Nonlinear Dynamics of Traffic Jams

We review current progress and challenges in modeling of nonlinear phenomena in traffic flow. We are particularly interested in the nonlinear dynamics of traffic jams: the transition to instabilities of traffic flow, the formation of traffic jams and the evolution of the traffic jams. A discrete dynamic system approach is proposed. The self- organized oscillatory behavior and chaotic behavior in traffic are identified and formulated. The results can help to explain the appearance of a phantom traffic jam observed in real traffic. It is also proved that coarser resolution results in richer dynamics of the traffic flow including traffic jams. This is exactly what the numerical simulations showed. It is clear that the spatial and temporal resolution affects traffic variable and assumptions in a considerable way. Continuum models run into the limits of describing discrete phenomena in a continuous way. Therefore a combination of continuum and discrete descriptions will be necessary to further improve and declare specific traffic phenomena.