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Balázs István (SZTE): Hopf bifurcation for Wright-type delay differential equations: the simplest formula, period estimates, and the absence of folds

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Thursday, 5. December 2019, 10:30 - 12:30

Abstract. 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 critical values $\mu_k$. 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 $k$th Hopf branch has no folds in a neighbourhood of the bifurcation point $\mu_k$ with radius $6.83\times10^{-3}(4k+1)$.
Location : Bolyai Intézet, I. emelet, Riesz terem, Aradi vértanúk tere 1., Szeged


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