Polynomial Liénard systems with a nilpotent global center

Abstract

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