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 C3. In this talk, we show that if we perturb any of these isochrones inside C3, 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 C2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in C2. We will see that if we perturb three of them inside C2, 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 will also be discussed.

This talk is based upon a joint work with Jordi Villadelprat (URV).