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.
Original languageEnglish
Pages (from-to)104-114
Number of pages11
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Issue number1
Publication statusPublished - 1 Dec 2021


  • asymptotic expansions
  • Dulac time
  • Period function
  • saddle-node unfolding
  • Asymptotic expansions
  • Saddle-node unfolding


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