Establishing definitive conditions for global centers in generalized polynomial Liénard systems

In this study, the primary objective is to establish a comprehensive and indispensable criterion for identifying generalized polynomial Liénard systems with a linear or nilpotent global center. Our study not only tackles these variations, but also solves the center-focus problem at infinity for the generalized polynomial Liénard system, a challenge that was previously approached in the study 'Polynomial Liénard Equations near infinity' in the Journal of Differential Equations (1999), volume 153, pages 1-29.Recently, a necessary and sufficient condition was proposed by Llibre and Valls (2022) for the generalized polynomial Liénard system with a linear global center. In comparison, the proposed conditions are relatively simple. Our criterion offers explicit formulations for all generalized polynomial Liénard differential systems of the fifth degree having a global center at the origin. Furthermore, we provide a distinct expression for a generalized polynomial Liénard differential system of indefinite degree, distinguished by a global center at the origin.