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 a recently-proposed car-following model; namely, the Reduced Classical Car-Following Model (RCCFM). Specifically, we analyse the RCCFM in three regimes -- no delay, small delay and arbitrary delay. The control-theoretic analyses for no-delay and small-delay regimes yield a sufficient condition for local stability of the RCCFM. Next, we derive the necessary and sufficient condition for local stability of the RCCFM for an arbitrary delay. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation. #R##N#In the context of self-driven vehicles, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the RCCFM, thus ensuring smooth traffic flow and good ride quality. Next, we characterise the rate of convergence of the RCCFM, and bring forth the interplay between non-oscillatory convergence and the rate of convergence of the RCCFM. Finally, we present some simulations that validate the theoretical analysis. #R##N#Qualitatively, since a Hopf bifurcation results in the emergence of limit cycles, we characterise the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincar\'{e} normal forms and the center manifold theory. These limit cycles manifest as back-propagating congestion waves. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations.
['Gopal Krishna Kamath', 'Krishna Jagannathan', 'Gaurav Raina']
The reduced classical car-following model: stability analyses and design guidelines