Structural Stability of Planar Quasihomogeneous Vector Fields

Abstract

Let \(H_{pqm}\) be the space of all planar \((p,q)\)-quasihomogeneous vector fields of weight degree \(m\) endowed with the coefficient topology. In this paper we characterize the set \(\Omega _{pqm}\) of all vector fields in \(H_{pqm}\) which are structurally stable with respect to perturbations in \(H_{pqm}\) in the Poincare disc, and determine the exact number of the topological equivalence classes in \(\Omega _{pqm}\) in terms of \(p,q\) and \(m\). This characterization is applied to give an extension of the Hartman–Grobmann Theorem at the origin for \((p,q)\) quasihomogeneous vector fields of weight degree greater than \(m\) starting with a term \(X_m \in \Omega _{pqm}\). This work is an extension of the Llibre et al.’s paper (J Differ Equ 125:490–520, 1996), where the homogeneous case was considered.