An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities

Abstract We consider polynomial vector fields X with a linear type and with homogenous nonlinearities. It is well-known that X has a center at the origin if and only if X has an analytic first integral of the form H = 1 2 ( x 2 + y 2 ) + ∑ j = 3 ∞ H j , where H j = H j ( x , y ) is a homogenous polynomial of degree j . The classical center-focus problem already studied by H. Poincare consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials in such a way that H is a first integral of X and consequently the origin of X will be a center. We study the particular case of centers which have a local analytic first integral of the form H = 1 2 ( x 2 + y 2 ) ( 1 + ∑ j = 1 ∞ ϒ j ) , in a neighborhood of the origin, where ϒ j is a convenient homogenous polynomial of degree j , for j ≥ 1 . These centers are called weak centers, they contain the class of center studied by Alwash and Lloyd, the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We give a classification of the weak centers for quadratic and cubic vector fields with homogenous nonlinearities.