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Summary: HOPF CALCULATIONS IN DELAYED CAR-FOLLOWING MODELS
G´abor St´ep´an
and G´abor Orosz
Department of Applied Mechanics, Budapest University
of Technology and Economics, Budapest, H-1111, Hungary
Mathematics Research Institute, University of Exeter,
Laver Building, North Park Road, Exeter, EX4 4QE, UK
Abstract: A nonlinear car-following model that includes the reaction-time delay of
drivers is considered. When investigating the linear stability of the uniform flow
solution, boundaries of Hopf bifurcations are determined in the parameter space.
Crossing these boundaries, oscillations may appear corresponding to travelling
wave solutions. Hopf normal form calculations prove robustly subcritical behavior
which leads to bistability between the stable uniform traffic flow and the stop-
and-go waves travelling against the flow of vehicles. Analogies with wheel shimmy
dynamics and machine tool vibrations are presented. Copyright c 2006 IFAC
Keywords: vehicular traffic, reaction-time delay, translational symmetry,
subcritical Hopf bifurcation, bistability
1. INTRODUCTION
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