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

TL;DR

This paper establishes that periodic boundary-value problems for delay differential equations, including state-dependent delays, are equivalent to finite-dimensional algebraic systems, enabling easier analysis of bifurcations such as Hopf bifurcations.

Contribution

It proves the local equivalence between periodic BVPs for SD-DDEs and algebraic systems, extending bifurcation theory tools to this class of equations.

Findings

Equivalence allows algebraic methods for SD-DDEs

Elementary proof of Hopf bifurcation for SD-DDEs

Extension of classical bifurcation results to state-dependent delays

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 (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).