Weak Singularities of Planar Vector Fields Under Weak Perturbation

Abstract

We characterize the elements of the set Hn of degree n homogeneous polynomial vector fields that are structurally stable with respect to perturbation in Hn, both on the plane and on the Poincare sphere. We use this information to characterize elements of the set Wn of smooth vector fields on ĝ2 beginning with terms of order n at (0, 0) that are structurally stable in a neighborhood of (0, 0) under perturbation in Wn. We also determine the set of elements of Hn that are determining for topological equivalence at (0, 0), in the sense that the topological type of the singularity at (0, 0) is invariant under the addition of higher order terms.