Theory of center-focus for a class of higher-degree critical points and infinite points
Here, the real planar autonomous differential systems with higher-degree critical points (including elementary critical points and infinite points) \[ \frac{dx}{dt}=\sum_{k=2n+1}^{\infty}X_{k}(x,y)=X(x,y),\quad \frac{dy}{dt}=\sum_{k=2n+1}^{\infty}Y_{k}(x,y)=Y(x,y), \] and \[ \frac{dx}{dt}=\sum_{k=0}^{2n+1}X_{k}(x,y),\quad \frac{dy}{dt}=\sum_{k=0}^{2n+1}Y_{k}(x,y), \] are studied, where \(X_{k}(x,y), Y_{k}(x,y)\) are homogeneous polynomials of degree \(k\) and the functions \(X(x,y), Y(x,y)\) are analytic at the origin. The results extend statements concerning the detection of center or focus, successor function, central integration, integration factor, focal values, values of singular points and bifurcation of limit cycles.
scientific article; zbMATH DE number 1782095