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

  • Gopal Krishna KamathEmail author
  • Krishna Jagannathan
  • Gaurav Raina
Original Paper


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 three regimes—no delay, small delay and arbitrary delay. First, we derive a sufficient condition for local stability of the CCFM in no-delay and small-delay regimes using control-theoretic methods. 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. This ensures smooth traffic flow and good ride quality. 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. We then study the nonlinear oscillations in system dynamics that emerge when the CCFM loses local stability via a Hopf bifurcation. To that end, we outline an analytical framework to establish the type of the Hopf bifurcation and the asymptotic orbital stability of the emergent limit cycles using Poincaré normal forms and the center manifold theory. Next, we numerically bring forth the supercritical nature of the bifurcation that result in asymptotically orbitally stable limit cycles. The analysis is complemented with stability charts, bifurcation diagrams and MATLAB simulations. Thus, using a combination of analysis and numerical computations, we highlight the trade-offs inherent among various system parameters and also provide design guidelines for the upper longitudinal controller of self-driven vehicles.


Transportation networks Car-following models Time delays Stability Convergence Hopf bifurcation limit cycles 



This work is undertaken as a part of an Information Technology Research Academy (ITRA), Media Lab Asia, project titled “De-congesting India’s transportation networks.” The authors are also thankful to Debayani Ghosh and Sreelakshmi Manjunath for many helpful discussions.

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest concerning the publication of this manuscript.

Ethical standards

The authors state that this research work complies with ethical standards.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Electrical EngineeringIndian Institute of Technology MadrasChennaiIndia

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