On the number of critical periods for planar polynomial systems

In this paper we get some lower bounds for the number of critical periods of families of centers which are perturbations of the linear one. We give a method which lets us prove that there are planar polynomial centers of degree ℓℓ with at least 2[(ℓ−2)/2]2[(ℓ−2)/2] critical periods as well as study concrete families of potential, reversible and Liénard centers. This last case is studied in more detail and we prove that the number of critical periods obtained with our approach does not increases with the order of the perturbation.