A sufficient and necessary condition of generalized polynomial Liénard systems with global centers

The aim of this paper is to give a sufficient and necessary condition of the generalized polynomial Liénard 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 Liénard 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 Liénard differential systems of degree 5 having a global center at the origin and the explicit expression of a generalized polynomial Liénard differential system of indefinite degree having a global center at the origin.

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