Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.