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 $\mathscr C_3.$ In this paper we prove that if we perturb any of these isochrones inside $\mathscr C_3,$ 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 $\mathscr C_2.$ Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside $\mathscr C_2,$ 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.