Geometric criterium in the center problem (Q727546)

Among other results the authors prove the following theorem and use it to solve the center-focus problem in various cases, particularly degenerate ones. Theorem. A smooth system of differential equations \(\dot x = F(x)\) on \({\mathbb R}^2\) with \(F(0) = 0\) admits a smooth Lyapunov function if and only if there exists a smooth system \(\dot x = X(x)\) that has a center at the origin and is such that \(X \wedge F\) does not change sign in a neighborhood of the origin.