On general algebraic mechanisms for producing centers in polynomial differential systems (Q948488)

This article deals with a classification of nondegenerate centers for the class of generalized polynomial Liénard systems \[ \dot{x} = P_{3}(x)y,{\quad}\dot{y} = P_{0}(x) + P_{1}(x)y + P_{2}(x)y^{2}, \] where the \(P_i(x)\) are polynomials in \(x\) over \(\mathbb R\) with \(P_{3}(0)\neq0,\) \(P_{0}(0)=0\) and \(P_{3}(0)P'_{0}(0)<0\). The authors consider two general mechanisms for producing centers: 1) by finding a first integral using the method of Darboux and 2) by producing a system by means of a pull-back of a nonsingular differential equation along a map of algebraic nature. The goal of the paper is to show how the mentioned two mechanisms yield necessary and sufficient conditions for a center at the origin in the above class of generalized polynomial Liénard systems. One of the main tools in this investigation is the change of variables of \textit{L. A. Cherkas} [Differ. Uravn. 10, 367--368 (1974; Zbl 0296.34020)].