A particular system of ordinary differential equations (ODEs) modeling the traffic flow associated with \(N\) cars on a circular road of length \(L\) is analyzed. This corresponds to the so-called follow-the-leader model. The corresponding flow may present car collisions: values of time, \(t_E\), where the positions of two cars coincide, \(x_j = x_{j+1}\). The trajectory in the model, then, becomes non-physical for \(t > t_E\). The same authors in L. Buřič and V. Janovský [Physica D 237, 28–49 (2008; Zbl 1173.90003)] proposed a new interpretation of the car collision by stating that car \(j\) passes (overtakes) car \(j+1\) with a positive velocity and elaborated a new follow-the-leader model by following this approach.
In the paper under review, this alternative model is formulated in new state variables as an ODE system with a discontinuous vector field, so that it constitutes indeed a particular example of a piecewise-smooth dynamical system. The corresponding flow is analyzed for the case \(N=3\), with particular attention to its asymptotic properties as \(t \to \infty\): limit cycles and their classification, specific oscillatory patterns and even bifurcations taking place in the space of parameters. Numerical tools are used for the dynamical simulations.