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

TL;DR

This paper investigates the upper bounds of critical periodic orbits near the outer boundary of potential systems, employing the Roussarie-Ecalle compensator to analyze bifurcations in specific families of centers.

Contribution

It introduces a novel application of the Roussarie-Ecalle compensator to bound bifurcations at the outer boundary of potential centers, extending previous Chebyshev system approaches.

Findings

Bounded the number of bifurcating critical orbits for specific potential systems.

Applied theoretical results to power-like and dehomogenized Loud's centers.

Enhanced understanding of bifurcation structures at the outer boundary.

Abstract

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.