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DCDS

Recently, in [9] we characterized the set of planar
homogeneous
vector fields that are structurally stable and we 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 $H_{m,n}$
of planar semi-homogeneous
vector fields $X = (P_m,Q_n)$, where $P_m$ and $Q_n$
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
$H_{m,n}$. 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
$\Delta_{m,n}$ that is dense in $H_{m,n}$
and whose elements are structurally stable.
We prove that there exist vector
fields in $\Delta_{m,n}$ that have at least
$(m+n)/2$ hyperbolic limit cycles.

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