Stability and instability in a class of car following model on a closed loop

A velocity-matching car following model is modified to represent the motion of n vehicles travelling on a closed loop. Each vehicle is given a preferred velocity profile, which it attempts to achieve while also attempting to maintain a zero relative velocity between itself and the vehicle in front. The crucial distinctive of the looped model, as opposed to ‘non-looped’ models, is that the last vehicle in the stream is itself being followed by the lead (first) vehicle. The model gives rise to a system of n coupled time delay differential equations which are solved approximately (using a Taylor series expansion in time delay) and numerically using a fourth-order Runge–Kutta routine. The stability of the model is considered and an analytic form of the stable region in parameter space is found in the limit as n approaches infinity.