Asymptotic development of an integral operator and boundedness of the criticality of potential centers

We study the asymptotic development at infinity of an integral operator. We use this development to give sufficient conditions in order to upper bound the number of critical periodic orbits that bifurcate from the outer boundary of the period function of planar potential centers. We apply the main results to two different families: the power-like potential family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$, $p>q$; and the family of dehomogenized Loud's centers.Comment: 33 pages

The monotonicity of the apsidal angle using the theory of potential oscillators

In a central force system the angle between two successive passages of a body through pericenters is called the apsidal angle. In this paper we prove that for central forces of the form $f(r)\sim \lambda r^{-(\alpha+1)}$ with $\alpha<2$ the apsidal angle is a monotonous function of the energy, or equivalently of the orbital eccentricity.Comment: 4 page

On the upper bound of the criticality of potential systems at the outer boundary using the Roussarie-Ecalle compensator

This paper is concerned with the study of the criticality of families of planar centers. More precisely, we study sufficient conditions to bound the number of critical periodic orbits that bifurcate from the outer boundary of the period annulus of potential centers. In the recent years, the new approach of embedding the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary has shown to be fruitful in this issue. In this work, we tackle with a remaining case that was not taken into account in the previous studies in which the Roussarie-Ecalle compensator plays an essential role. The theoretical results we develop are applied to study the bifurcation diagram of the period function of two different families of centers: the power-like family $\ddot x=x^p-x^q$, $p,q\in\mathbb{R}$ with $p>q$; and the family of dehomogenized Loud's centers.Comment: 4 figure

Periodic oscillators, isochronous centers and resonance

An oscillator is called isochronous if all motions have a common period. When the system is forced by a time-dependent perturbation with the same period the dynamics may change and the phenomenon of resonance can appear. In this context, resonance means that all solutions are unbounded. The theory of resonance is well known for the harmonic oscillator and we extend it to nonlinear isochronous oscillators.Comment: 28 page