Criticality via first order development of the period constants

In this work we study the criticality of some planar systems of polynomial differential equations having a center for various low degrees n. To this end, we present a method which is equivalent to the use of the first non-identically zero Melnikov function in the problem of limit cycles bifurcation, but adapted to the period function. We prove that the Taylor development of this first order function can be found from the linear terms of the corresponding period constants. Later, we consider families which are isochronous centers being perturbed inside the reversible centers class, and we prove our criticality results by finding the first order Taylor developments of the period constants with respect to the perturbation parameters. In particular, we obtain that at least 22 critical periods bifurcate for n=6, 37 for n=8, 57 for n=10, 80 for n=12, 106 for n=14, and 136 for n=16. Up to our knowledge, these values improve the best current lower bounds.