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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 98, pp. 1--28.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/98\hfil Structural Stability]
{Structural stability of polynomial second order differential
  equations with periodic coefficients}

\author[Adolfo W. Guzman\hfil EJDE-2004/98\hfilneg]
{Adolfo W. Guzm\'an}


\curraddr{Departamento de Matem\'atica, Universidade Federal de Vi\c cosa \\
  Campus Universit\'ario CEP 36571-000. Vi\c cosa - MG. Brasil\\
  \newline
  Departamento de Matem\'atica Aplicada,
  IME, Universidade de S\~ao Paulo \\
  Rua do Mat\~ao, 1010 - Cidade Universit\'aria  
  CEP 05508-090 S\~ao Paulo -  SP - Brasil}
\email{guzman@ime.usp.br, guzman@ufv.br}


\date{}
\thanks{Submitted April 29, 2004. Published August 9, 2004.}
\thanks{Supported by Grant 02/13419-5 from FAPESP, Brazil}
\subjclass[2000]{37C20}
\keywords{Singularity at infinity; compactification; structural stability;
\hfill\break\indent second order differential equation}


\begin{abstract}
 This work characterizes the structurally stable second order
 differential equations  of the form
 $x''= \sum_{i=0}^{n}a_{i}(x)(x')^{i}$ where
 $a_{i}:\Re \to \Re$ are $C^r$ periodic functions.
 These equations have naturally the cylinder $M= S^1\times \Re$
 as the phase space and are associated to the vector fields
 $X(f) = y \frac{\partial}{\partial x}
  + f(x,y) \frac{\partial}{\partial y}$,  where
 $f(x,y)=\sum_{i=0}^n a_i(x) y^i \frac{\partial}{\partial y}$.
 We apply a compactification to $M$ as well as to $X(f)$ to
 study the behavior at infinity. For $n\geq 1$, we define a set
 $\Sigma^{n}$ of $X(f)$ that is open and dense and characterizes
 the class of structural differential equations as above.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

 We denote by $\mathcal{E}^{n,r}$ the space of vector fields
\[
X(f) = y \frac{\partial}{\partial x} + \sum_{i=0}^{n} a_{i}(x)
y^{i} \frac{\partial}{\partial y}
\]
defined on $M=S^{1}\times\Re$ where  $a_{i}(x)$ are $C^{r}$
periodic functions, $r\geq 1$ and $n\geq 1$.

A vector field $X(f)$ is associated  naturally, it is in fact
equivalent,  to the second order differential equation
$$E_{f}:x''= f(x,x') \quad \text{where}\quad
f(x,y)=\sum_{i=0}^{n}a_{i}(x)y^{i}.
$$
We endow  $\mathcal{E}^{n,r}$ with the structure in which $X(f)$
is identified with the $n+1$-tuple $(a_0(x), \dots, a_n(x))$ of
its coefficient functions and the norm is defined by
$$||X(f)||= \sup_{1 \leq k \leq r}\{ |\frac{d^k}{d x^k}a_i(x)| :
x\in S^1, 0\leq i \leq n \}.$$

The aim of this paper is to characterize the vector fields $ X(f)
\in \mathcal{E}^{n,r}$ (therefore, $E_f$) that are structurally
stable under small   perturbations in the space
$\mathcal{E}^{n,r}$. See precise definition below.

We establish the  structural stability of $X(f) \in
\mathcal{E}^{n,r} $ on the open surface $M$ using a
compactification of the type $ u=x$ and $ v= \arctan(y)$
(cylindrical compactification). We denote the  compactifications
of $X(f)$ and $M$ by $\widetilde {X}(f)$ and $\widetilde{M}$
respectively. In section \ref{sec:compactification}, we find the
following expressions:
\\
For $n=1,2,3$, $\widetilde{X}(f)=  \sin(v)\frac{\partial}
{\partial u} + \sum_{i=0}^{n} a_{i}(u) \cos^{3-i}(v) \sin^{i}(v)
 \frac{\partial}{\partial v}$;
\\
for $n\geq 4$, $\widetilde{X}(f) =
\sin(v) \cos^{n-3}(v)\frac{\partial} {\partial u}+ \sum_{i=0}^{n}
a_{i}(u) \sin^{i}(v) \cos^{n-i}(v) \frac{\partial}{\partial v}$.
\\
 This allows us to
understand the behavior of $X(f)$ at infinity (i.e at the ends) of
$M$ by studying $\widetilde{X}(f)$ near the boundary of
$\widetilde{M}$. Thus we find  that on $\partial \widetilde{M}$,
$\widetilde{X}(f)$ has: periodic orbits when $n=1,2$; tangency
points when  $n=3$; hyperbolic singularities when  $n=4$; and
semi-hyperbolic or nilpotent singularities when $n>4$. See section
\ref{sec:behavior}.


The characterization  of structurally stable vector fields $X(f)$
on $M$ is expressed in terms of $\widetilde{X}(f)$ on
$\widetilde{M}$. For that, we give the following definition:

\begin{definition} \label{def1} \rm
A vector field $X(f)$ is structurally stable in $M$ if there is a
neighborhood $\mathcal{U}$ in $\mathcal{E}^{n,r}$ such that
$\forall X(g) \in \mathcal{U}$, there exists a homeomorphism $h_g :
\widetilde{M} \to \widetilde{M}$ which maps trajectories
of $\widetilde{X}(f)$ onto trajectories of $\widetilde{X}(g)$,
preserving orientation and $\partial \widetilde{M}$.
\end{definition}

In section \ref{sec:definition}, we define for each $n\geq 1$ a
subset $\Sigma^{n}$ of $\mathcal{E}^{n,r}$ such that if $X(f) \in
\Sigma^{n}$, its  compactification $\widetilde{X}(f)$ has generic
properties with  respect to singularities, to periodic orbits and
to connections of singularity separatrix on $\widetilde{M}$. In
this way, we extend  the conditions of Peixoto M.M and Peixoto M.C
(see \cite{peixoto-peixoto}, also \cite{Sotomayor2}) that
characterize the $C^1$-structurally stable systems on closed
surfaces, with singularities on the boundary. We recall that these
conditions include tangencies at the boundary of a closed surface
which insure the $C^1$ structural stability.


The main results of this paper can be formulated as follows:

\begin{theorem}[Genericity] \label{theo:genericity}
$\Sigma^{n}$ is open and   dense
in $\mathcal{E}^{n,r}$ where $r \geq 2$ for $n =1$ or $n \geq 5$
and  $r\geq 1$ for $n=2,3$ or $4$.
\end{theorem}

\begin{theorem}[Characterization] \label{theo:characterization}
$X(f) \in \mathcal{E}^{n,r}$ is structurally stable if and only if $X(f)\in
\Sigma^{n}$ where $r \geq 2$ for $n =1$ or $n \geq 5$
and  $r\geq 1$ for $n=2,3$ or $4$.
\end{theorem}

We prove  Theorems \ref{theo:genericity} and
\ref{theo:characterization} in sections \ref{sec:genericity} and
\ref{sec:characterization} respectively.
In section  \ref{sec:discussion}, we present a discussion of the
sources that motivate this work. Our results make a link between
the works of Sotomayor \cite{sotomayor} and Barreto \cite{barreto}
and Shahshahani \cite{Shahshahani} dedicated to
$C^{r}$-structurally stable second order differential equations.
The first author considered $E_f$ with the uniform topology on
compact regions of $\Re^2$ and $M$;  the second two authors
considered $E_f$  with the Whitney topology on the whole $M$.


\section{Compactification}\label{sec:compactification}

In this section, we define a vector field on the cylinder
$\widetilde{M}= S^{1}\times [\frac{\pi}{2}, \frac{\pi}{2}]$
induced by $X(f) \in \mathcal{E}^{n,r}$, where $f(x,y) =
\sum_{i=0}^n  a_i(x) y^{i}$, and we describe $X(f)$ in coordinate
neighborhoods of the infinity of $M$. We denote by
$\widetilde{M}^{\circ} = S^1 \times (\frac{\pi}{2},
\frac{\pi}{2})$.

Let $\mathcal{C}: M \to \widetilde{M}^{\circ}$ be a
diffeomorphism defined by
$$ \mathcal{C}(x,y)= (x, \arctan(y)).$$
The ends of $M$ are transformed into the circles $C_{\pm
\frac{\pi} {2}} = S^1 \times \{\pm \frac{\pi} {2}\}$.

Now, we induce the vector field $\mathcal{C}_{\ast}(X(f))$ on
$\widetilde{M}^{\circ}= S^1 \times (\frac{\pi}{2}, \frac{\pi}{2}) $ by $X(f)$
as follows
$$ \mathcal{C}_{\ast}(X(f))(u,v)= D\mathcal{C}(x,y) \cdot
X(f)(x,y)$$ where $(u,v) = \mathcal{C}(x,y)$ and
$D\mathcal{C}(x,y)$ is the derivative of $\mathcal{C}$ at $(x,y)$.

Thus we obtain
\[
\mathcal{C}_{\ast}(X(f))(u,v) = \frac{\sin(v)}{\cos(v)}
  \frac{\partial}{\partial u} + \sum \limits_{i=0}^{n} a_{i}(u) \sin^{i}(v)
  \cos^{-i+2}(v) \frac{\partial}{\partial v}.
  \]
Then the following vector field
$$ \widetilde{X}(f) = \begin{cases}
  \cos(v) \cdot \mathcal{C}_{\ast}(X(f)) & \text{for } n=1,2,3 \\
  \cos(v)^{n-2} \cdot \mathcal{C}_{\ast}(X(f)) & \text{for } n \geq 4 \\
\end{cases} $$
can be extended to the whole $\widetilde{M}$.

 We call $\widetilde{X}(f)$ the cylindrical
compactification of $X(f)$. The explicit expressions of
$\widetilde{X}(f)$  are:
\\
for $n=1,2,3$,
\begin{equation}
 \widetilde{X}(f)=  \sin(v)\frac{\partial}
{\partial u} + \sum_{i=0}^{n} a_{i}(u) \cos^{3-i}(v) \sin^{i}(v)
 \frac{\partial}{\partial v}\label{eq:xfn123}
\end{equation}
and for $n\geq 4$,
\begin{equation}
 \widetilde{X}(f) = \sin(v) \cos^{n-3}(v)\frac{\partial} {\partial u}+
 \sum_{i=0}^{n} a_{i}(u) \sin^{i}(v) \cos^{n-i}(v) \frac{\partial}{\partial
 v}.\label{eq:xfn4}
\end{equation}

In the sequel, we write $X(f)$ in coordinate neighborhoods of the ends
of $M$. Under the map $\Upsilon: U^{\pm} \to
\widetilde{U}^{\pm}$ defined by  $\Upsilon(x,y) = (x,\frac{1}{y})$
for $y\neq 0$, the sets
\begin{gather*}
U^{+}=\{(x,y)\in M: x\in S^{1},y > y_0 \},\\
U^{-}=\{(x,y)\in M: x\in S^{1},y < -y_0 \}
\end{gather*}
(where $y_0 \in \Re^{+}$) are transformed into
\begin{gather*}
\widetilde{U}^{+} = \{ (x,y) \in M: x\in S^{1}, 0 \leq y <
y_0^{-1} \},\\
\widetilde{U}^{-} = \{ (x,y) \in M: x\in S^{1}, - y_0^{-1} < y
\leq 0 \}
\end{gather*}
respectively. The ends of $M$ are represented by the circle
$\widetilde{C}_{0} = S^{1}\times\{0\} \subset
\widetilde{U}^{\pm}$.

Now,   we induce the vector field $ \Upsilon_{\ast}(X(f))$ on
$\widetilde{U}^{\pm} \setminus \widetilde{C}_{0}$  by $X(f)$ as
follows
$$ \Upsilon_{\ast}(X(f))(u,v)= D\Upsilon(x,y) \cdot
X(f)(x,y)$$ where $(u,v) = \Upsilon(x,y)$.
Then the vector field
$$
\widetilde{X}_1(f)= \begin{cases}
  v \cdot \Upsilon_{\ast}(X(f)) & \text{for } n=1,2,3 \\
  v^{n-2} \cdot \Upsilon_{\ast}(X(f)) & \text{for } n \geq 4 \\
\end{cases}
$$
can be extended to the whole $\widetilde{U}^{\pm}$.
The explicit expressions of
$\widetilde{X}_1(f)$ are
\begin{gather}
\text{for } n=1,2,3, \quad \widetilde{X}_1(f)=
\frac{\partial}{\partial u}-
 \sum_{i=0}^{n} a_{i}(u) v^{3-i}\frac{\partial} {\partial v}
\label{eq:xf1n123}\\
\text{for } n\geq 4, \quad\widetilde{X}_1(f)=  v^{n-3}\frac{\partial}{\partial
  u}-\sum_{i=0}^{n} a_{i}(u)v^{n-i}\frac{\partial}{\partial
  v}.\label{eq:xf1n4}
\end{gather}

\section{Behavior of $X(f)$ at infinity}\label{sec:behavior}

In this section, we study the behavior of $X(f)$ near infinity
by means of its cylindrical compactification $\widetilde{X}(f)$.

\begin{proposition}\label{pro:PeriodicOrbit}
 Let $X(f) \in \mathcal{E}^{n,r}(M)$, where   $f(x,y)= \sum_{i=0}^{n} a_{i}
 (x)y^{i}$, $r\geq 2$ and  $n=1,2$. Then
 \begin{enumerate}
 \item[(a)] For $n=1$, $C_{\pm\frac{\pi}{2}}$ are periodic orbits
 of $\widetilde{X}(f)$  with the first and second derivatives of
 the Poincar\'e  map equal to $1$ and  $\pm 2\int_{0}^{\tau} a_{1}(s)ds$
 respectively.
  \item[(b)] For $n=2$, $C_{\pm\frac{\pi}{2}}$  are  periodic
  orbits of $\widetilde{X}(f)$  with the first derivative of the
  Poincar\'e map equal to  $\exp(\mp \int_{0}^{\tau} a_{2}(s) ds)$.
\end{enumerate}
\end{proposition}
\begin{proof}
 (a) Let $\gamma_{\pm}(u)= (\pm u, \pm \frac{\pi}{2})$ be the periodic orbits
 of  $\widetilde{X}(f)$ on $C_{\pm\frac{\pi}{2}}$ where $u \in
 [0,\tau]$ and  $\tau$ is the period of $a_i(u)$.
 Taking the change of variables $u = s$ and $v = \pm \frac{\pi}{2} - \eta$,
 $\widetilde{X}(f)$ is associated to the differential equations
\begin{equation}
      \begin{gathered}
        \frac{d s}{dt} =  \pm \cos(\eta) \\
        \frac{d \eta}{dt} = a_{0}(s) \sin^3(\eta) \pm a_{1}(s)
        \sin^2(\eta) \cos(\eta)
      \end{gathered}
\end{equation}
Dividing the last equation by the first, we obtain
\begin{equation}
  \label{eq:Rseta}
  \frac{d \eta}{d s} =  R(s, \eta)
\end{equation}
where $R(s,\eta) = \pm a_0(s) \frac{\sin^3(\eta)}{\cos(\eta)} + a_1(s)
   \sin^2(\eta)$ and $R(s, 0) = 0$.

Let  $\eta = f(s; s_0 , \eta_0)$ be a solution of (\ref{eq:Rseta})
with the initial condition $f(s_0; s_0, \eta_0) = \eta_0.$ Without
loss of generality we can consider $s_0 = 0$.

Now, we consider the function  $d(\eta_0)= \phi(\eta_0) - \eta_0$
where $\phi(\eta_0) = f(\tau; 0, \eta_0)$ is the first return map,
which is defined on  an  arc normal to $\gamma_{\pm}$.
The integral expressions for $d'(\eta_0)$ and $d^{\prime
\prime}(\eta_0)$ can be found in \cite{Andronov} page 252. Their calculation
is included here for the sake of completeness.  From the relation $ \frac{d
  \eta}{d s} = R(s, f(s; 0,\eta_0)) = \frac{d}{d s} f(s; 0, \eta_0)$, we obtain
\begin{gather}
  \label{eq:deriva1feta0}
  \frac{d}{d s}(\frac{\partial f}{\partial \eta_0}) = \frac{\partial}
  {\partial \eta} R(s, f(s; 0, \eta_0)) \cdot \frac{\partial f}{\partial
  \eta_0}(s; 0, \eta_0) \\
\label{eq:deriva2feta0}\begin{aligned}
  \frac{d}{d s}(\frac{\partial^2 f}{\partial \eta^{2}_{0}}) &=
 \frac{\partial^2} {\partial \eta^2} R(s, f(s; 0, \eta_0)) \cdot
 (\frac{\partial f}{\partial \eta_0}(s; 0, \eta_0))^2  \\
 &\quad + \frac{\partial}{\partial \eta} R(s, f(s; 0, \eta_0)) \frac{\partial^2
 f}{\partial \eta_0^2}(s; 0, \eta_0)
\end{aligned}
\end{gather}
The solutions of  (\ref{eq:deriva1feta0}) and  (\ref{eq:deriva2feta0}) are
given by
\begin{gather}
  \label{eq:solderiva1feta0}
  \frac{\partial f}{\partial \eta_0}(s; 0, \eta_0) =  \exp(\int_{0}^{s}
  \frac{\partial} {\partial \eta} R(t, f(t; 0, \eta_0)) dt)\\
  \label{eq:solderiva2feta0}
\begin{aligned}
  \frac{\partial^2 f}{\partial \eta^{2}_{0}}(s;0,\eta_0) & =
 \exp(\int_{0}^{s} \frac{\partial} {\partial \eta} R(t, f(t; 0, \eta_0)) dt)
 \cdot \int_{0}^{s} \frac{\partial^2}{\partial \eta^2} R(t, f(t; 0, \eta_0))\\
 &\quad\cdot (\frac{\partial f}{\partial \eta_0}(t; 0, \eta_0))^2 \cdot
 \exp(-\int_{0}^{t} \frac{\partial}{\partial \eta} R(\tilde{t}, f(t; 0,
 \eta_0)) d \tilde{t}) dt
\end{aligned}
\end{gather}
The partial derivatives of $R(s, \eta)$ with respect to $\eta$
are:
 \begin{gather}
  \label{eq:derivaReta} \frac{\partial R}{\partial \eta}(s, \eta)  =
  \pm a_{0}(s)(3\sin^2(\eta) + \frac{\sin^4(\eta)}{\cos^2(\eta)}) + 2 a_1(s)
  \sin(\eta) \cos(\eta) \\
  \label{eq:deriva2Reta} \begin{aligned}
 \frac{\partial^2 R}{\partial \eta^2}(s, \eta) & =
   \pm a_{0}(s)(6 \frac{\sin(\eta)}{ \cos^3(\eta)} + 4
  \frac{\sin^3(\eta)}{\cos(\eta)} + 2 \frac{\sin^5(\eta)}{\cos^3(\eta)}) \\
 &\quad   + 2 a_1(s)(\cos^2(\eta)- \sin(\eta)).
 \end{aligned}
\end{gather}
Since $\eta_{0} = 0$ and  $f(s; 0, \eta_0) = \eta_0$, it follows  from
 (\ref{eq:solderiva1feta0})-(\ref{eq:deriva2Reta}),
\begin{gather*}
  d(\eta_0) = f(\tau;0,\eta_0) - \eta = 0\\
  d'(\eta_0) = 0 \\
  d''(\eta_0) = \pm 2 \int_{0}^{\tau} a_1(s) ds
\end{gather*}
(b) It follows from a straightforward computation of the divergence of
\begin{align*}
\widetilde{X}(f) &=  \sin(v)\frac{\partial}{\partial u}+(a_{0}(u)\cos^{3}
(v)+a_{1}(u)\sin(v)\cos^{2}(v)\\
&\quad + a_{2}(u)\sin^{2}(v)\cos(v))\frac{\partial}{\partial v}
\end{align*}
i.e. for a periodic orbit  $\gamma_{\pm} = (\pm u , \pm \frac{\pi}{2}))$,
$\mathop{\rm div}D\widetilde{X}(\gamma_{\pm}) = \mp a_2(u)$.
\end{proof}


We say that $\widetilde{X}(f)$ is \emph{transversal to} $\partial
\widetilde{M}$ at $(u,v)$ if
$$\widetilde{X}(f)\cdot b(u,v)  \doteq \langle \widetilde{X}(f)(u,v),
\nabla b(u,v)\rangle \neq 0,
$$
where $b(u,v)$ is a $C^2$ function such
that $b(u,v) =0$ on $\partial \widetilde{M}$, $b(u,v) > 0$ on
$\widetilde{M}^{\circ}$ (interior of  $\widetilde{M}$) and $\nabla
b(u,v) \neq 0$ at $(u,v) \in \partial \widetilde{M}$.

When $ \widetilde{X}(f)\cdot b(u,v) =0$, we say that
$\widetilde{X}(f)$ is tangent to $\partial \widetilde{M}$.
Moreover, we say that the tangency is \emph{parabolic} if
$$\widetilde{X}^2(f)  \cdot b(u,v) \doteq \widetilde{X}(f)
\cdot (\widetilde{X}(f) \cdot b(u,v)) \neq 0.$$

\begin{proposition}
\label{pro:Tangencia} Let $\widetilde{X}(f) \in \mathcal{E}^{3,r}
(\widetilde{M})$, where $f(x,y)=\sum_{i=0}^{3}a_{i}(x)y^{i}$ and
$r \geq 1$. Then the  trajectories of $\widetilde{X}(f)$ are
transversal to the circles $C_{\pm\frac{\pi}{2}}$,  except at
points $(u_{\ast},\pm\frac{\pi}{2})$ in which they are tangent,
with $a_3(u_{\ast})=0$. The tangency is parabolic if
$a_{3}'(u_{\ast})\neq 0$. The tangency is external (resp.
internal) in $C_{+\frac{\pi}{2}}$ and internal (resp. external) in
$C_{-\frac{\pi}{2}}$ when $a_{3}'(u_{\ast})>0$ (resp.
$a_{3}'(u_{\ast})< 0$).
\end{proposition}

\begin{proof}
We consider the function $b:\widetilde{M} \to\Re$ defined
by $b(u,v)=\cos(v).$ It satisfies $b(u,v)>0$ on
$\widetilde{M}^{\circ}$ and $b(u,v)=0$ for $(u,v)\in
C_{\pm\frac{\pi}{2}}$.

Here, the  condition of the transversality of $\widetilde{X}(f)$
to $C_{\pm\frac{\pi}{2}}$ is given by
$$ \widetilde{X}(f) \cdot b(u,v)= -\sum_{i=0}^{3} a_{i}(u)
\sin^{i+1}(v) \cos^{3-i}(v) = - a_{3}(u) \neq 0
$$
for $(u,v)\in C_{\pm\frac{\pi}{2}}$.

The  condition of parabolic tangency is given by  $\widetilde{X}(f)\cdot
b(u,v) = 0$ and
$\widetilde{X}^{2}(f) \cdot b(u,\pm\frac{\pi}{2}) = \mp
a_{3}'(u) \mp a_{3}(u) a_{2}(u) \neq 0$.
Hence we have a parabolic tangency if $a_{3}(u_{\ast})=0$ and
$a_{3}^{\prime }(u_{\ast})\neq 0$, where $u_{\ast}$ is a root of
$a_3(u)$.
\end{proof}

We say that a singularity, $(u,v)$, of $\widetilde{X}(f)$ is
\emph{semi-hyperbolic} if $D\widetilde{X}(f)(u,v)$ has exactly one
zero eigenvalue. Also we say that a singularity, $(u,v)$ of
$\widetilde{X}(f)$ is \emph{nilpotent} if $D\widetilde{X}(f)(u,v)$
is nilpotent.

\begin{proposition}
\label{zerosdeXTf}Let $X(f) \in \mathcal{E}^{n,r}(M)$, where
$f(x,y)=\sum_{i=0}^{n} a_{i}(x)y^{i}$, $r\geq 1$ and $n \geq 4$.  Then
\begin{enumerate}
\item For $n=4$, $\widetilde{X}(f)$ has singularities,
    $(u_{\ast},\pm\frac{\pi}{2})$, on $C_{\pm \frac{\pi}{2}}$ where
$u_{\ast}$ is root of $a_{4}(u)$. The singularities are hyperbolic
if
    $a_{4}'(u_{\ast}) \neq 0$ and $a_{3}(u_{\ast})\neq 0$,
    semi-hyperbolic if
    $a_{4}'(u_{\ast}) = 0$ and  $a_{3}(u_{\ast}) \neq 0$, and
    nilpotent if $a_{4}'(u_{\ast}) =  0$  and $a_{3}(u_{\ast}) = 0$.

\item For $n>4$, $\widetilde{X}(f)$ has singularities,
    $(u_{\ast},\pm\frac{\pi}{2})$,  on $C_{\pm \frac{\pi}{2}}$ where
$u_{\ast}$ is a root of $a_{n}(u)$. The singularities are semi-hyperbolic if
    $a_{n-1}(u_{\ast}) \neq 0$, and nilpotent if $a_{n-1}(u_{\ast}) = 0$ and
    $a_{n}'(u_{\ast}) \neq 0$.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) For $n=4$, the Jacobian matrix of
$$
\widetilde{X}(f) =\sin(v) \cos(v) \frac{\partial}{\partial
  u} + \sum \limits_{i=0}^{4} a_{i}(u) \sin^{i}(v) \cos^{4-i}(v)
  \frac{\partial}{\partial   v}
$$
at a singularity   $(u_{\ast},\pm\frac{\pi}{2})$ is
\[
D\widetilde{X}(f)(u_{\ast},\pm\frac{\pi}{2})=
\begin{pmatrix}
0 & -1\\
a_{4}'(u_{\ast}) & -a_{3}(u_{\ast})
\end{pmatrix}
\]
The proof is finished by analyzing
$det(D\widetilde{X}(f))=a_{4}'(u_{\ast})$ and
$\text{trace}(D\widetilde {X}(f))=-a_{3}(u_{\ast})$.

\noindent (2). For $n>4$, $ \widetilde{X}(f) = \sin(v) \cos^{n-3}(v) \frac{\partial}
  {\partial   u} + \sum \limits_{i=0}^{n} a_{i}(u) \sin^{i}(v) \cos^{n-i}(v)
  \frac{\partial}{\partial v}$
has the Jacobian matrix at $(u_{\ast},\pm\frac{\pi}{2})$
\[
D\widetilde{X}(f)(u_{\ast},\pm\frac{\pi}{2})=
\begin{pmatrix}
0 & 0\\
(\pm1)^{n}a_{n}'(u_{\ast}) & (\pm1)^{n}a_{n-1}(u_{\ast})
\end{pmatrix}
\]
Hence this singularity is semi-hyperbolic when
$a_{n-1}(u_{\ast})\neq0$ and nilpotent when $a_{n-1}(u_{\ast}) =
0$ and $a_{n}'(u_{\ast}) \neq 0$.
\end{proof}

\begin{remark} \label{rmk1} \rm
The Propositions \ref{pro:PeriodicOrbit}-\ref{zerosdeXTf} also hold if we
consider $\widetilde{X}_1(f)$  given by (\ref{eq:xf1n123}) and (\ref{eq:xf1n4})
instead of $\widetilde{X}(f)$. In this
case, the singularities  or periodic orbits of $\widetilde{X}_1(f)$ lie on
$\widetilde{C}_0$.
\end{remark}

For simplicity, in the following propositions, we use
$\widetilde{X}_{1}(f)$ as $(\ref{eq:xf1n4})$ to describe the phase
portrait of the semi-hyperbolic singularities of $X(f)$ at
infinity when $n \geq 4$.

\begin{proposition}\label{pro:SemiHiper4}
Let $X(f) \in \mathcal{E}^{4,r}$, where
$f(x,y)=\sum_{i=0}^{4}a_{i}(x)y^{i}$, $r \geq 2$. Let
$(u_{\ast},0)$ be a semi-hyperbolic singularity of
$\widetilde{X}_1(f)$ in $\widetilde{C}_{0}$ and let $k \in N$, $k
\geq 2$, such that $$a_{4}^{(k)}(u_{\ast}) \neq 0 \, \text{ and }
\, a_{4}^{(j)}(u_{\ast}) = 0, \, \text{ for } j < k.$$ Then
$(u_{\ast}, 0)$ is one of the following topological types:
\begin{enumerate}
\item[(a)] a node, if $k$ is odd and $a_4^{(k)}(u_{\ast}) > 0$,
figure \ref{fig:SeNsemihiperbolica} (a);

\item[(b)] a saddle, if $k$ is odd and $a_4^{(k)}(u_{\ast}) < 0$,
figure \ref{fig:SeNsemihiperbolica}  (b);

\item[(c)] a saddle-node, if $k$ is even, figure
\ref{fig:SNsemihiperbolica}.
\end{enumerate}
\end{proposition}

\begin{figure}[htbp]
  \begin{center}
   \psfrag{u}[h][B]{$u$}
   \psfrag{v}[h][B]{$v$}
   \psfrag{Wu}[h][B]{$W_u$}
   \psfrag{Ws}[h][B]{$W_s$}
   \psfrag{Wc}[h][B]{$W_c$}
   \includegraphics[height=1.8in, width=3.5in]{fig1.eps} %{Sela-No-Semihip4.eps}
    \caption{Phase portraits of semi-hyperbolic singularities in
   $\widetilde{C}_0$ for $n =4$. (a) Node , (b)  Saddle.}
    \label{fig:SeNsemihiperbolica}
  \end{center}
  \begin{center}
   \psfrag{x}[h][B]{$u$}
   \psfrag{y}[h][B]{$v$}
   \psfrag{Wu}[h][B]{$W_u$}
   \psfrag{Ws}[h][B]{$W_s$}
   \psfrag{Wc}[h][B]{$W_c$}
   \includegraphics[height=1.7in,width=3.5in]{fig2.eps}%{SelaNo-Semihip4.eps}
    \caption{Phase portraits  of semi-hyperbolic singularities saddle-node in
   $\widetilde{C}_0$ for
   $n = 4$:  (a) $a_3(u_{\ast}) > 0$, (b) $a_3(u_{\ast}) < 0$.}
    \label{fig:SNsemihiperbolica}
  \end{center}
\end{figure}
\begin{proof}

We can suppose that $u_{\ast} = 0$. We calculate the restriction of
$$\widetilde{X}_1(f)=  v \frac{\partial}{\partial u}-  \sum_{i=0}^{4} a_{i}(u)
 v^{4-i}\frac{\partial} {\partial v}$$
to the center manifold  $W_c$ in  $(u_{\ast},0)$ when
 $a_{4}'(u_{\ast}) = 0$ and  $a_{3}(u_{\ast}) \neq 0$.

The center manifold of  $\widetilde{X}_1(f)$ is tangent to eigenspace $T_c$
associated to null eigenvalue  and is spanned  by the vector $(1,0)$. Then
$W_c$ is the graph of a $C^{r}$-function $h: \Re \to \Re$,
$$W_c =\{(u,v) \in \Re^2 : v = h(u) \}.$$ By the  condition of tangency
of $W_c$, $h(u_{\ast})= h'(u_{\ast})=0$.

The restriction of  $\widetilde{X}_1(f)$ to the center manifold is of the form
\begin{equation}
     u' =  h(u) \label{eq:uprime}
\end{equation}
Replacing $v= h(u)$ in the second component of
$\widetilde{X}_1(f)$, we obtain $ \phi(u) = h'(u) h(u) +
\sum_{i=0}^{4} a_{i}(u) h^{4-i}(u) =0$ for all $u$.

Let $k \in \mathcal{N}$, $k \geq 2$ such that  $a_{4}^{(k)}(u_{\ast}) \neq 0$
and $a_{4}^{(j)}(u_{\ast}) = 0$ as  $j < k$.
Writing $h(u)= h_{2} u^2 + \cdots + h_{k} u^k+ \cdots$, we have
\begin{gather*}
 \phi(u_{\ast}) =  0, \quad \phi'(u_{\ast}) =  0,
    \quad   \phi''(u_{\ast}) =  2 a_3(u_{\ast}) h_2 = 0, \\
   \phi '''(u_{\ast}) =  3! (2 h_2^{2} +
    a_3(u_{\ast})  h_3) = 0, \,  \dots, \\
  \begin{aligned}
    \phi^{(k)}(u_{\ast}) &=   k! (a^{(k)}_4(u_{\ast} + a_3(u_{\ast})  h_{k})
       + a'_3(u_{\ast}) h_{k-1} + \cdots + a^{(k-2)}_3(u_{\ast})    h_2) \\
      &\quad  + A_{3k}(h_2, \cdots, h_{k-1}) + A_{2k}(h_2, \cdots, h_{k-2}) +
    A_{1k}(h_2, \cdots, h_{k-3}) \\
    &\quad  + A_{0k}(h_2, \dots, h_{k-4}) = 0\end{aligned}
\end{gather*}
where  $A_{ik}(h_2,\dots) = \frac{d^k}{d u^k}(a_i \cdot
h^{4-i})(u_{\ast})$ as  $i= 0,\dots, 3$.

We solve these equations with respect to $h_i$ as follows
$$h_2 = h_3 = \cdots = h_{k-1} = 0 \, \, \text{ and } \, \, h_k = -
\frac{a_4^{(k)}}{a_3(u_{\ast})}.$$
Then $ h(u) \equiv \alpha u^k + O(u^{k+1})$, where  $\alpha = -
\frac{a_4^{(k)}(u_{\ast})}{a_3(u_\ast)}$. Hence, (\ref{eq:uprime}) is of the
form
$$u' = \alpha u^k + O(u^{k+1}).$$
The proposition follows by analyzing the sign of $\alpha$ and the
orientation of the hyperbolic manifold (unstable $W_u$ or stable
$W_s$) which is tangent to $v= -a_{3}(u_{\ast}) u$.\end{proof}


\begin{proposition}\label{pro:SemiHiper5}
Let $X(f) \in \mathcal{E}^{n,r}$, where
$f(x,y)=\sum_{i=0}^{n}a_{i}(x)y^{i}$, $r \geq 2$ and $n > 4$.  Let
$(u_{\ast},0)$ be a semi-hyperbolic singularity  of
$\widetilde{X}_1(f)$ and let $k \in N$, $k \geq 1$, such that
$$a_{n}^{(k)}(u_{\ast}) \neq 0 \,\text{ and } a_{n}^{(j)}(u_{\ast}) = 0 \,
\text{ for } j < k.$$ Then $(u_{\ast},0)$  is one of the following topological
types:
\begin{enumerate}
\item[(a)] a node, if $n$ is even, $k$ is odd and
$a_n^{(k)}(u_{\ast}) > 0$, figures
\ref{fig:Sela-NoSemihiperbolica5} (a) and \ref{fig:Sela-NoSemihiperbolica5}
(b);

\item[(b)] a saddle, if $n$ is even, $k$ is odd and
$a_n^{(k)}(u_{\ast}) < 0$, figures
\ref{fig:Sela-NoSemihiperbolica5} (c) and \ref{fig:Sela-NoSemihiperbolica5}
$d)$;


\item[(c)] a  saddle-node, if $(n-3)\cdot k$ is even, figures
\ref{fig:SelaNoSemihiperbolica52} and
\ref{fig:SelaNoSemihiperbolica51}.
\end{enumerate}
\end{proposition}

\begin{figure}[htbp]
  \begin{center}
   \psfrag{u}[h][B]{$u$}
   \psfrag{v}[h][B]{$v$}
   \psfrag{Wu}[h][B]{$W_u$}
   \psfrag{Ws}[h][B]{$W_s$}
   \psfrag{Wc}[h][B]{$W_c$}
   \includegraphics[height=1.7in,width=3.5in]{fig3.eps}% {Sela-No-Semihip5k1.eps}
    \caption{Phase portraits of semi-hyperbolic singularities saddle and
    node  in  $\widetilde{C}_0$  for $n \geq 5$}
    \label{fig:Sela-NoSemihiperbolica5}
  \end{center}
\end{figure}

\begin{figure}[htbp]
  \begin{center}
   \psfrag{u}[h][B]{$u$}
   \psfrag{v}[h][B]{$v$}
   \psfrag{Wu}[h][B]{$W_u$}
   \psfrag{Ws}[h][B]{$W_s$}
   \psfrag{Wc}[h][B]{$W_c$}
   \includegraphics[height=1.7in,width=3.5in]{fig4.eps}%{SelaNo-Semihip5k2.eps}
    \caption{Phase portraits of semi-hyperbolic singularities saddle-node
    for  in  $\widetilde{C}_0$ $n\geq 5$ and $k \geq 2$}
    \label{fig:SelaNoSemihiperbolica52}
  \end{center}
%\end{figure}
%\begin{figure}[htbp]
  \begin{center}
   \psfrag{u}[h][B]{$u$}
   \psfrag{v}[h][B]{$v$}
   \psfrag{Wu}[h][B]{$W_u$}
   \psfrag{Ws}[h][B]{$W_s$}
   \psfrag{Wc}[h][B]{$W_c$}
   \includegraphics[height=1.7in,width=3.5in]{fig5.eps}%{SelaNo-Semihip5k1.eps}
    \caption{Phase portraits of semi-hyperbolic singularities saddle-node  in
   $\widetilde{C}_0$ for $n\geq 5$ and $k=1$}
    \label{fig:SelaNoSemihiperbolica51}
  \end{center}
\end{figure}

\begin{proof}
We use the method of the center manifold as in Proposition
\ref{pro:SemiHiper4}. We find that the restriction of
$\widetilde{X}_1(f)$ to the center manifold, $W_c$, is of the form
$$g(u) = \alpha^{n-3} u^{k(n-3)} + O(u^{k+1(n-2)})$$
where  $\alpha = - \frac{a_n^{(k)}(u_{\ast})}{a_{n-1}(u_\ast)}$.

The proof is finished by analyzing the sign of $\alpha$ and the
orientation of the hyperbolic manifold (unstable $W_u$ or stable
$W_s$) which is tangent to $v$-axis.
\end{proof}

\section{Definition of the sets  $\Sigma^{n}$}\label{sec:definition}

According to section \ref{sec:behavior},  the behaviors of $\widetilde{X}(f)$
on $C_{\pm  \frac{\pi}{2}}$, under non-degeneracy conditions on the
 periodic  orbits, singularities and tangencies, split in the following
 cases.

\begin{description}
 \item[A] $C_{\pm \frac{\pi}{2}}$ are periodic orbits of
 $\widetilde{X}(f)$ with the first and
 second derivatives of the Poincar\'e map equal  $1$ and $\pm \int_{0}^{\tau}
 a_1(s) ds$ respectively. This occurs when $n=1$.
\item[B] $C_{\pm \frac{\pi}{2}}$ are hyperbolic periodic orbits of
 $\widetilde{X}(f)$ with second derivative of the Poincar\'e map equals $\pm 2
 \int_{0}^{\tau} a_2(s) ds$. This occurs when  $n=2$.
 \item[C] the trajectories of $\widetilde{X}(f)$ are either transversal or
   tangent to $C_{\pm\frac{\pi}{2}}$. This case occurs when $n= 3$.
 \item[D] $\widetilde{X}(f)$ has hyperbolic singularities in
 $C_{\pm \frac{\pi}{2}}$. This cases occurs when $n=4$.
 \item[E] $\widetilde{X}(f)$ has semi-hyperbolic  singularities in $C_{\pm
 \frac{\pi}{2}}$. This occurs when $n \geq 5$.
\end{description}


With these cases in mind, in the subsections
\ref{sec:genericidade1}-\ref{sec:genericidade5}, we define for the
corresponding $n$ the
set $\Sigma^{n} \subset \mathcal{E}^{n,r}$ and  we prove its density in
the space $\mathcal{E}^{n,r}$.

Throughout this section, we denote by $\widetilde{X}(f)$ and
$\widetilde{M} = S^1 \times [-\frac{\pi}{2}, \frac{\pi}{2}]$ the
cylindrical  compactification of $X(f)$ and $M$ respectively. Also
we consider the following notation:

$C_{\pm \frac{\pi}{2}} = S^1\times \{ \pm \frac{\pi}{2}\}$, $C_0=
S^1 \times {0}$,

$\Delta \widetilde{X}(f) = \text{det}D\widetilde{X}(f)$ and
$\sigma\widetilde{X}(f) = \text{trace}D\widetilde{X}(f)$
where $D\widetilde{X}(f)$ is the Jacobian matrix of
$\widetilde{X}(f)$.

In the first  subsection, we present several lemmas which hold for all
cases.

\subsection*{Preliminary lemmas}

 The first lemma is a
particular and easy case of Sard's Theorem \cite{Sard}.

\begin{lemma}\label{lema:sard}
Let $h:I \to \Re$ be a $C^1$ function. The set of critical values of
$h$, given by Crit$(h) = \{ h(x) : h'(x) =0 \}$, has zero Lebesgue
measure in $\Re$.
\end{lemma}


\begin{lemma}\label{Lema:B1Xfn1}
Let $X(f) \in \mathcal{E}^{n,r}(M)$
with  $r\geq 2$ and $n\geq 1$. Then the set
\begin{align*}
B^n_{1}(X(f)) = \{  & \mu_{0} \in \Re : \widetilde{X}(f +
\mu_{0}) \text{ has some singularity}\\
 & (u_{\ast}, 0)\in C_0 \text{ with } \Delta \widetilde{X}(f + \mu_{0}) = 0 \}
\end{align*}
has Lebesgue measure zero in $\Re$.
\end{lemma}

\begin{proof} Let $f(x,y) = \sum_{i=0}^{n} a_i(x) y^{i}$.
 The  set of critical value of $-a_{0}$ is given by
 $$\mathop{\rm Crit}(-a_{0})=\{ \mu_{0} \in \Re : \exists x_{\mu_0} \text{ such that }
 -a_{0}(x_{\mu_0})= \mu_0 \text{ e } a_{0}'(x_{\mu_0}) = 0 \}.$$  By
 Lemma \ref{lema:sard}, $Crit(-a_{0})$ has zero
 Lebesgue measure in  $\Re$.
On the other hand, we can write
$$ \begin{array}{ll} Crit(-a_{0}) = & \{ \mu_0 \in \Re : \exists (x_{\mu_0},0)
  \in C_0 \text{ such that } \widetilde{X}(f + \mu_{0})(x_{\mu_0},0) = (0,0)
  \\
  & \text{ with }   \Delta \widetilde{X}(f + \mu_{0})(x_{\mu_0},0) = 0 \}.
  \end{array} $$
It follows that  $B^n_{1}(X(f))= Crit(-a_{0})$.
\end{proof}

\begin{lemma}\label{Lema:B2Xfn1}
Let $X(f) \in \mathcal{E}^{n,r}(M)$,
with $r \geq 1$, $n\geq 1$,  and $\mu_{0} \notin B^n_{1}(X(f))$. Then the set
\begin{align*}
B^n_{2}(X(f);\mu_{0})= &  \{ \mu_{1} \in \Re: \widetilde{X}( f + \mu_{0} +
\mu_{1} y ) \text{ has some }\\
&  \text{ non-hyperbolic singularity} \}
\end{align*}
has zero Lebesgue measure in $\Re$.
\end{lemma}

\begin{proof} Let $f(x,y) = \sum_{i=0}^{n} a_i(x) y^{i}$.
 For $\mu_{0} \notin B^n_{1}(X(f))$, all singularities of $\widetilde{X}(f +
 \mu_0)$ in $C_0$, say $(u_{\ast},0)$, satisfy $\Delta\widetilde{X}(f
 + \mu_{0})(u_{\ast}, 0) \neq 0$.

Thus $\widetilde{X}(f + \mu_{0} +\mu_1 y)$ has in $C_{0}$ a finite
number of singularities. For each one, $(u_{\ast},0)$, there is a
single value  $\mu_{1}^{\ast} \in \Re$ such that
$\sigma\widetilde{X}(f + \mu_{0} + \mu_{1}^{\ast} y ) = 0$, since
$\sigma\widetilde{X}(f+ \mu_{0} + \mu_{1} y ) = a_{1}(u) +
\mu_{1}$. Hence, $(u_{\ast},0)$ is a non-hyperbolic singularity of
$\widetilde{X}(f + \mu_0 + \mu_{1}^{\ast} y)$.

Then $B^n_{2}(X(f);\mu_{0})$ can be written of the form
\[ B^n_{2}(X(f);\mu_{0}) = \{ \mu_1 \in \Re : a_1(x) + \mu_1 =0, a_0(x) +
\mu_0 =0 \text{ e } a'_0(x) >0 \}
\]
It follows that  $ B^n_{2}(X(f);\mu_{0})$ is a finite set and has
therefore zero measure in $\Re$.\end{proof}

\begin{lemma}\label{Lema:B3Xfn1}
Let $X(f) \in \mathcal{E}^{n,r}(M)$, $n\geq 1$. Then
for $\mu_{0} \in \Re$ and $r\geq 1$, the set
\begin{align*}
B^n_{3}(X(f);\mu_{0})=\{& \mu_{1} \in\Re:
\widetilde{X}(f+ \mu_{0}+ \mu_{1} y)  \text{ has some non-hyperbolic }\\
& \text{periodic orbit contained in } M \}
\end{align*}
has zero Lebesgue measure  in $\Re$.
\end{lemma}
\begin{proof}
A period orbit $\gamma$ of $\widetilde{X}(f + \mu_{0} + \mu_{1} y)$ is
one of the following types:
\begin{itemize}
\item[(a)] homotopic to zero. It contains in its interior a singularity and
  cuts  the $x$-axis transversally.

\item[(b)] non-homotopic to zero. It circles the cylinder without
intercepting the $x$-axis.
\end{itemize}

For each type we find expressions for the derivatives of the
first return map with respect to a parameter.

We consider $X(f)$ with $f(x,y) = \sum_{i=0}^{n} a_i(x) y^{i}$ and  $n \geq
1$.

Case (a). For $\mu_{0} \in \Re$, we consider a periodic orbit of
$\widetilde{X}(f + \mu_{0} + \mu_{1} y)$,
$\gamma(t,p,\mu_{1})=(\varphi(t,p,\mu_{1}), \psi(t,p,\mu_{1}))$
through $p=(x_{0},0)$ with period $\tau=\tau(x_{0},\mu_{1})$.

Let $\pi(x,\mu_{1})$ be the Poincar\'e map defined in an interval
$I\subset C_{0}$, where $x_{0}\in I$, and associated to $X(f +
\mu_{0} + \mu_{1} y)$.

The derivatives of $\pi(x,\mu_{1})$ with respect to $x$ and $\mu_{1}$ were
 calculated in
\cite{Andronov} by Andronov \emph{et al.} Here, we write them  in
terms of the coefficients of $f(x,y)+\mu_{0}+\mu_{1}y$ as follows
\begin{gather*}
\frac{\partial\pi}{\partial x}(x_{0},\mu_{1}) = \prod_{i=1}^{n}\exp(\int
_{0}^{\tau(x_{0},\mu_{1})}(ia_{i}(\varphi(s))\psi^{i-1}(s)+\mu_{1})ds)
\\
\begin{aligned}
\frac{\partial\pi}{\partial\mu_{1}}(x_{0},\mu_{1})
&=  \frac{\frac{\partial
\pi}{\partial x}(x_{0},\mu_{1})}{-|\gamma^{'}(0,x_{0},\mu_{1})|}
{\displaystyle\int_{0}^{\tau(x_{0},\mu_{1})}}
{\textstyle\prod_{i=1}^{n}}
\exp(-\int_{0}^{s}(ia_{i}(\varphi(t))\psi^{i-1}(t) \\
 &\quad +\mu_{1})dt)\psi^{2}(s)ds.
\end{aligned}
\end{gather*}
Since $\frac{\partial\pi}{\partial\mu_{1}}(x_{0},\mu_{1}) \neq 0$
(and by the Implicit Function  Theorem) there is a function
$\mu_1(x)$ defined in a neighborhood $I_{x_{0}}$ of $x_0$ such
that $\pi(x,\mu_{1}(x))-x=0$ for $\forall x \in I_{x_0}$. The
derivative of the last equation with respect to $x$ is
\[
\frac{\partial\pi}{\partial x}(x_{0},\mu_{1})+\frac{\partial\pi}{\partial
\mu_{1}}(x_{0},\mu_{1})\frac{\partial\mu_{1}}{\partial x}-1=0.
\]
Thus $ \gamma $ is non-hyperbolic if and only if $\frac {\partial \mu_{1}}
{\partial x} = 0$.

We remark that for each $x \in I_{x_0}$, $\mu_1(x)$ is a value
 of the parameter such that  $X(f + \mu_0 +  \mu_1(x) y)$ has a periodic
orbit through $(x,0)$. This periodic orbit is non-hyperbolic when
$\mu_1(x)$ is a critical value.

Then, for each $\mu_{0}\in \Re$ fixed, the set of critical values of
$\mu_{1}(x)$ is written as
\begin{align*}
O_{1} = & \{ \mu_{1}(x) \in \Re : X(f+ \mu_{0} + \mu_{1}(x) y)
\text{ has a  homotopic to zero periodic orbit}\\
&  \text{  non-hyperbolic through } (x,0)\}.
\end{align*}
Case (b). Let $\gamma(t) = (\varphi(t),\psi(t))$ be a non-homotopic to zero
 periodic orbit, with period $\tau$, through $(0,y_{0})$ where  $y_{0} \neq 0$.
The Poincar\'e map, $\pi(y,\mu_{1})$, is defined by
\begin{equation}\label{eq:Piymu1}
\pi(y,\mu_{1})=\Psi(\tau,y,\mu_{1})
\end{equation}
where $\Psi (x,y,\mu_{1})$ is a solution of differential equation
\[
\frac{d\Psi}{dx}(x,y,\mu_{1})=\frac{1}{\Psi}\left(
  \sum_{i=0}^{n}a_{i}(x)\Psi^{i}+\mu
_{1}\Psi\right)  =F(x, y ,\mu_{1})
\]
with the initial condition  $\Psi(0,y,\mu_{1}) =y$.

The derivative of $\Psi(x,y,\mu_{1})$ with respect to $\mu_{1}$ is the
solution of linear equation
\[
\frac{d}{dx}(\frac{\partial\Psi}{\partial\mu_{1}})=D_{2}F(x,\Psi(x,y,\mu
_{1}),\mu_{1})\frac{\partial\Psi}{\partial\mu_{1}}(x,y,\mu_{1})+D_{3}%
F(x,\Psi(x,y,\mu_{1}),\mu_{1})
\]
where  $D_{2}F(x,y,\mu_{1})=-\frac{a_{0}(x)}{y^{2}}$ and
$D_{3}F(x,y,\mu_{1})=1$.

By equation (\ref{eq:Piymu1}) the derivative of $\pi(y,\mu_{1})$
with respect to $\mu_{1}$ is given by
$$\frac{\partial\pi}{\partial\mu_{1}} (y,\mu_{1})= \exp(\mathcal{I}(\tau,y,
\mu_{1})) \int_{0}^{\tau} \exp(- \mathcal{I}(s,y, \mu_{1})) ds $$
where $\mathcal{I}(t, y, \mu_{1})= \int_{0}^{t} D_{2}F(s,\Psi(s,y,
\mu_{1}),\mu_{1})ds$

Similarly to (a), we find a function $\mu_1: I_{y_0} \to
\Re$ where $I_{y_0}$ is  an interval of $y$-axis containing $y_0$
and without intersection with the $x$-axis such that
$\pi(y,\mu_{1}(y))-y = 0$. Hence, $\gamma$ is non-hyperbolic if
and only if $ \frac{\partial\mu_{1}}{\partial y} = 0$.

 The critical value set of $\mu_{1}(y)$ is given by
\begin{align*}
O_{2} = & \{\mu_{1}(\xi )\in\Re:X(f+\mu_{0}+\mu_{1}(\xi) y)\text{
has a non-hyperbolic periodic orbit}\\
&  \text{  circling the cylinder through } (0, \xi) \}
\end{align*}

The proof of $i)$ is complete by observing that
\[
B^1_{3}(X(f);\mu_{0}) = O_{1}\cup O_{2}
\]
and by applying the Sard Lemma to sets $O_1$ e $O_2$.
\end{proof}


\subsection{Case \bf{A}}\label{sec:genericidade1}

In this case, $X_{\mu}(f)$ denotes the vector field $X(f+ \mu_0 +
\mu_1 y)$ where $\mu = (\mu_0, \mu_1) \in \Re^2$.

\begin{definition}\label{def:n1} \rm
Let $\Sigma^1$ be the set of $C^r$-vector
fields  $X(f) \in \mathcal{E}^{1,r}$ with $r \geq 2$ such that
$\widetilde{X}(f)$ satisfies:
\begin{enumerate}
\item the singularities are hyperbolic and contained in $C_{0}$.
\item  the periodic orbits in the  $\widetilde{M}^{\circ}$  are
    hyperbolic and  the periodic orbits in  $C_{\pm \frac{\pi}{2}}$
    are semi-stable.
\item no saddle connection.
\end{enumerate}
\end{definition}

Next, we give  the measure of the complementary set of
$\Sigma^{1}$ in the parameter space $\Re^{2}$.
\begin{theorem} \label{Teo:BXfn1}
Let $X(f)\in \mathcal{E}^{1,r}$, with $r\geq 2$. Then the set
\[
B^1(X(f))  =  \{ \mu \in \Re^{2}: X_{\mu}(f ) \notin \Sigma^{1} \}
\]
has zero  Lebesgue measure in $\Re^{2}$.
\end{theorem}

We have divided the proof in a sequence of lemmas.

\begin{lemma}\label{Lema:B4Xfn1}
Let $X(f) \in \mathcal{E}^{1,r}$. Then for $r\geq 2$, the set
\begin{align*}
B^1_{4}(X(f)) = \{& \mu_{1} \in\Re: \widetilde{X}(f+
\mu_{1} y)  \text{ has some non semi-hyperbolic }\\ &
\text{periodic orbit in } C_{\pm \frac{\pi}{2}} \}
\end{align*}
has zero Lebesgue measure  in $\Re$.
\end{lemma}

\begin{proof} We observe that the periodic orbits of
$\widetilde{X}(f + \mu_0 + \mu_1 y)$ in $C_{\pm \frac{\pi}{2}}$ do
not depend of $\mu_0$. Moreover, the second derivative of the
Poincar\'e map of these periodic orbits is $\pi'' = \pm 2
\int_0^{\tau} a_1(s) ds + 2 \mu_1 \tau$ (see Proposition
 \ref{pro:PeriodicOrbit}).

Thus  $\pi'' = 0$ only for a finite number of  $\mu_1$.  Then
$B_4^1(X(f))$ has  zero Lebesgue measure in $\Re$.
\end{proof}


\begin{lemma}\label{Lema:B5Xfn1}
Let $X(f)\in \mathcal{E}^{1,r}$, $r\geq 1$ and  $\mu_{0} \notin
B^1_{1}(X(f))$. Then the set
\[ B^1_{5}(X(f);\mu_{0}) = \{ \mu_{1} \in \Re:
\widetilde{X}(f + \mu_0 + \mu_1 y)\text{ has not the condition } 3 \text{ of }
\Sigma^{1}\}
\]
has zero Lebesgue measure in $\Re$.
\end{lemma}

\begin{proof}
Fix $\mu_{0} \notin B_{1}(X(f))$. The  number of singularities of
$\widetilde{X}(f + \mu_{0})$ is finite. Thus the saddle
connections of $\widetilde{X}_{\mu}(f)$ also are finite in number
for values of $\mu_{1}$.

We claim that all connections can be broken with perturbations of the form
$\widetilde{X}(f + \mu_0 + \mu_1 y)$.
Suppose that for  $\mu_{1}^{\ast}$, $\widetilde{X}(f + \mu_{0} +
\mu_{1}^{\ast}y)$  has a trajectory $\widehat{pq}$ in the superior part of
$\widetilde{M}$ connecting the saddles $p$ and $q$ in  $C_{0}$. We denote by
$S_E(q)$  the stable separatrix of $q$ and by  $S_I(p)$ the unstable
separatrix of $p$.  See figure \ref{fig:conexaoselan12}.

\begin{figure}[htbp]
  \begin{center}
   \psfrag{C0}[h][B]{$C_0$}
   \psfrag{C+pi}[h][B]{$C_{+\frac{\pi}{2}}$}
   \psfrag{C-pi}[h][B]{$C_{-\frac{\pi}{2}}$}
   \psfrag{p}[h][B]{$p$}
   \psfrag{q}[h][B]{$q$}
   \psfrag{su}[h][B]{$S_I(p)$}
   \psfrag{ss}[h][B]{$S_E(q)$}
   \includegraphics[height=1.9in,width=2.5in]{fig6.eps}%{conexaoselan12.eps}
    \caption{Breaking connection of saddles $p$ and $q$ for $\mu_1 >
   \mu_1^{\ast}$ }
    \label{fig:conexaoselan12}
  \end{center}
\end{figure}

Let $\mathop{\rm sep}(\mu_1)$ be  the  separation function of
$S_I(p,\widetilde{X}(f+\mu_0 + \mu_1 y))$ and
$S_E(q,\widetilde{X}(f+\mu_0 + \mu_1 y))$. It is defined on a
transversal section to the trajectory that links the saddles. The
derivative of  $\mathop{\rm sep}(\mu_1)$ with respect to  $\mu_1$ is of the
form
\begin{align*}
 \mathop{\rm sep}_{\mu_1}(\mu_1^{\ast}) &=  \int_{-\infty}^{+ \infty}
  \exp(-\int_{0}^{t} \mathop{\rm div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y)(u(s),v(s)))ds)\\
 &\quad \cdot \widetilde{X}(f+\mu_0 + \mu_1^{\ast} y) \wedge
  \frac{d}{d \mu_1}\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y)(u(s),v(s)) dt
  \end{align*}
where  $\wedge$ is the vectorial product defined by
$(v_1,v_2) \wedge (w_1, w_2) = - \det \begin{pmatrix} v_1 & v_2 \\ w_1
    & w_2 \end{pmatrix}$ and
 $(u(s),v(s))$ is an orbit  connecting the saddles $p$ and $q$.
For a treatment of  the integral formula of $\mathop{\rm sep}_{\mu_1}(\cdot)$
we refer the reader to Guckeheimer-Holmes
\cite{guckeheimer-holmes} and Chicone \cite{chicone}.

In our case,
\begin{equation}  \label{eq:derivadaseparacao}
\begin{aligned}
 \mathop{\rm sep}_{\mu_1}(\mu_1^{\ast}) &=  - \int_{- \infty}^{+\infty} \exp(-\int_{0}^{t}
\mathop{\rm div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast}
y)(u(s),v(s)))ds) \\ & \quad  \times \sin^2(v(t)) \cos^2(v(t)) dt.
\end{aligned}
\end{equation}
Since  $v(t) \in [-\frac{\pi}{2}, \frac{\pi}{2}]$,  the integrand
in (\ref{eq:derivadaseparacao}) is a non-negative function.
Therefore $\mathop{\rm sep}_{\mu_1}(\mu_1^{\ast}) < 0$. Then the connection of
$p$ and $q$ saddles is broken, without another connection to
arise.
Thus  $B^1_5(X(f); \mu_0)$ is a discrete set. This ends the proof.
\end{proof}


\begin{proof}[Proof of Theorem \ref{Teo:BXfn1}]
The set  $B^1(X(f))$ is the union of the following sets
   \begin{gather*}
\mathcal{B}_1 =  B^1_{1}(X(f)) \times \Re, \quad
 \mathcal{B}_2 =  \bigcup_{\mu_0 \in \Re-B^1_{1}(X(f))} \{ \mu_0 \}
   \times  B^1_{2}(X(f),  \mu_0),  \\
 \mathcal{B}_3 =  \bigcup_{ \mu_0 \in \Re} \{ \mu_0 \} \times B^1_{3}(X(f);
  \mu_0), \quad
 \mathcal{B}_4 =  \Re \times B^1_{4}(X(f)), \\
 \mathcal{B}_5 =  \bigcup_{\mu_0 \in \Re-B^1_{1}(X(f))} \{
 \mu_0   \}  \times B^1_{5}(X(f),  \mu_0),
\end{gather*}
where  $B_{i}^1(X(f), \cdot)$, $i=1, \dots,5$ are given by Lemmas
 \ref{Lema:B1Xfn1}-\ref{Lema:B5Xfn1}
 respectively.

  Each set  $\mathcal{B}_i$:
\begin{itemize}
\item contains  parameters $(\mu_0, \mu_1) \in \Re^2$ such that
$X(f + \mu_0 + \mu_1 y)$ violates at least a condition of
$\Sigma^1$.

\item is  measurable. Because its complement in $\Re^2$ is open.

\item has measure zero in $\Re^2$. Because $\mathcal{B}_1$ and  $\mathcal{B}_4$
  are  products of $\Re$ times a zero measure set in $\Re$. To calculate the
  measure of
  $\mathcal{B}_i$ for $i=2,3$ and $5$, we use Fubini's Theorem
  \cite{Royden}:
$$ \int_{\Re^2} \chi(\mathcal{B}_i) d\mu_0 d\mu_1 = \int (\int
\chi(\cdot, B_i^1(X(f),\mu_0)) d\mu_1) d\mu_0 = 0$$
where  $\chi(\cdot)$ is the characteristic function of sets in $\Re^2$.
\end{itemize}
Then  $B^1(X(f))$ is  measurable with zero Lebesgue measure in $\Re^2$.
\end{proof}

\subsection{Case B}\label{sec:genericidade2}
In this case, $X_{\mu}(f)$ denotes the vector field $X(f+ \mu_0 +
\mu_1 y + \mu_2 y^2)$ where $\mu =(\mu_0, \mu_1, \mu_2) \in \Re^3$.

\begin{definition} \rm
Let $\Sigma^{2}$ be the set of $C^r$-vector fields $X(f) \in
\mathcal{E}^{2,r}$ with $r \geq 1$ such that $\widetilde{X}(f)$
satisfies:
\begin{enumerate}
\item the singularities are hyperbolic and contained in $C_{0}$.

\item the periodic orbits are hyperbolic and contained  in  $\widetilde{M}$.

\item no saddle connection.
\end{enumerate}
\end{definition}

We now give  the measure of the complement of $\Sigma^{2}$ in
the parameter space $\Re^{3}$.

\begin{theorem} \label{Teo:BXfn2}
Let $X(f)\in \mathcal{E}^{2,r}$, with $r \geq 1$. Then the set
\[
B^2(X(f))  =  \{ \mu \in \Re^{3}: X_{\mu}(f ) \notin \Sigma^{2} \}
\]
has zero Lebesgue measure in $\Re^{3}$.
\end{theorem}

We have divided the proof in a sequence of lemmas.


\begin{lemma}\label{Lema:B4Xfn2}
Let $X(f) \in \mathcal{E}^{2,r}$ with $r\geq 1$. Then, the set
\begin{align*}
B^2_{4}(X(f)) = \{& \mu_{2} \in \Re:  \widetilde{X}(f +
\mu_{2} y^2)  \text{ has some non semi-hyperbolic }\\
&\text{periodic orbit in } C_{\pm \frac{\pi}{2}} \}
\end{align*}
has zero Lebesgue measure  in $\Re$.
\end{lemma}

\begin{proof} Let  $f(x,y) = a_0(x) + a_1(x) y + a_2(x) y^2$.
By  Proposition \ref{pro:PeriodicOrbit},  $\widetilde{X}(f)$ has two
periodic orbits of period $\tau$ (the same period of function
$a_i(u)$) on $C_{\pm \frac{\pi}{2}}$ with first derivative of the
Poincar\'e maps equals to $\exp(\mp \int_{0}^{\tau} a_2(u) du)$.
Then, $\widetilde{X}(f+ \mu_2 y^2)$ has a periodic orbit on $C_{\pm
\frac{\pi}{2}}$ non-hyperbolic if and only if  $\mu_2 = \pm
\frac{1}{\tau} \int_{0}^{\tau} a_2(u) du$. It follows that
$B^2_{4}(X(f))$ is discrete and has therefore zero measure in
$\Re$.\end{proof}


We remark that for every  $(\mu_0, \mu_1) \in
\Re^2$, $B^2_{4}(X(f +\mu_0 +
  \mu_1 y)) = B^2_{4}(X(f))$.

\begin{lemma}\label{Lema:B5Xfn2}
Let $X(f)\in \mathcal{E}^{2,r}(M)$, where $r\geq 1$ and let  $\mu_{0} \notin
B^2_{1}(X(f))$. Then the set
\[
B^2_{5}(X(f);\mu_{0}) = \{ \mu_{1} \in \Re:
\widetilde{X}(f + \mu_0 + \mu_1 y)\text{ has not the condition  3  of }
\Sigma^{2}\}
\]
has zero Lebesgue measure in $\Re$.
\end{lemma}
For the proof of this lemma; see proof of  Lemma \ref{Lema:B5Xfn1}.



\begin{proof}[Proof of Theorem \ref{Teo:BXfn2}]
The set $B^2(X(f))$ is union of following sets:
 \begin{gather*}
\mathcal{B}_1 =  B^2_{1}(X(f)) \times \Re^2,\quad
\mathcal{B}_2 =  \bigcup_{\mu_0 \in \Re-B^2_{1}(X(f))} \{ \mu_0 \}
   \times  B^2_{2}(X(f),  \mu_0) \times \Re, \\
\mathcal{B}_3 =  \bigcup_{ \mu_0 \in \Re} \{ \mu_0 \} \times
  B^2_{3}(X(f); \mu_0) \times \Re,\quad
\mathcal{B}_4 =  \Re^2 \times B^2_{4}(X(f)),\\
 \mathcal{B}_5 =  \bigcup_{\mu_0 \in \Re-B^2_{1}(X(f))} \{ \mu_0 \}
 \times B^2_{5}(X(f),  \mu_0) \times \Re,
 \end{gather*}
where  $B_{i}^2(X(f), \cdot)$, $i=1, \dots,5$, are given by Lemmas
 \ref{Lema:B1Xfn1}-\ref{Lema:B3Xfn1},
\ref{Lema:B4Xfn2} and \ref{Lema:B5Xfn2} respectively.

 Each $\mathcal{B}_i$
\begin{itemize}
\item contains  parameters $(\mu_0, \mu_1, \mu_2) \in \Re^3$ such that
$X(f + \mu_0 + \mu_1 y + \mu_2 y^2)$ violates at least a condition of
$\Sigma^2$.

\item is measurable. Because its complement in $\Re^3$ is open.

\item has zero measure in  $\Re^3$. Because $\mathcal{B}_1$ and
  $\mathcal{B}_4$ are  product of $\Re^2$ times a zero measure set in
  $\Re$. To measure  $\mathcal{B}_i$ for $i=2,3$ and $5$, we apply the
  Fubini Theorem as follows
$$ \int_{\Re^3} \chi(\mathcal{B}_i) d\mu_0 d\mu_1 d\mu_2 = \int_{\Re^2} (\int
\chi(\cdot, B_i^2(X(f),\mu_0), \cdot) d\mu_1) d\mu_0 d\mu_2 = 0$$
where $\chi(\cdot)$ is characteristic function in $\Re^3$.
\end{itemize}
Then $B^2(X(f))$ has zero measure in $\Re^3$.
\end{proof}

\subsection{Case C}\label{sec:genericidade3}
Here $X_{\mu}(f)$ denotes the vector field $X(f+ \mu_0 + \mu_1 y +
  \mu_{2} y^{3})$ where $\mu =(\mu_0, \mu_1, \mu_2) \in \Re^3$.
\begin{definition}
Let $\Sigma^{3}$ be  the set of $X(f) \in \mathcal{E}^{3,r}$,
$r\geq 1$ such that $\widetilde{X}(f)$ satisfies:
\begin{enumerate}
\item the singularities are hyperbolic and contained in $C_{0}$.

\item the periodic orbits  are hyperbolic and contained in
  $\widetilde{M}^{\circ}$.

\item the tangency points of $\widetilde{X}(f)$
with $C_{\pm\frac{\pi}{2}}$ are parabolic.

\item
\begin{enumerate}
\item there are no saddle connection of $\widetilde{X}(f)$.

\item the separatrix of the singularity points \ in  $C_{0}$ can
be only transversal to $C_{\pm\frac{\pi}{2}}$.

\item The trajectories are tangent to $C_{\pm\frac{\pi}{2}}$ at
most one point.
\end{enumerate}
\end{enumerate}
\end{definition}

\begin{theorem} \label{Teo:BXfn3}
Let $X(f)\in\mathcal{E}^{3,r}$ with  $r\geq 1$. Then the set
\[
B^3(X(f))  =  \{ (\mu_0, \mu_1, \mu_2) \in \Re^{3}:  X(f + \mu_0 +
\mu_1 y + \mu_2 y^3) \notin \Sigma^{3} \}
\]
has   Lebesgue measure zero in $\Re^{3}$.
\end{theorem}

The proof of this Theorem needs the following lemmas.

\begin{lemma}\label{Lema:B4Xfn3}
Let $X(f) \in \mathcal{E}^{3,r}$ with $r\geq 1$. Then, the set
\begin{align*}
B^3_{4}(X(f)) = \{ \mu_{2} \in \Re: & \text{ }
\widetilde{X}(f + \mu_{2} y^3)  \text{ has not the condition 3 of } \Sigma^3 \}
\end{align*}
has  Lebesgue measure zero in $\Re$.
\end{lemma}

\begin{proof}
 Let $X(f)$ with $f(x,y) = \sum_{i=0}^3 a_{i}(x) y^{i}$.
By Proposition \ref{pro:Tangencia},   $(u_{\ast}, \pm \frac{\pi}{2}) \in C_{\pm
  \frac{\pi}{2}}$ is a tangency point of $\widetilde{X}(f + \mu_{2} y^3)$ if
$a_{3}(u_{\ast}) + \mu_2 = 0$. Moreover the tangency is parabolic when
$a_{3}'(u_{\ast}) \neq 0$.

The set of critical values of $-a_{3}(u)$,
$$Crit(-a_{3}(u)) = \{ \mu_{2} \in \Re : \exists \, (u_\ast, 0) \text{ such
  that } a_{3}(u_{\ast}) + \mu_2 = 0 \text{ e } a_{3}'(u_{\ast}) = 0
  \},$$
describes the set of  $\mu_2 \in \Re$ such that  $X(f + \mu_{0} +
\mu_{1} y + \mu_{2} y^3)$ does not satisfy the condition  $3$ of $\Sigma^3$.

By Sard's Lemma, $B^{3}_4(X(f); \mu_0, \mu_1)$ has Lebesgue
measure zero in $\Re$.\end{proof}

\begin{lemma}\label{Lema:B5Xfn3}
Let $X(f)\in \mathcal{E}^{3,r}$, $r\geq 1$,  $\mu_{0} \notin
B^3_{1}(X(f))$ and $\mu_{2} \notin B^3_{4}(X(f))$. Then the set
\[ B^3_{5}(X(f);\mu_{0}, \mu_{2}) = \{ \mu_{1} \in \Re:
\widetilde{X}(f + \mu_0 + \mu_1 y + \mu_2 y^3) \text{ has not the
condition } 4 \text{ of } \Sigma^{3} \}
\]
has zero Lebesgue measure in $\Re$.
\end{lemma}
\begin{proof}
Fix  $\mu_{0} \notin B^3_{1}(X(f))$ and  $\mu_{2} \notin B^3_{3}(X(f))$ as in
Lemmas \ref{Lema:B1Xfn1} and \ref{Lema:B4Xfn3}.

There are three types of connections (see figure \ref{fig:conexaoseparan3}).
\begin{itemize}
\item[(a)] Connections of saddles in  $C_{0}$.
\item[(b)] A saddle  separatrix that is tangent to  $C_{+\frac{\pi}{2}}$ or
  $C_{-\frac{\pi}{2}}$ with parabolic tangency.
\item[(c)] A trajectory with two parabolic tangencies to  $C_{\pm
    \frac{\pi}{2}}$.
\end{itemize}

  \begin{figure}[htbp]
  \begin{center}
   \psfrag{p}[h][B]{$p$}
   \psfrag{q}[h][B]{$q$}
   \psfrag{C0}[h][B]{$C_{0}$}
   \psfrag{C+pi}[h][B]{$C_{+\frac{\pi}{2}}$}
   \psfrag{C-pi}[h][B]{$C_{-\frac{\pi}{2}}$}
   \includegraphics[height=1.9in,width=3.5in]{fig7.eps}%{conexaoseparan3.eps}
    \caption{Saddle connections in $C_0$ and tangency points in $C_{\pm
   \frac{\pi}{2}}$.}
    \label{fig:conexaoseparan3}
  \end{center}
\end{figure}

The saddle connections of $\widetilde{X}(f + \mu_{0} + \mu_{1} y +
\mu_{2}y^{3})$ happen in a number finite of
 $\mu_{1}$, since the number of singularities is finite.

The Lemma \ref{Lema:B4Xfn1} applies straightforwardly to  (a), (b)
and (c). In fact, we can break these connections with
perturbations of the form $\widetilde{X}(f+ \mu_0 + \mu_1 y +
\mu_2 y^3)$ where   $\mu_0$ and  $\mu_2$ are fixed. The derivative
of the  separation function of the stable and unstable manifolds  is
\begin{equation}  \label{eq:derivadaseparacao3}
\begin{aligned}
 \mathop{\rm sep}_{\mu_1}(\mu_1^{\ast})
 & =  - \int_{- \infty}^{+\infty} \exp(-\int_{0}^{t}
\mathop{\rm div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y + \mu_2 y^3)(u(s),v(s)))
ds)  \\
&\quad\times \sin^2(v(t)) \cos^2(v(t)) dt.
\end{aligned}
\end{equation}
For (b) and (c) we must consider the time $T$ taken  by a
trajectory from  $q \in \widetilde{M}^{\circ}$ to a tangency point
$p \in C_{\pm   \frac{\pi}{2}}$. Thus we have
\begin{equation}  \label{eq:derivadaseparacao3tan}
\begin{aligned}
 \mathop{\rm sep}_{\mu_1}(\mu_1^{\ast})
 &=  - \frac{1}{|\widetilde{X}(f)(q)|} \int_{0}^{T}  \exp(-\int_{0}^{t}
\mathop{\rm div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y
+ \mu_2 y^3)(u(s),v(s))) ds) \\
&\quad \times \sin^2(v(t)) \cos^2(v(t)) dt.
\end{aligned}
\end{equation}

The integrands in  (\ref{eq:derivadaseparacao3}) and
(\ref{eq:derivadaseparacao3tan}) are non-negative functions for
$v(t) \in [-\frac{\pi}{2}, \frac{\pi}{2}]$. Since $\mathop{\rm
sep}_{\mu_1}(\mu_1^{\ast}) < 0$, the connections are broken and no
other one  can arise. Then $B_4(X(f); \mu_0, \mu_2)$ is a discrete
subset of $\Re$ and has zero Lebesgue measure.\end{proof}


\begin{proof}[Proof of  Theorem \ref{Teo:BXfn3}]
The set  $B^3(X(f))$ is union of
\begin{gather*}
\mathcal{B}_1 =  B^3_{1}(X(f)) \times \Re^2,\quad
\mathcal{B}_2 =  \bigcup_{\mu_0 \in \Re-B^3_{1}(X(f))} \{ \mu_0 \}
   \times  B^3_{2}(X(f),  \mu_0) \times \Re,\\
\mathcal{B}_3 =  \bigcup_{ \mu_0 \in \Re} \{ \mu_0 \} \times
  B^3_{3}(X(f); \mu_0) \times \Re,\quad
\mathcal{B}_4 =  \Re^2 \times B^3_{4}(X(f)),\\
 \mathcal{B}_5 =  \bigcup_{(\mu_0, \mu_2) \in S}  \{ \mu_0 \}
 \times B^3_{5}(X(f), \mu_0, \mu_2) \times  \{ \mu_2 \},
\end{gather*}
where $S = \Re-B^3_{1}(X(f)) \times \Re-B^3_{4}(X(f))$ and  $B_{i}^3(X(f),
\cdot)$, $i=1, \dots,5$, are given by the Lemmas
 \ref{Lema:B1Xfn1}-\ref{Lema:B3Xfn1} and
\ref{Lema:B4Xfn3}-\ref{Lema:B5Xfn3} respectively.

 Each  $\mathcal{B}_i$:
\begin{itemize}
\item contains  $(\mu_0, \mu_1, \mu_2) \in \Re^3$ such that the
$X(f + \mu_0 + \mu_1 y + \mu_2 y^3)$ violates at least some condition of
$\Sigma^3$.

\item is mensurable. Because its  complement in $\Re^3$ is open.

\item  has zero measure in $\Re^3$. Because $\mathcal{B}_1$ and
  $\mathcal{B}_4$ are  products of $\Re^2$ times a set of zero measure in
  $\Re$. For $\mathcal{B}_i$ with  $i=2,3$ e $5$, we apply the  Fubini Theorem.
\end{itemize}
This completes the proof.\end{proof}


\subsection{Case \bf{D}}\label{sec:genericidade4}
In this case, $X_{\mu}(f)$ denotes the vector field $X(f+ \mu_0 +
\mu_1 y +
  \mu_{2} y^{3} + \mu_3 y^4)$ where $\mu =(\mu_0, \mu_1, \mu_2, \mu_3) \in
  \Re^4$.

\begin{definition} \rm
Let $\Sigma^{4}$ be the set of $X(f) \in \mathcal{E}^{4,r}$ with
$r \geq 1$ such that $\widetilde{X}(f)$  satisfies:
\begin{enumerate}
\item the singularities $(u_{\ast}, \cdot) \in C_{0} \cup C_{\pm
    \frac{\pi} {2}}$ are hyperbolic and the  eigenvalues associated to
    singularities  in $ C_{\pm\frac{\pi}{2}}$ are distinct.

\item the periodic orbits are hyperbolic and contained in
$\widetilde{M}^{\circ}$.

\item there are no connection of singularity separatrix. More specifically
 \begin{enumerate}
 \item no  connection of saddle that  belongs to $C_0$ or  $C_{\pm
 \frac{\pi}{2}}$,
 \item separatrices of saddle in $C_0$ or in  $C_{\pm \frac{\pi}{2}}$ are not
 weak manifolds of a  node in $C_{\pm \frac{\pi}{2}}$.
  \end{enumerate}
\end{enumerate}
\end{definition}

\begin{theorem} \label{Teo:BXfn4}
Let $X(f)\in\mathcal{E}^{4,r}$, where  $r\geq 1$. Then
\[
B^4(X(f))  =  \{ \mu \in \Re^{4}:\ X(f + \mu_0 + \mu_1 y + \mu_2
y^3 + \mu_3 y^4) \notin \Sigma^{4} \}
\]
has Lebesgue measure  zero in $\Re^{4}$.
\end{theorem}

The proof of this Theorem needs the following lemmas.

\begin{lemma}\label{Lema:B1Xfn4}
Let $X(f) = X(\sum^{4}_{i=0} a_{i}(x) y^{i}) \in
\mathcal{E}^{4,r}$ with $r \geq 1$. Then, the set
\begin{align*}
B^4_{4}(X(f)) = \{ & \mu_{3} \in \Re : \widetilde
{X}(f + \mu_{3} y^4)  \text{ has some singularity in }\\
 &  C_{\pm \frac{\pi}{2}}  \text{ with } \Delta \widetilde{X}(f + \mu_{3}y^4 )
 = 0 \}
\end{align*}
has Lebesgue measure zero in $\Re$.
\end{lemma}

\begin{proof}
The set $Crit(-a_{4})= \{\mu_{3}= -a_{4}(x) : a_{4}'(x) =
0 \}$ determines the set of  $\mu_{3}$ such that $\widetilde{X}(f+
\mu_{3}y^{4})$ has singularities in $C_{\pm \frac{\pi}{2}}$  with
$\Delta \widetilde{X}(f + \mu_{3}y^{4}) = 0$. By Sard's theorem,
this set has Lebesgue measure  zero in $\Re$.
\end{proof}

\begin{lemma}\label{Lema:B2Xfn4}
Let $X(f)= X(\sum^{4}_{i=0} a_{i}(x) y^{i}) \in \mathcal{E}^{4,r}$
with  $r \geq 1$, and let  $\mu_{0} \notin B^4_{1}(X(f))$ and
$\mu_{3} \notin B^4_{2}(X(f))$. Then the set
 \begin{align*}
B^4_{5}(X(f);\mu_{3})=   \{ &\mu_{2}  \in \Re : \widetilde{X}(f +
\mu_{2} y^3 + \mu_{3} y^4) \text{ has some  hyperbolic, with equal }\\
& \text{eigenvalues, or non-hyperbolic singularity  in  } C_{\pm \frac{\pi}{2}} \}
\end{align*}
has Lebesgue measure zero in $\Re$.
\end{lemma}

\begin{proof}
Let $ \mu_3 \notin B^4_4(X(f))$. Since
\begin{equation}\label{eq:ValorProprioIgual}
  a_3(u_{\ast}) +  \mu_2 = 2 \sqrt{a_4'(u_{\ast})},
\end{equation}
 $\widetilde{X}(f +  \mu_2 y^3 + \mu_3 y^4)$ has a hyperbolic
singularity $(u_{\ast}, \pm \frac{\pi}{2})$ with equal eigenvalues. Then
there exists a single value $\mu_2^{\ast}$ that satisfies the condition
(\ref{eq:ValorProprioIgual}).

On the other hand, for each non-hyperbolic singularity $(u_{\ast}, \pm
\frac{\pi}{2})$ of $\widetilde{X}(f + \mu_2 y^3 + \mu_3 y^4)$, there
exists a unique value $\mu_2^{\ast}$ such that
$\sigma(\widetilde{X}(f + \mu_2^{\ast}  y^3 + \mu_3 y^4))=
-(a_3(u_{\ast}) +  \mu_2) = 0$.
It follows that $B^{4}_5(X(f); \mu_3)$ is discrete  in $\Re$ and
has therefore zero Lebesgue measure.\end{proof}

\begin{lemma}\label{Lema:B4Xfn4}
Let $X(f) \in \mathcal{E}^{4,r}$ with $r\geq 1$, $\mu_{0} \notin
B^4_{1}(X(f))$ and  $\mu_{3} \notin B^4_{4}(X(f))$. Then, the set
\begin{align*}
B^4_{6}(X(f); \mu_{0},\mu_{3}) = \{& \mu_{1} \in \Re:
\widetilde{X}(f + \mu_0 + \mu_1 y + \mu_3 y^4)  \text{ has not }
\\ & \text{ the condition 3 of } \Sigma^4 \}
\end{align*}
has Lebesgue measure  zero in $\Re$.
\end{lemma}

\begin{proof}
All singularity connections of  $\widetilde{X}(f)$ can be broken
with perturbations of the form  $\widetilde{X}(f+ \mu_0 + \mu_1 y
+ \mu_3 y^4)$.
The derivative of the separation function is of the form
\begin{equation}
\begin{aligned}
\mathop{\rm sep}_{\mu_1}(\mu_1^{\ast}) & =  - \int_{-
\infty}^{+\infty} \exp(-\int_{0}^{t} \mathop{\rm
div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y + \mu_3 y^4)) ds) \\
&\quad \times \sin^2(v(t)) \cos^3(v(t)) dt.
\end{aligned} \label{eq:derivaSeparacao4}
\end{equation}
The integrand in (\ref{eq:derivaSeparacao4}) is non-negative for $v(t) \in
[-\frac{\pi}{2}, \frac{\pi}{2}]$.
Then  $\mathop{\rm sep}_{\mu_1}(0) < 0$. This ends the proof.\end{proof}


\begin{proof}[Proof of  Theorem \ref{Teo:BXfn4}]
The set  $B^4(X(f))$ is union of
\begin{gather*}
\mathcal{B}_1 =  B^4_{1}(X(f)) \times \Re^3,\quad
\mathcal{B}_2 =  \bigcup_{\mu_0 \in \Re-B^4_{2}(X(f))} \{ \mu_0 \}
   \times  B^4_{3}(X(f), \mu_0) \times \Re^2, \\
\mathcal{B}_3 =  \bigcup_{ \mu_0 \in \Re} \{ \mu_0 \} \times
  B^4_{3}(X(f); \mu_0) \times \Re^2,\\
\mathcal{B}_4 = \Re^3 \times B^4_{4}(X(f)),\quad
\mathcal{B}_5 =  \Re^2 \times B^4_5(X(f), \mu_3) \times \bigcup_{\mu_3
   \in \Re-B^4_{4}(X(f))} \{ \mu_3 \}, \\
 \mathcal{B}_6 =  \bigcup_{(\mu_0, \mu_3) \in S} \{ \mu_0 \}
 \times B^4_{6}(X(f), \mu_0, \mu_3) \times \Re \times \{ \mu_3 \},
 \end{gather*}
where $ S = \Re-B^4_{1}(X(f)) \times \Re-B^4_{4}(X(f))$ and
$B_{i}^4(X(f), \cdot)$, $i=1, \dots, 6$, are given by the  Lemmas
\ref{Lema:B1Xfn1}-\ref{Lema:B3Xfn1} and
\ref{Lema:B1Xfn4}-\ref{Lema:B4Xfn4}.

  Each  $\mathcal{B}_i$
\begin{itemize}
\item contains   $(\mu_0, \mu_1, \mu_2, \mu_3) \in \Re^4$ such that
$X(f + \mu_0 + \mu_1 y + \mu_2 y^3 + \mu_3 y^4)$ violates at least a condition
of $\Sigma^4$.

\item is measurable. Because its complement in $\Re^4$ is open.

\item has measure  zero in $\Re^4$. Because  $\mathcal{B}_1$ and
  $\mathcal{B}_4$ are   products of $\Re^3$ times a set of  measure zero in
  $\Re$. For other $\mathcal{B}_i$, we apply the Fubini
  Theorem.
\end{itemize}
This completes the proof.\end{proof}

\subsection{Case \bf{E}}\label{sec:genericidade5}
In this case, $X_{\mu}(f)$ denotes the vector field $X(f+ \mu_0 +
\mu_1 y + \mu_{2} y^{n-1})$, where $\mu =(\mu_0, \mu_1, \mu_2 )
\in \Re^3$ and $n \geq 5$.

\begin{definition}\label{def:n5} \rm
Let  $\Sigma^n$ be the set of
   $X(f) \in \mathcal{E}^{n,r}$  for  $n \geq 5$ and  $r\geq 2$ such that
   $\widetilde{X}(f)$ satisfies:
\begin{enumerate}
 \item the singularities  $(u_{\ast}, 0) \in C_{0}$ are  hyperbolic and the
  singularities  $(u_{\ast}, \pm \frac{\pi}{2}) \in C_{\pm \frac{\pi}{2}}$
are   semi-hyperbolic.
 \item the periodic orbits are hyperbolic and
contained in  $\widetilde{M}$.
 \item no connections of singularity separatrix. More specifically
 \begin{enumerate}
 \item no connection of saddles that  belong to $C_0$,
 \item saddle separatrices in $C_0$ are not invariant manifolds of
 singularities in $C_{\pm \frac{\pi}{2}}$.
  \item no connection between singularities that belong to  $C_{\pm
  \frac{\pi}{2}}$ by invariant manifolds.
 \end{enumerate}
\end{enumerate}
\end{definition}

\begin{theorem} \label{Teo:BXfn5}
Let $X(f)\in\mathcal{E}^{n,r}$ with $n \geq 5$ and $r \geq 2$.
Then the set
\[
B^n(X(f))  =  \{ (\mu_0, \mu_1, \mu_2) \in \Re^{3} :  X(f + \mu_0
+ \mu_1 y + \mu_2 y^{n-1}) \notin \Sigma^{n} \}
\]
has zero Lebesgue measure  in $\Re^{3}$.
\end{theorem}

\begin{lemma}\label{Lema:B1Xfn5}
Let $X(f) = X(\sum^{n}_{i=0} a_{i}(x) y^{i}) \in
\mathcal{E}^{n,r}$ with $n \geq 5$, $r \geq 1$. Then the set
\begin{align*}
B^n_{4}(X(f)) = \{ & \mu_{2} \in \Re : \widetilde
{X}(f + \mu_{2} y^{n-1}) \text{ has some singularity in }\\
 &  C_{\pm \frac{\pi}{2}}  \text{ with } \sigma \widetilde{X}(f +
 \mu_{2}y^{n-1} )  = 0 \}
\end{align*}
has zero Lebesgue measure in $\Re$.
\end{lemma}

\begin{proof}
Each singularity $(u, \pm \frac{\pi}{2})$ of  $\widetilde{X}(f +
\mu_2 y^{n-1})$ that satisfies  $a_{n-1}(u) + \mu_2 = 0$,
determines a non semi-hyperbolic singularity. Then,
$B^n_{4}(X(f))$ must be a discrete set with measure zero in
$\Re$.
\end{proof}

\begin{lemma}\label{Lema:B4Xfn5}
Let $X(f) \in \mathcal{E}^{n,r}$  with $n \geq 5$, $r\geq 2$,
$\mu_{0} \notin B^n_{1}(X(f))$ and  $\mu_{2} \notin
B^n_{2}(X(f))$. Then the set
\begin{align*}
B^n_{5}(X(f); \mu_{0},\mu_{2}) = \big\{ &\mu_{1} \in \Re:
\widetilde{X}(f + \mu_0 + \mu_1 y + \mu_2 y^{n-1}) \\
& \text{ does not satisfying property 3 of } \Sigma^n \big\}
\end{align*}
has zero Lebesgue measure in $\Re$.
\end{lemma}

\begin{proof}
It follows by the same way as in Lemma \ref{Lema:B4Xfn4}. In this case,  the
derivative of the separation function of the stable and unstable manifolds is
of  the form
\begin{equation}   \label{eq:derivadaseparacao5}
 \begin{aligned}
\mathop{\rm sep}_{\mu_1}(\mu_1^{\ast})  &=  - \int_{-
\infty}^{+\infty} \exp(-\int_{0}^{t} \mathop{\rm
div}(\widetilde{X}(f+\mu_0 + \mu_1^{\ast} y + \mu_2 y^{n-1})) ds) \\
& \quad \times \sin^2(v(t)) \cos^{n-1}(v(t)) dt.
\end{aligned}
\end{equation}
The integrand in (\ref{eq:derivadaseparacao5}) is a non-negative
function.\end{proof}

\begin{proof}[Proof of Theorem \ref{Teo:BXfn5}]
The set  $B^n(X(f))$ is union of:
\begin{gather*}
\mathcal{B}_1 =  B^n_{1}(X(f)) \times \Re^2,\quad
\mathcal{B}_2 =  \bigcup_{\mu_0 \in \Re - B_1^n(X(f))} \{ \mu_0 \}
   \times B^n_2(X(f), \mu_0) \times \Re, \\
\mathcal{B}_3 =  \bigcup_{ \mu_0 \in \Re} \{ \mu_0 \} \times
  B^n_{3}(X(f); \mu_0) \times \Re,\quad
\mathcal{B}_4 =  \Re^2 \times  B^n_{4}(X(f)),\\
\mathcal{B}_5 =  \bigcup_{ (\mu_0, \mu_2) \in S} \{ \mu_0 \}
\times B^n_{5}(X(f); \mu_0, \mu_2) \times \{ \mu_2 \}
\end{gather*}
where $S = \Re - B_1^n(X(f)) \times
\Re - B^n_{4}(X(f))$ and $B_{i}^n(X(f), \cdot)$, $i=1, \dots,5$,
are given by Lemmas
  \ref{Lema:B1Xfn1}-\ref{Lema:B3Xfn1} and \ref{Lema:B1Xfn5}-\ref{Lema:B4Xfn5}.

  Each  $\mathcal{B}_i$
\begin{itemize}
\item contains   $(\mu_0, \mu_1, \mu_2) \in \Re^3$ such that
  $X(f + \mu_0 + \mu_1 y + \mu_2 y^{n-1})$ violates at least a condition of
  $\Sigma^{n}$.

\item is measurable. Because its  complement  in $\Re^3$ is open.

\item has measure zero in $\Re^3$. (As in the proof of Theorem
\ref{Teo:BXfn4} by the Fubini Theorem).
\end{itemize}
This completes the proof.\end{proof}


\begin{remark}\label{rem:concluding} \rm
Theorems \ref{Teo:BXfn1} to \ref{Teo:BXfn5} express the measure
of the complementary set of $\Sigma^n$ for $n\geq 1$. They  may  be
summarized by stating that
\[
B^n(X(f))  =  \{ (\mu_0, \mu_1, \mu_{n-1},\mu_{n}) :  X(f + \mu_0
+ \mu_1 y + \mu_{n-1} y^{n-1} +  \mu_{n} y^{n}) \notin \Sigma^{n}
\}
\]
has zero Lebesgue measure. However, we observe that for $n >4$ three
parameters only are sufficient. In the present work, the
proof has been divided in  cases {\bf A}-{\bf E} to make the
presentation more accessible.
\end{remark}

\section{Genericity  of $\Sigma^{n}$.}\label{sec:genericity}
In this section, we prove the genericity of $\Sigma^{n}$.

\begin{proof}[Proof of Theorem \ref{theo:genericity}]
 The density of $\Sigma^{n}$ follows straightforwardly  from Theorems
 $\ref{Teo:BXfn1}$-$\ref{Teo:BXfn5}$ for corresponding $n$.
 In fact, for $X(f) \in \mathcal{E}^{n,r}$ and $\forall \, \epsilon > 0$ we can
 find $X_{\mu}(f) \in \Sigma^{n}$ such that $|\widetilde{X}_{\mu}(f) -
 \widetilde{X}(f)| < \epsilon$. $\mu \in \Re^k$ where $k=2$ for $n=1$; $k=3$
 for $n=2,3$ and $\geq 5$; and $k=4$ for $n=4$.

To see that $\Sigma^{n}$ is open in $\mathcal{E}^{n,r}$ we write
$X(f) = y \frac{\partial}{\partial x} + \sum_{i=0}^{n} a_{i}(x)
y^{i}\frac{\partial}{\partial y}$. In $C_{0}$, a singularity of
$\widetilde{X}(f)$, say $(u_{\ast}, 0)$, satisfies
$a_{0}(u_{\ast})=0$ and $a'_{0}(u_{\ast}) \neq 0$. In
$C_{\pm \frac{\pi}{2}}$, we must consider  the following cases:
\begin{enumerate}
\item $n=3$, since $\widetilde{X}(f)$ has  a parabolic tangency at
$(u_{\ast}, \cdot)$, it satisfies $a_{3}(u_{\ast}) = 0$ and
$a'_{3}(u_{\ast}) \neq 0$.

\item $n=4$, a singularity of $\widetilde{X}(f)$, say $(u_{\ast},
  \cdot)$, satisfies $a_{4}(u_{\ast})=0$ and $a'_{4}(u_{\ast}) \neq 0$,
  and, also, $a_3(u_{\ast}) \neq 2 \sqrt{a'_{4}(u_{\ast})}$ if
$a'_{4}(u_{\ast})>0$.

\item $n \geq 5$, a singularity of $\widetilde{X}(f)$, say
$(u_{\ast},
  \cdot)$, satisfies $a_{n}(u_{\ast})=0$ and $a_{n-1}(u_{\ast})\neq 0$.
\end{enumerate}
In all those cases, we can choose  $\mu_{0}, \mu_{1}, \mu_{2}$ and $\mu_{3}$
sufficiently  small such that
$X( \sum_{i=0}^{n} b_{i}(x) y^{i})$, where $b_{0} = a_{0} +
\mu_{0}$, $b_{1} = a_{1} + \mu_{1}$, $b_{n-1} = a_{n-1} + \mu_{2}$
and $b_{n} = a_{n} + \mu_{3}$, has singularities or tangencies of
the same type than $X(f)$.

If $X(f)$ has a hyperbolic periodic orbit, $X(g)$ with $b_{1} =
a_{1}(u) + \mu_{1}$  will also have a hyperbolic periodic orbit
for small values of $\mu_{1}$.
For $n=1$ and $2$, we obtain the same type of periodic orbits in $C_{\pm
  \frac{\pi}{2}}$, taking $b_1 = a_1 + \mu_1$ and $b_2 = a_2 + \mu_2$.
\end{proof}


\section{Characterization}\label{sec:characterization}

We will prove the necessity of Theorem \ref{theo:characterization}
in detail. For the sufficiency, we will only touch on some  aspects
with respect to  new canonical regions and to the building of
homeomorphism.

\subsection{Necessity}\label{sec:necessity}
 There is a neighborhood $\mathcal{U}$ of
$X(f)$ in $\mathcal{E}^{n,r}$ such that for $X(g) \in \mathcal{U}$
there is a homeomorphism $h_{g}:\widetilde{M} \to
\widetilde{M}$
 that transforms trajectories of $\widetilde{X}(f)$
in trajectories of $\widetilde{X}(g)$.

By density of $\Sigma^{n}$, there is $X(\sum_{i=0}^{n} b_i(x)
y^{i}) \in \Sigma^{n} \cap \mathcal{U}$ such that it is topologically
equivalent to $X(f)$. Then $X(f)$ inherits the following
properties of $X(\sum_{i=0}^{n} b_i(x) y^{i})$:
\begin{itemize}
\item In $C_0$, $X(f)$ has a finite number of singularities and  all
  topologically equivalent to saddles,  focus or nodes.
\item In $C_{\pm \frac{\pi}{2}}$, $X(f)$  has a finite number of
tangencies (when $n=3$) or of singularities (when $n \geq 4$). The
tangencies are internal or external.

\item The periodic orbits are finite in number, all are attractors
or repellors and are contained in the interior of $\widetilde{M}$.
\item $X(f)$ has no connection of singularity separatrix, nor
separatrix of saddle tangent to $C_{\pm \frac{\pi}{2}}$.
\end{itemize}
The following properties of $X(f)$ remain  to be proved.

\noindent{\bf (a)} The singularities in $C_0$ are all hyperbolic. In fact,
if $\Delta \widetilde{X}(f) = 0$  at some singularity, say
$(u_{\ast},0)$, we can consider $\widetilde{X}(f+ \epsilon
\beta(u) (u-u_{\ast}))$ where  $\beta$ is a non-negative periodic
function on a neighborhood $V$ of $u_{\ast}$ and with the compact
support  contained in $V$,  $\beta(u_{\ast}) = 1$ and  $V$ does
not  contain  other singularities. Consequently,  $\Delta
\widetilde{X}(f)(u_{\ast},0) =  -\epsilon \beta(u_{\ast})$. Then,
by choosing the adequate sign for  $\epsilon$, $(u_{\ast}, 0)$ is
also a singularity of  $\widetilde{X}(f+\epsilon \beta(u) (u-
u_{\ast}))$, but it has the different  index  as singularity    of
$\widetilde{X}(f)$. This contradicts the structural stability of
$X(f)$. \smallskip

\noindent{\bf (b)} In  $C_{\pm \frac{\pi}{2}}$, depending on
$n$, we have hyperbolic and semi-hyperbolic singularities, or
parabolic tangency points.

For $n=3$, suppose that $(u_{\ast}, \frac{\pi}{2})$ is a point of
  non-parabolic tangency but it is internal or external. That is,
  $(u_{\ast},\frac{\pi}{2})$ satisfies $a_3(u_{\ast}) = 0$ and
  $a_3'(u_{\ast}) = 0$.

Let $V$ be a neighborhood of  $u_{\ast}$ in $\Re$  such that it
contains no  other  tangency points of $\widetilde{X}(f)$. For
$\epsilon$ to be
 small, $\widetilde{X}(f + \epsilon (x-u_{\ast}) \beta(x) y^3)$ is
 sufficiently close
 to $X(f)$, where  $\beta(x)$ is a non-negative $C^r$
 periodic  function with support contained in $V$.  Since
$\widetilde{X}(f + \epsilon (x-u_{\ast}) \beta(x) y^3) \cdot b
(u_{\ast}, \frac{\pi}{2}) = 0$, $(u_{\ast}, \frac{\pi}{2})$ is a
tangency point. Moreover the tangency is parabolic, since
$\widetilde{X}^2(f + \epsilon (x-u_{\ast}) \beta(x) y^3) \cdot b
(u_{\ast}, \frac{\pi}{2}) =
 \epsilon \neq 0$, where $b: \widetilde{M} \to \Re$ is defined by
 $b(u,v)= \cos(v)$, see the Proposition \ref{pro:Tangencia}.

Then, if $(u_{\ast}, \frac{\pi}{2})$ is an external (respectively
internal) tangency point of $\widetilde{X}(f)$, we take $\epsilon$
positive (respectively, negative) such that $\widetilde{X}(f +
\epsilon (x-u_{\ast}) \beta(x) y^3)$ has at $(u_{\ast},
\frac{\pi}{2})$ an internal (respectively, external) tangency
point, contradicting the structural stability of $X(f)$.

For $n =4$, the hyperbolicity of all singularities of
$\widetilde{X}(f)$ in $C_{\pm \frac{\pi}{2}}$ can be shown in the
same way that (a), considering the system $\widetilde{X}(f +
\epsilon (x- u_{\ast}) \beta(x) y^{4})$ close to
$\widetilde{X}(f)$ and $(u_{\ast}, \pm \frac{\pi}{2})$ is a
singularity of $\widetilde{X}(f)$.

We need to treat here only the case that $D\widetilde{X}(f)$ at  a singularity
of $\widetilde{X}(f)$ has equal eigenvectors.
In the fact, if $(u_{\ast}, \pm \frac{\pi}{2})$ is one such
singularity then $a_4'(u_\ast)
>0$ and  $a_3(u_{\ast}) = 2 \sqrt{a_4' (u_{\ast})}$. Thus
taking $X(f + \epsilon (x- u_{\ast}) \beta(x) y^{3})$ sufficiently
near to $\widetilde{X}(f)$, where $\beta(x)$ is a non-negative
periodic function with support in a neighborhood of $u_{\ast}$ and
without any other singularity, we find that $(u_{\ast}, \pm
\frac{\pi}{2})$ is also a singularity of $X(f + \epsilon (x-
u_{\ast}) \beta(x) y^{3})$, but with non-equal eigenvalues. More
specifically, if $\epsilon < 2 \sqrt{a_4'(u_{\ast})} -
a_3(u_{\ast})$, the singularity is a focus and if $\epsilon > 2
\sqrt{a_4'(u_{\ast})} - a_3(u_{\ast})$ the singularity is
a hyperbolic node. Therefore, $X(f + \epsilon (x- u_{\ast})
\beta(x) y^{3})$ and $X(f)$ can not be topologically equivalent.

For $n \geq 5$, if $(u_{\ast}, \pm \frac{\pi}{2})$ is a
singularity of $\widetilde{X}(f)$ non-semi-hyperbolic then
$a_n(u_{\ast}) = 0$ and $a_{n-1}(u_{\ast}) = 0$. Now, $X(f +
\epsilon (x- u_{\ast}) \beta(x) y^{n-1})$ sufficiently close to
$\widetilde{X}(f)$ has $(u_{\ast}, \pm \frac{\pi}{2})$ as a
semi-hyperbolic singularity. This contradicts the structural
stability of $X(f)$. \smallskip


\noindent{\bf (c)} The periodic orbits in
$\widetilde{M}^{\circ}$ are all hyperbolic. In fact, we
suppose that  $\gamma(s) =(\varphi(s), \xi(s))$ is a periodic
orbit  of $X(f)$ with period $\tau$ such that $\chi(\gamma, X(f))
=  0$. We will obtain a field $X(f+g)$ sufficiently  near to
$X(f)$ in which $\gamma$ is a periodic orbit but $\chi(\gamma,
X(f+g)) \neq 0$.

Then, we search a function $g(x,y) = b_{0}(x) + b_{1}(x) y$, where
 $b_0$ and $b_1$ are $C^r$-periodic such that
\begin{itemize}
\item[(i)] $X(f+g)$ is near to  $X(f)$, we say at a distance
$|\epsilon| \neq 0$.

 \item[(ii)] $X(f+g)(\gamma) = X(f)(\gamma)$ (i.e. $\gamma$ is a periodic
 orbit of both vector  fields).
\item[(iii)] $\chi(\gamma, X(f+g)) \neq 0$
\end{itemize}

The conditions $i), ii),$ and $iii)$ determine:
\begin{itemize}
\item[$\ast$] $\sup |b_{i}^{(k)}(x)| <  |\epsilon| $ for $i=1,2,
0\leq k \leq r$ and $x\in S^{1}$.

 \item[$\ast$] $b_0(\varphi(s)) + b_{1}(\varphi(s)) \xi(s) = 0$.

 \item[$\ast$] $b_{1}(\varphi(s))$ is not identically null in $\Re$.
\end{itemize}

The task is now to find a function $b_{1}: \Re \to \Re$
which is $C^{r}$-periodic and not identically null. Thus we can
choose $b_0:\Re \to \Re$ $C^{r}$-periodic such that
$b_0(\varphi(s)) = -b_{1}(\varphi(s)) \xi(s)$ for  $s\in \Re$.

We should consider the  following three situations:
\begin{enumerate}
\item $b_0$  can not  be null. Because, in otherwise, $\xi(s) =0$
for $s \in \Re$ and $C_0$ should be a periodic orbit of $X(f)$,
contradicting the fact that all periodic orbits only can intercept
$C_0$ at isolated points.

\item If $\xi(s)$ is not a non-null constant, say $\xi_0$, then we
can put $b_{1}(x) = \epsilon \neq 0$ and $b_{0}(x) = -\epsilon
\xi_{0}$ both constant. Thus, $g(x,y) = -\epsilon \xi_0 + \epsilon
y$ defines a perturbation $|\epsilon|$-close to $X(f)$ such that
$\chi(\gamma, X(f+g)) = \epsilon$.

\item If $\xi(s)$ is not constant, we can choose some $s_0 \in (0,
\tau)$ such that $\xi(s_0) \neq 0$ and  $\varphi(s_0) \neq 0$. Let
$W= \varphi([0, \tau]) \subset \Re$. Then there is an open
neighborhood $U$ of $s_0$ contained in $[0,\tau]$ such that
$\varphi_{|U}$ has differentiable inverse (since
$\varphi'(s_0) = \xi(s_0)\neq 0$).

We suppose that $\xi(s_0)>0$. We denote $\widetilde{U} =
\varphi(U) \subset W$ and let $V=(\alpha, \beta)$ be a
neighborhood of $\varphi(s_0)$ such that $V \subset \overline{V}
\subset \widetilde{U}$. (See figure \ref{fig:orbitas}).

\begin{figure}[htb]
\begin{center}
\psfrag{xi}[h][B]{$\xi(s_0)$}
 \psfrag{phi}[h][B]{$\varphi(s_0)$}
 \psfrag{alfa}[h][B]{$\alpha$}
 \psfrag{beta}[h][B]{$\beta$}
 \psfrag{a}[h][B]{$a$}
 \psfrag{b}[h][B]{$b$}
 \psfrag{gamma}[h][B]{$\gamma(s) = (\varphi(s),\xi(s))$}
 \psfrag{tau}[h][B]{$\tau$}
 \psfrag{U}[h][B]{$U$} \psfrag{c0}[h][B]{$C_0$}
 \psfrag{s0}[h][B]{$s_0$}
 \includegraphics[height=2.8in,width=4in]{fig8.eps}%{orbitas.eps}
 \caption{(a) Homotopic to zero periodic orbit. (b) Periodic
orbit  circling the cylinder. The points $a$ and $b$ are identified}
\label{fig:orbitas}
\end{center}
\end{figure}

Now, we can take a function  $b_{1}:\Re \to \Re$ with the
following properties:
\begin{itemize}
\item[$\ast$] $b_{1}(x) > 0$ for $x \in V$ and $b_{1}(x) = 0$ for
$x\in W-V$,

\item[$\ast$] $C^r$  with $r\geq 1$,

\item[$\ast$] periodically extended to whole $\Re$.
\end{itemize}

Next, we define $b_0: \Re \to \Re$ by $b_{0}(x) =
-b_{1}(x) \varphi'(\varphi^{-1}(x))$ for $x \in V$ and
$b_{0}=0$ for  $x \in W-V$. $b_{0}$ can be extended periodically
to all $\Re$ and of $C^{r}$ class.
It follows that $g(x,y) = \epsilon b_0(x) + \epsilon b_{1}(x) y$ is a
perturbation of $f(x,y)$, as we wanted. In fact, $X(f+g)$ has
$\gamma$ as a period orbit with characteristic index
$$\chi(\gamma, X(f+g)) = \epsilon \int_{\varphi^{-1}(\alpha)}
^{\varphi^{-1}(\beta)} b_{1}(\varphi(s)) ds \neq 0.$$
\end{enumerate}
It follows from (1)-(3) that if $\gamma$ is attractor for
$X(f)$ then by taking $\epsilon > 0$, $X(f+g)$ has at least two periodic orbits more than $X(f)$. This
contradicts the  structural stability of $X(f)$. The proof of (c)
is complete.
Finally  (a), (b) and (c) together with the properties described
above show that $X(f) \in \Sigma^n$.

\subsection{Sufficiency}\label{sec:sufficiency}

The proof of the sufficiency of Theorem
\ref{theo:characterization} is essentially the same given by
Peixoto-Peixoto in \cite{peixoto-peixoto} and later by Sotomayor
in \cite{Sotomayor2}. Here we indicate some important aspects of
this Theorem.

Every $X(f) \in \Sigma^n$ determines a decomposition of
$\widetilde{M}$ in connected components, called  \emph{canonical
regions}, whose boundary are formed by separatrices (i.e. an arc of
trajectory, a singularity, a limit cycle, a saddle separatrix and
a portion of $\partial \widetilde{M}$).

An  \emph{attractor} (resp. a \emph{source}) of $\widetilde{X}(f)$
associated to a canonical region $R \subset \widetilde{M}$ is a
node, a focus or a limit cycle of $\widetilde{X}(f)$ or is an arc
of $\partial R$ where the trajectories in $R$ tend as
$t\to + \infty$ (resp. $t\to - \infty$).
A \emph{critical region of a singularity} $p$ of
$\widetilde{X}(f)$ is a neighborhood  $D$ of $p$ such that the
systems sufficiently near to  $\widetilde{X}(f)$ have a single
singularity and of the same type as  $p$.
A \emph{critical region of a  limit cycle} $\Gamma$ of
$\widetilde{X}(f)$ is a ring $A$ contained $\Gamma$ such that the
systems sufficiently close to $\widetilde{X}(f)$ have a limit
cycle and of the same stability that $\Gamma$.

Every canonical region has in its boundary only a source and an
attractor. This property plays a very important role in the
classification of canonical regions. See \cite{peixoto-peixoto}.
We denote by $\alpha$, $\omega$ and  $\gamma$  a source, an
attractor and a separatrix of  $\widetilde{X}(f)$ respectively. A
separatrix can be a point (e.g. a singularity or an external
tangency).

Then we classify  a canonical region  $R$ by the number and type
of separatrices contained in $\partial R$, as follows (see figure
\ref{fig:regcanonica}).
\begin{itemize}
\item \emph{Type} $i$. $\partial R = \{ \alpha, \omega \}$.

\item \emph{Type} $ii$. $\partial R = \{ \alpha, \omega, \gamma_1 \}$
  where  $\gamma_1$ is a  trajectory that is transversal to  $\partial M$, and
  $\alpha$ and  $\omega$ belong to  $\partial M$.

\item \emph{Type} $iii$. $\partial R = \{ \alpha, \omega, \gamma_1, p_1 \}$
  where $\gamma_1$ is a trajectory with internal tangency to $\partial M$ and
  $p_1$ is a point of  external tangency.

\item \emph{Type} $iv$. $\partial R = \{ \alpha, \omega, \gamma_1,
  \gamma_2\}$ where $\gamma_1$ and $\gamma_2$ are arcs of trajectories.
\item \emph{Type} $v$. $\partial R = \{ \alpha, \omega, p_1, \gamma_1,
  \gamma_2\}$ where  $p_1$ is a saddle, $\gamma_1$ and  $\gamma_2$ are
  trajectories tending to $p_1$.

\item \emph{Type} $vi$. $\partial R = \{ \alpha, \omega, p_1, \gamma_1,
   \gamma_2, \gamma_3\}$ where  $p_1$ is a saddle, $\gamma_1$, $\gamma_2$ are
   trajectories tending to $p_1$ and $\gamma_3$ is a trajectory that does not
   tend to  $p_1$.

\item \emph{Type} $vii$. $\partial R = \{ \alpha, \omega, p_1, \gamma_1,
  \gamma_2, \gamma_3 \}$ where $p_1$ is a saddle, $\gamma_1$, $\gamma_2$ and
  $\gamma_3$  are trajectories tending to  $p_1$.

\item \emph{Type} $viii$. $\partial R = \{ \alpha, \omega, p_1, \gamma_1,
 \gamma_2, p_2, \gamma_3 \}$ where  $p_1$ and $p_2$ are saddles, $\gamma_1$,
 $\gamma_2$ trajectories tending to $p_1$ and  $\gamma_3$ trajectory tending to
 $p_2$.

\item \emph{Type} $ix$. $\partial R = \{ \alpha, \omega, p_1, \gamma_1,
 \gamma_2, p_2, \gamma_3, \gamma_4 \}$ where $p_1$ and $p_2$ are saddles,
 $\gamma_1$,  $\gamma_2$ trajectories tending to  $p_1$, $\gamma_3$ and
 $\gamma_4$ are trajectories tending to  $p_2$.
\end{itemize}

\begin{figure}[htbp]
  \begin{center}
   \psfrag{alpha}[h][B]{$\alpha$}
   \psfrag{omega}[h][B]{$\omega$}
   \psfrag{gamma1}[h][B]{$\gamma_1$}
   \psfrag{gamma2}[h][B]{$\gamma_2$}
   \psfrag{gamma3}[h][B]{$\gamma_3$}
   \psfrag{gamma4}[h][B]{$\gamma_4$}
   \psfrag{dM}[h][B]{$\partial \widetilde{M}$}
   \psfrag{p1}[h][B]{$p_1$}
   \psfrag{p2}[h][B]{$p_2$}
   \includegraphics[height=3.0in,width=4in]{fig9.eps} %{regioescanonicas.eps}
    \caption{Canonical regions classified by the number and the type of
   separatrices.}
    \label{fig:regcanonica}
  \end{center}
\end{figure}

In  \cite{peixoto-peixoto}, Peixoto and  Peixoto showed five types
of canonical regions of $C^1$ vector fields in the plane. Later,
Sotomayor in \cite{Sotomayor2} extended to seven regions
considering external and internal tangencies of trajectories (that
are not saddle separatrices) with the boundary.

We remark that the types  $ii$, $v$ and $viii$ are new ones, which arise
by considering singularities on the boundary of $\widetilde{M}$.
In Table \ref{tab:regioescanonicas} we compare the classifications
of Peixoto-Peixoto, Sotomayor and ours.

\begin{table}[htbp]
  \centering
  \begin{tabular}{llcc}
 & Type & Peixoto-Peixoto & Sotomayor \\
\hline
$i.$ & $\partial R = \{ \alpha, \omega \}$ & I  & $1$ \\
$ii.$ & $ \partial R = \{ \alpha, \omega, \gamma_1 \}$ &   &  \\
$iii.$ & $ \partial R = \{ \alpha, \omega, \gamma_1, p_1 \}$ &   & $6$ \\
$iv.$ & $\partial R = \{ \alpha, \omega, \gamma_1, \gamma_2\}$ & II & $7$ \\
$v.$ & $ \partial R = \{ \alpha, \omega, p_1, \gamma_1,\gamma_2\}$ &  &  \\
$vi.$ & $ \partial R = \{ \alpha, \omega, p_1, \gamma_1, \gamma_2, \gamma_3\}$
&
III & $4, 5$ \\
$vii.$ & $ \partial R = \{ \alpha, \omega, p_1, \gamma_1, \gamma_2, \gamma_3
\}$ & IV & $3$ \\
$viii.$ & $ \partial R = \{ \alpha, \omega, p_1, \gamma_1, \gamma_2, p_2,
\gamma_3 \}$ &  & \\
$ix. $ & $\partial R = \{ \alpha, \omega, p_1, \gamma_1, \gamma_2, p_2,
\gamma_3, \gamma_4 \}$ & V & $2$
 \end{tabular}
 \caption{Classification of canonical regions with the notation  given by
   Peixoto-Peixoto
 ($I$-$V$), by Sotomayor ($1$-$7$) and by us ($i$-$ix$).}
  \label{tab:regioescanonicas}
\end{table}


Now we define a homeomorphism between same type canonical regions  of
two vector fields of $\Sigma^n$ which are sufficiently close to each
other. We present details of homeomorphism between two canonical regions $R$
and $\widetilde{R}$ of the type $v$. For that we consider  the function
$Z_{AB}: AB \to [0,1]$ defined by $Z_{AB}(m)=   \frac{l(Am)}{l(AB)}$
where $l(AB)$ is the length of arc $AB$ that  links the   points  $A$ and $B$,
and  $m \in AB$ (see figure \ref{fig:region-v}).

\begin{figure}[htbp]
  \begin{center}
   \psfrag{a}[h][B]{$\alpha$}
   \psfrag{w}[h][B]{$\omega$}
   \psfrag{a1}[h][B]{$\widetilde{\alpha}$}
   \psfrag{w1}[h][B]{$\widetilde{\omega}$}
   \psfrag{g1}[h][B]{$\gamma_1$}
   \psfrag{g2}[h][B]{$\gamma_2$}
   \psfrag{g11}[h][B]{$\widetilde{\gamma}_1$}
   \psfrag{g21}[h][B]{$\widetilde{\gamma}_2$}
   \psfrag{dM}[h][B]{$\partial \widetilde{M}$}
   \psfrag{p1}[h][B]{$p_1$}
   \psfrag{p11}[h][B]{$\widetilde{p}_1$}
   \psfrag{R}[h][B]{$R$}
   \psfrag{R1}[h][B]{$\widetilde{R}$}
   \psfrag{A0}[h][B]{$A_0$}
   \psfrag{A01}[h][B]{$\widetilde{A}_0$}
   \psfrag{B0}[h][B]{$B_0$}
   \psfrag{B01}[h][B]{$\widetilde{B}_0$}
   \includegraphics[height=2.0in,width=3in]{fig10.eps} %{region-v.eps}
    \caption{Canonical regions of the type $v$.}
    \label{fig:region-v}
  \end{center}
\end{figure}


Let $D$ (respectively $\widetilde{D}$) be a critical region of
$\alpha \in R$ (resp. $\widetilde{\alpha} \in \widetilde{R}$)  and let $B_0 =
\gamma_1 \cap \partial D$ (resp. $\widetilde{B}_0 = \widetilde{\gamma}_1 \cap
\partial \widetilde{D}$) and $A_0 = \gamma_2 \cap \omega$
(resp. $\widetilde{A}_0 = \widetilde{\gamma}_2 \cap
\widetilde{\omega}$). Hence, the boundary of $R$-$D^{\circ}$ (respectively
$\widetilde{R}$-$\widetilde{D}^{\circ}$) is formed by the
singularity $p_1$ (resp. $\widetilde{p_1}$), the arcs of
trajectory $p_1A_0$ (resp. $\widetilde{p_1}\widetilde{A}_0$) and
$B_0p_1$ (resp. $\widetilde{B}_0\widetilde{p_1}$) and the arcs
$B_0B_0$ (i.e. $\partial D$) and $A_0p_1$ (resp.
$\widetilde{B}_0\widetilde{B}_0$ and
$\widetilde{A}_0\widetilde{p_1}$).

Each arc of $\partial (R$-$D^{\circ})$ is mapped to the
corresponding one of $\partial
(\widetilde{R}$-$\widetilde{D}^{\circ})$ by
$$ \varphi(A) = \begin{cases}
   Z^{-1}_{\widetilde{A}_0 \widetilde{p_1}} \circ Z_{A_0p_1}(A) & \text{if } A
   \in A_0p_1 \\
 Z^{-1}_{\widetilde{p_1} \widetilde{A}_0} \circ Z_{p_1A_0}(A) & \text{if } A
   \in p_1A_0 \\
 Z^{-1}_{\widetilde{B}_0 \widetilde{p_1}} \circ Z_{B_0p_1}(A) & \text{if } A
   \in B_0p_1 \\
  \end{cases}
$$
We extend $\varphi$ to interior of $R$-$D^{\circ}$ as follows. For
$P \in (R$-$D^{\circ})^{\circ}$, there is  a trajectory $BA$
through $P$ where $B \in B_0B_0$ and $A \in A_0p_1$. Hence,
$$ \varphi(P) = Z^{-1}_{\widetilde{B} \widetilde{A}} \circ Z_{BA}(P).$$

The function $\varphi: R$-$D^{\circ} \to
\widetilde{R}$-$\widetilde{D}^{\circ}$ is bijective.
 $\varphi^{-1}$ is defined in the same way. The continuity of $\varphi$ follows
 from  that of  solutions with respect to the initial values.

On the other hand, the homeomorphism of the critical region $D$, say
$\psi$, also is built  using the function $Z_{AB}$. See
\cite{peixoto-peixoto, Sotomayor2} for more details. We observe
that $\varphi$ and $\psi$ coincide in $\partial D$, since both are
defined in the same way. This finishes the building of the
homeomorphism.


\section{Concluding remarks}\label{sec:discussion}

A study of the structural stability of second order differential
equation $E_{f}$ on $M = S^{1}\times \Re$ for $C^1$ functions
$f(x,y)$ periodic in $x$ and with the  Whitney topology was carried out by
Barreto in
\cite{barreto} (see also Shahshahani \cite{Shahshahani}). Below we
review the conditions  for the structurally
stable equations $E_f$ proposed by Barreto:

\begin{enumerate}
\item All singularities are hyperbolic and, therefore, finite in number

\item If a  trajectory $\lambda$  has a saddle as its
$\alpha$-limit (resp. $\omega$-limit), then every trajectory in
some tubular neighborhood of $\lambda$ has  the same
$\omega$-limit (resp. $\alpha$-limit). In particular, no
trajectory joins saddle points.

\item All periodic orbits are hyperbolic and, therefore,  countable in number.
Only a finite number of these intersect $C_0$.

\item  The $\alpha$ and  $\omega$-limit set of any trajectory can be
only singularities, periodic orbits or infinite.
\end{enumerate}


Among  the conditions above, $(2)$ is the most difficult, if not
impossible, to verify in concrete cases. It corresponds to the
asymptotic behavior of trajectory near infinity. In this context,
Camacho \emph{ et  al.} in \cite{CKM-Nitecki} and Kotus \emph{ et
al.} in \cite{KK-Nitecki} and \cite{Kotus} formulated analogous
conditions of those by  Peixoto for  $C^r$ vector fields on open
surfaces with finite genus  and a countable space of ends. They
distinguish a behavior at infinity of the type ``saddle at
infinity'' formed by two unbounded semi-trajectories with
prolongational limit sets contained in the space of ends.


On compact regions of $\Re^2$ and $M$, in \cite{sotomayor},
Sotomayor  established  a characterization theorem for
$C^{1}$-structurally stable second order differential equations,
using the uniform topology in the space of equations and the
tangency conditions between  the orbits and the boundary of the
regions.

The purpose of our work is to establish a link between
\cite{barreto} and \cite{sotomayor}, obtaining conditions that
allow verification in a simple calculatory fashion, when
restricted to the class $\mathcal{E}^{n,r}$. This is done by
applying  a compactification to the open surface $M$ and to the
equation $E_f$, as explained in section
\ref{sec:compactification}. This compactification allows us to obtain a large
class of stable equations $E_f$ that have behaviors at infinity
not exhibited by the conditions studied in \cite{barreto}, and also to obtain
some  patterns of behavior at infinity described  in terms of
the  tangency conditions between  the orbits and the boundary of
the compactification of $M$, as  given in \cite{sotomayor}.

Other compactifications present discouraging results. For example, by applying
a two-point compactification, we would be able   to induce a field with two
singularities at the infinite. In order to know the  topological type of those
singularities, we would need to apply a blow-up \cite{dumortier}. We would
expect  the same topological types  already calculated in section
\ref{sec:behavior}.

Another example is the known Poincar\'e compactification that is
applied successfully  when the vector field on $\Re^2$
\begin{enumerate}
\item  is  polynomial in the variables $x$ and $y$ (see  \cite{Sotomayor2},
  \cite{gonzales},  \cite{dumortier1}, \cite{DH}, \cite{Llibre-Perez}, among
  other works); or

\item has the Lojasiewicz property at infinity (see \cite{teruel}).
\end{enumerate}


The vector field $X(f)$ is not completely polynomial. However, it  can
  be proven that $X(f)$ has the property of Lojasiewicz at infinity. Then,
  $M$ and $X(f)$ can be compactified by the Poincar\'e  compactification as a
  compact cylinder, and a vector field with two lines of singularities on the
  border of the cylinder.

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\end{document}