Centers of quasi-homogeneous polynomial planar systems

TL;DR

This paper classifies centers in quasi-homogeneous polynomial planar systems of degrees 0 to 4, exploring their reversibility and integrability, and identifying new types of polynomial centers with unique properties.

Contribution

It provides a comprehensive classification of centers for low-degree quasi-homogeneous polynomial systems, including new cases that are neither reversible nor integrable.

Findings

Identified polynomial centers that are neither orbitally reversible nor analytically integrable.

Extended the understanding of center types in polynomial systems.

Provided explicit classifications for degrees 0 to 4.

Abstract

In this paper we determine the centers of quasi-homogeneous polynomial planar vector fields of degree 0, 1, 2, 3 and 4. In addition, in every case we make a study of the reversibility and the analytical integrability of each one of the above centers.Wefind polynomial centers which are neither orbitally reversible nor analytically integrable, this is a new scenario in respect to the one of non-degenerate and nilpotent centers.