Hopf Bifurcation Calculations in Delayed Systems with Translational Symmetry

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 has 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.

Authors and Affiliations

1. Bristol Center for Applied Nonlinear Mathematics, Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom

   G. Orosz

2. Department of Applied Mechanics, Budapest University of Technology and Economics, Pf. 91, Budapest H-1521, Hungary

   G. Stépán

3. Research Group on Dynamics of Vehicles and Machines, Hungarian Academy of Sciences, Pf. 91, Budapest H-1521, Hungary

   G. Stépán

Corresponding authors

Correspondence to G. Orosz or G. Stépán.

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