Bifurcation of critical periods from Pleshkan's isochrones

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities C-3. In this paper we prove that if we perturb any of these isochrones inside C-3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers C-2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in C-2. We prove that if we perturb three of them inside C-2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper.