Asymptotic development of an integral operator and boundedness of the criticality of potential centers

We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions in order to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family $\ddot{x}=x^p-x^q$, $p,q\in \mathbb{R}$, $p>q$; and the family of dehomogenized Loud’s centers.