Microscopic traffic follow-the-leader models are described by 2N - dimensional nonlinear ODEs, where N is the number of cars. A prototype is given by Bando (1995). We assume periodic boundary conditions to restrict our analysis to a circular road of length L, where bottleneck like roadwork are taken into account. The model shows a rich bifurcation scenario — Hopf bifurcations, folds and Neimark-Sacker bifurcation, By performing a numerical bifurcation and continuation analysis with respect to the two parameters ε (bottleneck strength) and L (the length of the circle) we encounter very interesting traffic dynamics like so-called Ponies-on-a-Merry-Go-Round-solutions (POMs) and quasi-POMs. We find the coexistence of stable POMs and quasi-POMs for the same parameter set where the bottleneck has qualitatively different influences on the dynamics. The analysis is based on the computation of fixed points and closed invariant curves of reduced Poincaré maps which correspond to POMs and quasi-POMs respectively.

There is an interesting macroscopic visualization of the microscopic dynamics as shown below for a quasi-POM. Here the single trajectories of the individual cars are shown colored according to their speed. We observed a time-periodicity of the macroscopic patterns of quasi-POMs. Using dynamical system theory for circle maps we are able to prove this observation.

References:
M. Bando, K. Hasebe, A. Nakayama, A. Shibata, Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Phys. Rev. E, 1995
I. Gasser, B. Werner, Dynamical Phenomena in Bottlenecks, Phil. Trans. R. Soc. A 2010 368, 4543-4562 (Special Issue Highway Traffic - Dynamics and Control)