A traffic flow model with overtaking as a Filippov system

We investigate microscopic models of road traffic. In particular, we consider a car-following model for a single-line traffic flow of NN identical cars on a circular road. The classical differentiable model breaks down at the time instant when two cars collide. However, the natural action of a driver would be to overtake a slower car. In our previous work, we proposed a model which simulates the overtaking. We observed a large variety of oscillatory solutions (oscillatory patterns) of the model. In the present contribution, we formulate our model as a particular Filippov system, i.e., ODEs with discontinuous right hand sides. Hence, we can identify our problem with a well defined solution class. We define oscillatory patterns as invariant objects of this Filippov system.