Abstract. First we present the simplest criterion to decide that the H opf bifurcations of the delay differential equation $x'(t)=-\mu f(x(t-1))$ are subcritical or supercritical, as the parameter $\mu$ passes through the critical values $\mu_k$. Generally, the first Lyapunov coefficient, that d etermines 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 redu ced to a simple inequality that is trivial to check. By comparing the magni tudes of $f''(0)$ and $f'''(0)$, we can immediately tell the direction of a ll the Hopf bifurcations emerging from zero, saving us from the usual lengt hy calculations.

The main result of the paper is that we obtain upper a nd lower estimates of the periods of the bifurcating limit cycles along the Hopf branches. The proof is based on a complete classification of the poss ible bifurcation sequences and the Cooke transformation that maps branches onto each other. Applying our result to Wright's equation, we show that the $k$th Hopf branch has no folds in a neighbourhood of the bifurcation point $\mu_k$ with radius $6.83\times10^{-3}(4k+1)$. DTSTAMP:20200924T010954Z DTSTART;TZID=Europe/Budapest:20191205T103000 DTEND;TZID=Europe/Budapest:20191205T123000 SEQUENCE:0 TRANSP:OPAQUE END:VEVENT END:VCALENDAR