Hopf bifurcation and period functions for Wright type delay differential equations

TL;DR

This paper provides a simple criterion for determining the direction of Hopf bifurcations in Wright-type delay differential equations, classifies bifurcation sequences, and analyzes period properties of resulting limit cycles.

Contribution

It introduces a straightforward criterion for Hopf bifurcation direction, classifies bifurcation sequences, and studies period functions in Wright-type delay differential equations.

Findings

Complete classification of bifurcation sequences

Local estimates of period functions

Relation to negative Schwarzian derivative

Abstract

We present the simplest criterion that determines the direction of the Hopf bifurcations of the delay differential equation $x'(t)=-\mu f(x(t-1))$, as the parameter $\mu$ passes through the critical values $\mu_k$. We give a complete classification of the possible bifurcation sequences. Using this information and the Cooke-transformation, we obtain local estimates and monotonicity properties of the periods of the bifurcating limit cycles along the Hopf-branches. Further, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.