Hopf bifurcation for Wright-type delay differential equations: The simplest formula, period estimates, and the absence of folds

Abstract

First we present the simplest criterion to decide that the Hopf bifurcations of the delay differential equation x′(t)=−μf(x(t−1)) are subcritical or supercritical, as the parameter μ passes through the critical values μk. Generally, the first Lyapunov coefficient, that determines the direction of the Hopf bifurcation, is given by a complicated formula. Here we point out that for this class of equations, it can be reduced to a simple inequality that is trivial to check. By comparing the magnitudes of f″(0) and f‴(0), we can immediately tell the direction of all the Hopf bifurcations emerging from zero, saving us from the usual lengthy calculations.The main result of the paper is that we obtain upper and lower estimates of the periods of the bifurcating limit cycles along the Hopf branches. The proof is based on a complete classification of the possible bifurcation sequences and the Cooke transformation that maps branches onto each other. Applying our result to Wright’s equation, we show that the kth Hopf branch has no folds in a neighbourhood of the bifurcation point μk with radius 6.83×10−3(4k+1).Finally, we show how our results relate to the often required property that the nonlinearity has negative Schwarzian derivative.