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 p0 of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider p0 being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of p0 the differential system can always be brought, by means of a change to (generalized) polar coordinates (r, θ), to an equation over a cylinder in which the singular point p0 corresponds to a limit cycle γ0. This equation over the cylinder always has an inverse integrating factor which is smooth and non–flat in r in a neighborhood of γ0. We define the notion of vanishing multiplicity of the inverse integrating factor over γ0. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point p0 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. 2000 AMS Subject Classification: 37G15, 37G10, 34C07