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

This talk introduces a recent result for giving a sufficient and necessary condition of the generalized  polynomial Lienard system with a global center (including linear type and nilpotent type). Recently, Llibre and Valls [J. Differential Equations, 330 (2022), 66-80] gave a sufficient and necessary condition of the generalized polynomial Lienard 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 Lienard differential system of indefinite degree having a global center at the origin.