The Hopf bifurcation of an equilibrium in dynamical systems consisting of $n$ equations with a single time delay and translational symmetry is investigated. The Jacobian belonging to the equilibrium of the corresponding delay-differential equations always have a zero eigenvalue due to the translational symmetry. This eigenvalue does not depend on the system parameters, while other characteristic roots may satisfy the conditions of Hopf bifurcation. An algorithm for this Hopf bifurcation calculation (including the center-manifold reduction) is presented. The closed form results are demonstrated for a simple model of cars following each other along a ring.
Additional information: Later published by Springer, (2004) Journal of Nonlinear Science, 14 (6), pp. 505-528. ISSN 0938-8974
Sponsorship: The authors greatly acknowledge the help of Robert Vertesi for programming and for describing his experiences in traffic jams. One of the authors (G.O.) acknowledges
with thanks discussions with Bernd Krauskopf and Eddie Wilson on tra±c dynamics. This re-
search was supported by the Hungarian National Science Foundation under grant no. OTKAT043368, by the Universities UK under an ORS Award, and by the University of Bristol under a Postgraduate Research Fellowship.