Focus-center problem of planar degenerate system (Q932369)

The authors consider an analytic system of differential equations on a neighborhood of \((0,0)\) in \(\mathbb{R}^2\), \[ \dot x = X(x,y),\quad \dot y = Y(x,y), \] in which each of \(X\) and \(Y\) begins with terms of order \(m\). The issue is to determine when the singularity at the origin is of focus or center type (but not fully determine which of these two types). They derive conditions based on the use of so-called generalized normal sectors. They illustrate applicability of their results with an example.