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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:
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:20240328T122904Z
DTSTART;TZID=Europe/Budapest:20191205T103000
DTEND;TZID=Europe/Budapest:20191205T123000
SEQUENCE:0
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