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

In this paper we consider the unfolding of saddle-node \[ X= \frac{1}{xU_a(x,y)}\Big(x(x^{\mu}-\varepsilon)\partial_x-V_a(x)y\partial_y\Big), \] parametrized by $(\varepsilon,\,a)$ with $\varepsilon \approx 0$ and $a$ in an open subset $A$ $ {\mathbb {R}}^{\alpha },$ study Dulac time $\mathcal {T}(s;\varepsilon,\,a)$ one its hyperbolic sectors. We prove (theorem 1.1) that derivative $\partial _s\mathcal tends to $-\infty$ as $(s,\,\varepsilon )\to (0^{+},\,0)$ uniformly on compact subsets $A.$ This result is addressed bifurcation critical periods Loud's family quadratic centres. regard show 1.2) no occurs from certain semi-hyperbolic polycycles.