On the period function of Newtonian systems

TL;DR

This paper investigates the period function of a class of planar Newtonian systems, providing conditions for isochronicity, characterizing centers, and introducing new families of isochronous centers.

Contribution

It offers a comprehensive characterization of isochronous centers for the system, including necessary and sufficient conditions, and introduces new families of such centers.

Findings

A sufficient condition for isochronicity at a center.

Necessary and sufficient conditions when functions are analytic.

New families of isochronous centers identified.

Abstract

We study the existence of centers of planar autonomous system of the form $$(S) \quad \dot x=y,\qquad \dot y = -h(x) - g(x)y - f(x)y^2.$$ We are interested in the period function $T$ around a center 0. A sufficient condition for the isochronicity of (S) at 0 is given. Such a condition is also necessary when $f,g,h$ are analytic functions. In that case a characterization of isochronous centers of system (S) is given. Some applications will be derived. In particular, new families of isochronous centers will be described