A class of recursive formulas on the calculation of focus values (Q1902250)

This note is the sketch of a work of the authors concerning the problem of distinguishing between a center and a focus for a planar system of differential equations \((*)\) \(\dot x= -y+ X(x, y)\), \(\dot y= x+ Y(x, y)\), where \(X\) and \(Y\) are analytic functions of \((x, y)\) in a neighborhood of the origin. From certain properties of a first integral and the solution series of \((*)\), the authors derive an algorithm for calculating focus values, in which the next focus value is determined by a linear combination of the preceding ones. This technique thus reduces the routine calculations to a great extent in some cases, as shown by examples in the paper.