Recently, in  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.
|Journal||Discrete and Continuous Dynamical Systems|
|Publication status||Published - 1 Oct 2000|
- Polynomial vector fields
- Semi-homogeneous systems
- Singular points
- Structural stability