Impact of Delayed Acceleration Feedback on the Reduced Classical Car-Following Model

Abstract

Time delays play an important role in determining the qualitative dynamical properties of a platoon of self-driven vehicles driving on a straight road. In this paper, we investigate the impact of Delayed Acceleration Feedback (DAF) on the dynamics of the Reduced Classical Car-Following Model (RCCFM). We first derive the Reduced Classical Car-Following Model with Delayed Acceleration Feedback (RCCFM-DAF). Next, we demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation. The analysis also yields the necessary and sufficient condition for local stability. We characterise the type of Hopf bifurcation and the asymptotic orbital stability of the emergent limit cycles for the RCCFM by using Poincare normal forms and the center manifold theory. We then use this analysis to infer requisite insights into the RCCFM-DAF by means of an appropriately defined linear transformation. The analysis is complemented with a stability chart and a bifurcation diagram. Our work reveals three effects of DAF on the RCCFM: (i) reduction in the stable region, (ii) increase in the frequency of the emergent limit cycles, and (iii) decrease in the amplitude of the emergent limit cycles. This, in turn, has two immediate repercussions: (i) decrease in resilience to the reaction delay, and (ii) an increase in the risk of a collision due to jerky vehicular motion.