Isochronicity conditions for some real polynomial systems

TL;DR

This paper investigates conditions for isochronicity in polynomial systems, identifying new centers and providing methods for linearization, with computational verification up to degree 10.

Contribution

It introduces new isochronous centers for polynomial systems and develops computational methods for their analysis using Urabe's theorem.

Findings

New isochronous centers for degree 4 and 5 polynomials

A linearizing change of coordinates for homogeneous perturbations

Existence of a unique isochronous center in Abel polynomial systems up to degree 10

Abstract

This paper focuses on isochronicity of linear center perturbed by a polynomial. Isochronicity of a linear center perturbed by a degree four and degree five polynomials is studied, several new isochronous centers are found. For homogeneous isochronous perturbations, a first integral and a linearizing change of coordinates are presented. Moreover, a family of Abel polynomial systems is also considered. By investigations until degree 10 we prove the existence of a unique isochronous center. These results are established using a computer implementation based on Urabe theorem.