Stability analysis of delayed car-following models Linear stability analysis of first-order delayed car-following models on a ring Sylvain Lassarre, 1 Michel Roussignol, 2 and Antoine Tordeux 3 1) IFSTTAR–GRETTIA, Descartes II, 2 rue de la Butte Verte, 93166 Noisy le Grand, France a) 2) Universit´ e Paris-Est–LAMA, 5 bvd Descartes, 77454 Marne-la-Vall´ ee, France b) 3) (Corresp. author) Universit´ e Paris-Est–LVMT, 19 rue Alfred Nobel, 77455 Marne-la-Vall´ ee, France c) The evolution of a line of vehicles on a ring is modeled by means of first-order car-following models. Three generic models describe the speed of a vehicle as a function of the spacing ahead and of the speed of the predecessor. The first model is a basic one with no delay. The second is a delayed car-following model with a strictly positive parameter for the driver / vehicle reaction time. The last model includes a reaction time parameter with an anticipation process where the delayed position of the predecessor is estimated. Explicit conditions for the linear stability of homogeneous configurations are calculated for each model. Two methods of calculus are compared : an exact one via Hopf bifurcations and an approximation by second-order models. The conditions describe stable areas for the parameters of the models that we interpret. The results notably show that the impact of the reaction time on the stability can be palliated by the anticipation process. PACS numbers: 05.45.-a 45.70.Vn 45.70.Qj 89.40.Bb Keywords: Car-following traffic flow model — First-order delayed differential equations system — Linear stability analysis — Hopf bifurcation a) Electronic mail: sylvain.lassarre@ifsttar.fr b) Electronic mail: michel.roussignol@univ-paris-est.fr c) Electronic mail: antoine.tordeux@enpc.fr 1