Finding periodic orbits  in state-dependent delay differential equations as roots of  algebraic equations

Jan Sieber

doi:10.3934/dcds.2012.32.2607

Article Contents

2012, Volume 32, Issue 8: 2607-2651. Doi: 10.3934/dcds.2012.32.2607

This issue Previous Article Preface Next Article Phase models and oscillators with time delayed coupling

Finding periodic orbits in state-dependent delay differential equations as roots of algebraic equations

Jan Sieber^1,

1.
Department of Mathematics, Lion Gate Building, University of Portsmouth, Portsmouth, PO1 3HF

Received: October 31, 2010

Revised: August 31, 2011

Published: July 2012

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Abstract

In this paper we prove that periodic boundary-value problems (BVPs) for delay differential equations are locally equivalent to finite-dimensional algebraic systems of equations. We rely only on regularity assumptions that follow those of the review by Hartung et al. (2006). Thus, the equivalence result can be applied to differential equations with state-dependent delays, transferring many results of bifurcation theory for periodic orbits to this class of systems. We demonstrate this by using the equivalence to give an elementary proof of the Hopf bifurcation theorem for differential equations with state-dependent delays. This is an extension of the Hopf bifurcation theorem by Eichmann (2006), along with an alternative proof.

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Mathematics Subject Classification: Primary: 34K13; Secondary: 34K17.

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