Non-bifurcation of critical periods from semi-hyperbolic polycycles of quadratic centres

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In this paper we consider the unfolding of saddle-node X=1xUa(x,y)(x(xμ−ε)∂x−Va(x)y∂y), parametrized by (ε,a) with ε≈0 and a in an open subset A of Rα, and we study the Dulac time T(s;ε,a) of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative ∂sT(s;ε,a) tends to −∞ as (s,ε)→(0+,0) uniformly on compact subsets of A. This result is addressed to study the bifurcation of critical periods in the Loud's family of quadratic centres. In this regard we show (theorem 1.2) that no bifurcation occurs from certain semi-hyperbolic polycycles.
Idioma originalEnglish
Pàgines (de-a)104-114
Nombre de pàgines11
RevistaProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volum153
Número1
DOIs
Estat de la publicacióPublicada - 1 de des. 2021

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