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

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

2010-10-12

arxiv.org/abs/1010.2391v7

Discrete and Continuous Dynamical Systems Series A 32(8), 2607-2651, 2012

Mathematics

Dynamical Systems

minor revision

Scientific paper

10.3934/dcds.2012.32.2607

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 (SD-DDEs), 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 alternative and extension to the original Hopf bifurcation theorem for SD-DDEs by Eichmann (2006).

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