On Chouikha's isochronicity criterion

TL;DR

This paper provides an alternative proof and new developments on Chouikha's criterion for determining when a center at the origin of a differential equation is isochronic, enhancing understanding of the criterion.

Contribution

It offers a new proof of Chouikha's isochronicity characterization and introduces additional insights into the subject.

Findings

Alternative proof of Chouikha's criterion

New theoretical developments in isochronicity characterization

Enhanced understanding of center conditions

Abstract

Recently A.R.Chouikha gave a new characterization of isochronicity of center at the origin for the equation $x"+g(x)=0$, where $g$ is a real smooth function defined in some neighborhood of $0 \in \R$. We describe another proof of his characterization and some new development of the subject.