Published online by Cambridge University Press: 01 December 2021
In this paper we consider the unfolding of saddle-node
with $\varepsilon \approx 0$
and $a$
in an open subset $A$
of $ {\mathbb {R}}^{\alpha },$
and we study the Dulac time $\mathcal {T}(s;\varepsilon,\,a)$
of one of its hyperbolic sectors. We prove (theorem 1.1) that the derivative $\partial _s\mathcal {T}(s;\varepsilon,\,a)$
tends to $-\infty$
as $(s,\,\varepsilon )\to (0^{+},\,0)$
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.