Car-following models with delayed feedback: Local stability and Hopf bifurcation

Reaction delays play an important role in determining the qualitative dynamical properties of a platoon of vehicles driving on a straight road. In this paper, we investigate the impact of delayed feedback on the dynamics of two widely-studied car-following models; namely, the classical car-following model and the optimal velocity model. We first conduct a control-theoretic analysis for both models and derive conditions that ensure local stability. We then demonstrate that the transition of traffic flow from the locally stable to the unstable regime occurs via a Hopf bifurcation. Qualitatively, this results in the emergence of limit cycles, which manifest as a back-propagating congestion wave. The analysis is complemented with stability charts and bifurcation diagrams. We also outline some of the implications that our results may have on the design of stable systems in the context of self-driven vehicles.

[1]  Rifat Sipahi,et al.  Analytical stability study of a deterministic car following model under multiple delay interactions , 2006 .

[2]  L C Davis.  Comment on "Analysis of optimal velocity model with explicit delay". , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[3]  Elen Twrdy,et al.  Optimal velocity functions for car-following models , 2010 .

[4]  Gábor Stépán,et al.  Traffic jams: dynamics and control , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[5]  M. Hoagland,et al.  Feedback Systems An Introduction for Scientists and Engineers SECOND EDITION , 2015 .

[6]  Robert Herman,et al.  Traffic Dynamics: Analysis of Stability in Car Following , 1959 .

[7]  Jianhong Wu,et al.  Introduction to Functional Differential Equations , 2013 .

[8]  Lucien Duckstein,et al.  Stability of Reciprocal-Spacing Type Car Following Models , 1967, Transp. Sci..

[9]  G. Stépán,et al.  Subcritical Hopf bifurcations in a car-following model with reaction-time delay , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  L. C. Davis,et al.  Modifications of the optimal velocity traffic model to include delay due to driver reaction time , 2003 .

[11]  K. Hasebe,et al.  Analysis of optimal velocity model with explicit delay , 1998, patt-sol/9805002.

[12]  Martin Treiber,et al.  How Reaction Time, Update Time, and Adaptation Time Influence the Stability of Traffic Flow , 2008, Comput. Aided Civ. Infrastructure Eng..

[13]  J. Silvester.  Determinants of block matrices , 2000, The Mathematical Gazette.

[14]  A. Schadschneider,et al.  Statistical physics of vehicular traffic and some related systems , 2000, cond-mat/0007053.

[15]  Dirk Roose,et al.  Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL , 2002, TOMS.

[16]  E. Montroll,et al.  Traffic Dynamics: Studies in Car Following , 1958 .

[17]  Samuel Greengard.  Smart transportation networks drive gains , 2015, Commun. ACM.

[18]  Davis Lc,et al.  Comment on "Analysis of optimal velocity model with explicit delay". , 2002 .

[19]  I. Gasser,et al.  Bifurcation analysis of a class of ‘car following’ traffic models , 2004 .

[20]  Gaurav Raina,et al.  FAST TCP: Some queueing models and stability , 2014, 2014 International Conference on Signal Processing and Communications (SPCOM).

[21]  Nakayama,et al.  Dynamical model of traffic congestion and numerical simulation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[22]  G. Ladas,et al.  Oscillation Theory of Delay Differential Equations: With Applications , 1992 .

[23]  R. E. Wilson,et al.  Car-following models: fifty years of linear stability analysis – a mathematical perspective , 2011 .

[24]  D. Helbing.  Traffic and related self-driven many-particle systems , 2000, cond-mat/0012229.

[25]  K. Engels,et al.  [Reaction time]. , 2021, Deutsche medizinische Wochenschrift.

[26]  Gaurav Raina,et al.  Local bifurcation analysis of some dual congestion control algorithms , 2005, IEEE Transactions on Automatic Control.

[27]  Pak Chiu Leung,et al.  Numerical bifurcation analysis of delay differential equations , 1982 .

[28]  D. Gazis,et al.  Nonlinear Follow-the-Leader Models of Traffic Flow , 1961 .

[29]  R. E. Wilson,et al.  Bifurcations and multiple traffic jams in a car-following model with reaction-time delay , 2005 .

[30]  David Jarrett,et al.  Stability analysis of the classical car-following model , 1997 .