Quasi-homogeneous linearization of degenerate vector fields

Abstract

We characterize the analytic planar vector fields orbitally equivalent to its quasi-homogeneous leader term, by means the existence of a class of inverse integrating factors. Such a class of inverse integrating factors is determined by providing a normal form of two-dimensional scalar functions. This fact allows us to give some relevant criteria on t-linearization, analytical integrability and characterization of the centers for several families of planar vector fields. We study the systems whose leader term is quadratic or cubic and we analyze also a class of nilpotent systems