Structural stability of planar quasi-homogeneous vector fields

In this work, we study the structural stability of planar quasi-homogeneous vector fields under quasi-homogeneous perturbations, and provide a complete classification. This study, which has been the subject of previous works, is only complete in the homogeneous case. The main tool in our analysis is a splitting of planar quasi-homogeneous vector fields into conservative-dissipative parts. Moreover, we describe the topological equivalence classes in the set of the structurally stable planar quasi-homogeneous vector fields. Finally, we include some examples where the equivalence classes are determined.