Bifurcation of critical periods from Pleshkan's isochrones

TL;DR

This paper investigates how small perturbations of specific cubic and quadratic isochrone centers affect the bifurcation of critical periods, establishing upper bounds and existence results for the number of bifurcating critical periods.

Contribution

It proves bounds on the number of critical periods bifurcating from perturbed cubic and quadratic isochrones and constructs perturbations realizing these bounds.

Findings

At most two critical periods bifurcate from cubic isochrones.

Exactly k critical periods can be realized for k=0,1,2 in cubic case.

At most one critical period bifurcates from quadratic 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 $\mathscr C_3.$ 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.