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

Abstract

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