Structural Stability of Planar Homogeneous Polynomial Vector Fields: Applications to Critical Points and to Infinity

Abstract

LetHmbe the space of planar homogeneous polynomial vector fields of degreemendowed with the coefficient topology. We characterize the setΩmof the vector fields ofHmthat are structurally stable with respect to perturbations inHmand we determine the exact number of the topological equivalence classes inΩm. The study of structurally stable homogeneous polynomial vector fields is very useful for understanding some interesting features of inhomogeneous vector fields. Thus, by using this characterization we can do first an extension of the Hartman–Grobman Theorem which allows us to study the critical points of planar analytical vector fields whosek-jets are zero for allk<munder generic assumptions and second the study of the flows of the planar polynomial vector fields in a neighborhood of the infinity also under generic assumptions.