Subcritical Hopf bifurcations in a car-following model with reaction-time delay

G Orosz, G Stepan

Research output: Working paper

117 Citations (Scopus)
369 Downloads (Pure)


A nonlinear car-following model of highway traffic is considered, which includes the reaction-time delay of drivers. Linear stability analysis shows that the uniform flow equilibrium of the system loses its stability via Hopf bifurcations and thus oscillations can appear. The stability and amplitudes of the oscillations are determined with the help of normal-form calculations of the Hopf bifurcation that also handles the essential translational symmetry of the system. We show that the subcritical case of the Hopf bifurcation occurs robustly, which indicates the possibility of bistability. We also show how these oscillations lead to spatial wave formation as can be observed in real-world traffic flows
Original languageEnglish
Publication statusPublished - 2005

Bibliographical note

Additional information: Preprint submitted to Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Sponsorship: This research was supported by the University of Bristol under a Postgraduate Research Scholarship and by the Hungarian National Science Foundation under grant no. OTKA T043368


  • subcritical Hopf bifurcation
  • reaction-time delay
  • translational symmetry
  • bistability
  • vehicular traffic
  • stop-and-go waves


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