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

Abstract

Abstract. We consider polynomial vector fields of the formwhere Xm = Xm(x, y) and Ym = Ym(x, y) are homogenous polynomials of degree m. It is well-known that X has a center at the origin if and only if X has an analytic first integral of the formwhereThe classical center-focus problem already studied by H. Poincaré 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 X m and Y m in such a way that H is a first integral of X and consequently the origin of X will be a center. Moreover, we study the case whenwhere Υ j is a convenient homogenous polynomial of degree j for j ≥ 1. The solution of the inverse center problem for polynomial differential systems with homogenous nonlinearities, provides a new mechanism to study the center problem, which is equivalent to Liapunov's Theorem and Reeb's criterion.

An inverse approach to the center problem

“…Weak condition for a center. The following condition weak condition for a center was due to Alwash and Lloyd [1,18], see also [18].…”

“…From (24) where ν = ν(x, y) is an arbitrary function. Denoting ν = 1 + Λ we get that differential equations (4) coincide with (18). On the other hand in view of the relations…”

“…but it is not isochronous see [5]. In fact in [18] we provide all the quadratic system with weak centers.…”

“…If the vector field is polynomial of degree m, then by Bezout Theorem the maximum number of singular points of system (18) is (m − 1) 2 + 1. (b) If in (18) we assume that Λ = ω(x 2 + y 2 ) − 1, and φ = λ(x 2 + y 2 ), then system (18) becomeṡ…”

Center problem for Λ–Ω differential systems

“…This kind of centers having first integral of the form (5) are called weak centers, they contain the uniform isochronous centers and the holomorphic isochronous centers (for a prof of these results see [11]), but they do not coincide with the all class of isochronous centers (see Remark 19 of [11]).…”