Global Hopf bifurcations of neutral functional differential equations with state-dependent delay

Xiuli Sun,Rong Yuan,Yunfei Lv

doi:10.3934/dcdsb.2018038

Abstract

A global Hopf bifurcation theory for a system of neutral functional differential equations (NFDEs) with state-dependent delay is investigated by applying the \begin{document}$S^{1}$\end{document} -equivariant degree theory. We use the information about the characteristic equation of the formal linearization with frozen delay to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system. The results are important in studying bifurcations of NFDEs with state-dependent delay.

Highlights

A great deal of research has been devoted to the Hopf bifurcation of functional differential equations of retarded type
We develop the Hopf bifurcation theory for neutral functional differential equations (NFDEs) with state-dependent delay by using S1-equivariant degree theory as the main mathematical tool
Our results provide a general tool and framework to study the Hopf bifurcation problem and, in particular, the global continuation of local bifurcation of periodic solutions of NFDEs with state-dependent delay from an equivariant degree point of view

Summary

Introduction

A great deal of research has been devoted to the Hopf bifurcation of functional differential equations of retarded type. If f (v, λ) is differentiable with respect to v, we are able to define singular points of system (3) through its linearization at the trivial solutions of (2) This is not so for the Hopf bifurcation problem of (1), as in the spaces of continuous periodic functions CT (R, RN ) = {x ∈ C(R, RN ) : x(t + T ) = x(t) for all t ∈ R} and CT (R, R) = {τ ∈ C(R, R) : τ (t + T ) = τ (t) for all t ∈ R} with a fixed period T , the composition operator χ : CT (R, RN ) × CT (R, R) → CT (R, RN ), χ(x, τ )(t) = x(t − τ (t)), t ∈ R, is generally not a C1 (continuously differentiable) map with respect to τ in the supremum norm.

It is clear that w

Suppose there exists a sequence

Let ω

Then we obtain

Conclusion