of NN identical cars on a circular road. In the classical setting, a car is not allowed to reach its leader. If it happens at a particular time tEtE, we propose to extend the model beyond this time tEtE: The idea is to define a physically reasonable leader for each car. Mathematically, the resulting model is a kind of Filippov system. The objective is to study a long time behavior of the model. In case N=3N=3, we observed several invariant patterns which apparently exhibit spatial–temporal symmetries. We managed to classify and identify two kinds of rotating waves. The classification exploits symbolic dynamical notions.