Structural stability of planar homogeneous polynomial vector fields: Applications to critical points and to infinity

Let Hm be the space of planar homogeneous polynomial vector fields of degree m endowed with the coefficient topology. We characterize the set Ωm of the vector fields of Hm that are structurally stable with respect to perturbations in Hm and we determine the exact number of the topological equivalence classes in Ωm. 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 whose k-jets are zero for all k < m under 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. © 1996 Academic Press, Inc.