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, θ), to an equation over a cylinder in which the singular point p ₀ corresponds to a limit cycle γ ₀. This equation over the cylinder always has an inverse integrating factor which is smooth and non-flat in r in a neighborhood of γ ₀. We define the notion of vanishing multiplicity of the inverse integrating factor over γ 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.