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

## TL;DR

This paper analyzes the asymptotic behavior of an integral operator to establish bounds on the number of bifurcating critical periodic orbits in planar potential centers, with applications to specific potential families.

## Contribution

It introduces new asymptotic analysis techniques to bound critical orbits in potential centers, extending understanding of bifurcation phenomena.

## Key findings

- Bounded the number of bifurcating critical orbits
- Applied results to power-like potential family
- Extended analysis to dehomogenized Loud's centers

## Abstract

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.