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This paper is concerned with the study of the criticality of families ofplanar centers. More precisely, we study sufficient conditions to bound thenumber of critical periodic orbits that bifurcate from the outer boundary ofthe period annulus of potential centers. In the recent years, the new approachof embedding the derivative of the period function into a collection offunctions that form a Chebyshev system near the outer boundary has shown to befruitful in this issue. In this work, we tackle with a remaining case that wasnot taken into account in the previous studies in which the Roussarie-Ecallecompensator plays an essential role. The theoretical results we develop areapplied to study the bifurcation diagram of the period function of twodifferent families of centers: the power-like family $\ddot x=x^p-x^q$,$p,q\in\mathbb{R}$ with $p>q$; and the family of dehomogenized Loud's centers.

On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator