Polynomial Liénard systems with a nilpotent global center

Abstract A center for a differential system $$\dot{\textbf{x}}=f(\textbf{x})$$ x ˙ = f ( ) in $${\mathbb {R}}^2$$ R 2 is singular point p having neighborhood U such that $$U\setminus \{p\}$$ U \ { p } filled with periodic orbits. global {R}}^2\setminus There are three kinds of centers, the centers Jacobian matrix Df ( ) has purely imaginary eigenvalues, nilpotent matrix, and degenerate zero matrix. For first class there several works studying when global. As far as we know no centers. One most studied classes systems polynomial Liénard systems. The objective this paper to study