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

Authors: Llibre J.¹; Perez del Rio J.S.²; Rodriguez J.A.²

Source: Journal of Differential Equations, Volume 125, Number 2, March 1996 , pp. 490-520(31)

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Abstract:

Let H m be the space of planar homogeneous polynomial vector fields of degree m endowed with the coefficient topology. We characterize the set Omega m of the vector fields of H m that are structurally stable with respect to perturbations in H m and we determine the exact number of the topological equivalence classes in Omega m . 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.

Language: English

Document Type: Research article

Affiliations: 1: Facultat de Ciencies, Universitat Autonoma de Barcelona, 08913 Bellaterra, Barcelona, Spain 2: Departamento de Matematicas, Universidad de Oviedo, Avda. Calvo Sotelo, s/n, Oviedo, 33007, Spain

Publication date: 1996-03-01