A sufficient and necessary condition of generalized polynomial Li\'enard systems with global centers

TL;DR

This paper establishes a comprehensive criterion for identifying when generalized polynomial Li'enard systems possess a global center, simplifying previous conditions and providing explicit examples for systems of degree 5 and indefinite degree.

Contribution

It offers a new, simpler necessary and sufficient condition for global centers in generalized polynomial Li'enard systems, extending prior work to include linear and nilpotent types.

Findings

Derived explicit conditions for degree 5 systems with global centers

Provided explicit examples of systems with indefinite degree and global centers

Simplified the criteria compared to previous results by Llibre and Valls

Abstract

The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Li\'enard system with a global center (including linear typer and nilpotent type). Recently, Llibre and Valls [J. Differential Equations, 330 (2022), 66-80] gave a sufficient and necessary condition of the generalized polynomial Li\'enard system with a linear type global center. It is easy to see that our sufficient and necessary condition is more easy by comparison. In particular, we provide the explicit expressions of all the generalized polynomial Li\'enard differential systems of degree 5 having a global center at the origin and the explicit expression of a generalized polynomial Li\'enard differential system of indefinite degree having a global center at the origin.