Bifurcation of critical periods from Pleshkan's isochrones

Abstract

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 𝒷3. In this paper we prove that if we perturb any of these isochrones inside 𝒷3, 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 𝒷2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in 𝒷2. We prove that if we perturb three of them inside 𝒷2, 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.