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CATEGORIES:{lang hu}Differenciálegyenletek szeminárium{/lang}{lang en}Differential equations seminar{/lang}
SUMMARY:Balázs István (SZTE): Hopf bifurcation for Wright-type delay differential equations: the simplest formula, period estimates, and the absence of folds
LOCATION:Bolyai Intézet, I. emelet, Riesz terem, Aradi vértanúk tere 1., Szeged
DESCRIPTION;ENCODING=QUOTED-PRINTABLE:\nAbstract. First we present the simplest criterion to decide that the Hopf
bifurcations of the delay differential equation $x'(t)=-\mu f(x(t-1))$ are
subcritical or supercritical, as the parameter $\mu$ passes through the cr
itical values $\mu_k$. Generally, the first Lyapunov coefficient, that dete
rmines the direction of the Hopf bifurcation, is given by a complicated for
mula. 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 magnitud
es 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.\nThe main result of the paper is that we obtain upper and low
er estimates of the periods of the bifurcating limit cycles along the Hopf
branches. The proof is based on a complete classification of the possible b
ifurcation sequences and the Cooke transformation that maps branches onto e
ach 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:20210117T073717Z
DTSTART;TZID=Europe/Budapest:20191205T103000
DTEND;TZID=Europe/Budapest:20191205T123000
SEQUENCE:0
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