Stability, convergence and Hopf bifurcation analyses of the classical car-following model

TL;DR

This paper analyzes how reaction delays affect the stability and dynamics of the classical car-following model, identifying conditions for stability, oscillations, and congestion waves, with implications for traffic flow and autonomous vehicle control.

Contribution

It provides new stability conditions for the CCFM under various delay regimes and characterizes the nature of bifurcations and limit cycles, informing traffic management and vehicle control design.

Findings

Derived stability conditions for no-delay, small-delay, and arbitrary delay regimes.

Identified Hopf bifurcation as the mechanism for transition to traffic congestion.

Characterized the type of bifurcation and stability of limit cycles.

Abstract

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles traversing a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of the Classical Car-Following Model (CCFM). Specifically, we analyze the CCFM in no delay, small delay and arbitrary delay regimes. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using. Next, we derive the necessary and sufficient condition for local stability of the CCFM 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, thus resulting in limit cycles in system dynamics. Physically, these limit cycles manifest as back-propagating congestion waves on highways. In the context of human-driven vehicles, our work provides phenomenological insight into the impact of reaction delays on the emergence and evolution of traffic congestion. In the context of self-driven vehicles, our work has the potential to provide design guidelines for control algorithms running in self-driven cars to avoid undesirable phenomena. Specifically, designing control algorithms that avoid jerky vehicular movements is essential. Hence, we derive the necessary and sufficient condition for non-oscillatory convergence of the CCFM. Next, we characterize the rate of convergence of the CCFM, and bring forth the interplay between local stability, non-oscillatory convergence and the rate of convergence of the CCFM. Further, to better understand the oscillations in the system dynamics, we characterize the type of the Hopf bifurcation and the asymptotic orbital stability of the limit cycles using Poincare normal forms and the center manifold theory. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations.