Car-following models with delayed feedback: Local stability and Hopf bifurcation

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles driving on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of two widely-studied car-following models; namely, the classical car-following model and the optimal velocity model. We first conduct a control-theoretic analysis for both models and derive conditions that ensure local stability. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation. Qualitatively, this results in the emergence of limit cycles, which manifest as a back-propagating congestion wave. The analysis is complemented with stability charts and bifurcation diagrams. We also outline some of the implications that our results may have on the design of stable systems in the context of self-driven vehicles.