Bifurcations and multiple traffic jams in a car-following model with reaction-time delay

We investigate an optimal velocity car-following model for n cars on a circular single-lane road, where reaction-time delay of drivers is taken into account. The stability of the uniform flow equilibrium is studied analytically, while bifurcating periodic solutions for different wave numbers are investigated with numerical continuation techniques. This reveals that the periodic solution with the smallest wave number may be stable, and all other periodic solutions are unstable. s increased, periodic solutions develop stop- and go-fronts that correspond to rapid deceleration and acceleration between regions of uniformly flowing and stagnant traffic. In terms of the positions of all cars on the ring these fronts are associated with traffic jams. All traffic jams form a traffic pattern that evolves under time, due to slow motion of the fronts. The traffic pattern corresponding to the stable periodic motion of cars is the only stable one. However, we find that other periodic orbits may be unstable only so weakly that they give rise to transient traffic jams that may persist for long times. Eventually, such traffic jams either merge with one another or disperse, until the stable traffic pattern is reached.