Header menu link for other important links
X
Stability, convergence and Hopf bifurcation analyses of the classical car-following model
Gopal Krishna Kamath, ,
Published in Springer Netherlands
2019
Volume: 96
   
Issue: 1
Pages: 185 - 204
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 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. © 2019, Springer Nature B.V.
About the journal
JournalData powered by TypesetNonlinear Dynamics
PublisherData powered by TypesetSpringer Netherlands
ISSN0924090X
Open AccessYes
Concepts (20)
  •  related image
    Approximation theory
  •  related image
    Convergence of numerical methods
  •  related image
    Design
  •  related image
    Economic and social effects
  •  related image
    Longitudinal control
  •  related image
    MATLAB
  •  related image
    Stability
  •  related image
    System theory
  •  related image
    Time delay
  •  related image
    Traffic congestion
  •  related image
    Vehicles
  •  related image
    Car following models
  •  related image
    Convergence
  •  related image
    HOPF BIFURCATION ANALYSIS
  •  related image
    LIMIT-CYCLE
  •  related image
    Longitudinal controllers
  •  related image
    Numerical computations
  •  related image
    OSCILLATORY CONVERGENCE
  •  related image
    Transportation network
  •  related image
    Hopf bifurcation