Structural stability of planar semi-homogeneous polynomial vector fields. Applications to critical points and to infinity

Jaume Llibre, Jesús S. Pérez Del Río, J. Angel Rodríguez

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8 Citations (Scopus)

Abstract

Recently, in [9] we characterized the set of planar homogeneous vector fields that are structurally stable and obtained the exact number of the topological equivalence classes. Furthermore, we gave a first extension of the Hartman-Grobman Theorem for planar vector fields. In this paper we study the structural stability in the set Hm,n of planar semi-homogeneous vector fields X = (Pm, Qn), where Pm and Qn are homogeneous polynomial of degree m and n respectively, and 0 < m < n. Unlike the planar homogeneous vector fields, the semi-homogeneous ones can have limit cycles, which prevents to characterize completely those planar semi-homogeneous vector fields that are structurally stable. Thus, in the first part of this paper we will study the local structural stability at the origin and at infinity for the vector fields in Hm,n. As a consequence of these local results, we will complete the extension of the Hartman-Grobman Theorem to the nonlinear planar vector fields. In the second half of this paper we define a subset Δm,n that is dense in Hm,n and whose elements are structurally stable. We prove that there exist vector fields in Δm,n that have at least m+n/2 hyperbolic limit cycles.
Original languageEnglish
Pages (from-to)809-828
JournalDiscrete and Continuous Dynamical Systems
Volume6
Issue number4
Publication statusPublished - 1 Oct 2000

Keywords

  • Polynomial vector fields
  • Semi-homogeneous systems
  • Singular points
  • Structural stability

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