\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 118, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/118\hfil Weight-homogeneous cubic polynomial centers]
{Limit cycles bifurcating from the periodic orbits of the
weight-homogeneous polynomial centers of weight-degree 3}

\author[J. Llibre, B. D. Lopes, J. R. de Moraes \hfil EJDE-2018/118\hfilneg]
{Jaume LLibre, Bruno D. Lopes, Jaime R. de Moraes}

\address{Jaume LLibre \newline
Departament de Matem\`atiques,
Universitat Aut\`onoma de Barcelona,
08193 Bellaterra, Barcelona,
Catalonia, Spain}
\email{jllibre@mat.uab.cat}

\address{Bruno D. Lopes \newline
IMECC--UNICAMP, CEP 13081-970, Campinas,
S\~ao Paulo, Brazil}
\email{brunodomicianolopes@gmail.com}

\address{Jaime R. de Moraes \newline
Curso de Matem\'atica - UEMS,
Rodovia Dourados-Itaum Km 12,
CEP 79804-970 Dourados, Mato Grosso do Sul,
Brazil}
\email{jaime@uems.br}

\thanks{Submitted November 16, 2016. Published May 17, 2018.}
\subjclass[2010]{34C07, 34C23, 34C25, 34C29, 37C10, 37C27, 37G15}
\keywords{Polynomial vector field; limit cycle; averaging method;
\hfill\break\indent weight-homogeneous differential system}

\begin{abstract}
 In this article we obtain two explicit polynomials, whose simple
 positive real roots provide the limit cycles which bifurcate from
 the periodic orbits of a family of polynomial differential centers
 of order 5, when this family is perturbed inside the class of all
 polynomial differential systems of order 5, whose average function
 of first order is not zero. Then the maximum number of limit cycles
 that bifurcate from these periodic orbits is 6 and it is reached.

 This family of  of centers  completes the study of the limit
 cycles which can bifurcate from periodic orbits of all centers
 of the weight-homogeneous polynomial differential systems of
 weight-degree 3 when perturbed in the class of all polynomial
 differential systems having the same degree and whose average
 function of first order is not zero.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction and statement of the main results}

One of the main goals in the qualitative theory of real planar
polynomial differential systems is the determination of their limit
cycles. Studying the number of limit cycles of a
polynomial differential system is strongly motivated by
Hilbert's 16-th problem (1900). For more details see \cite{HI} and
\cite{SS}.

Many authors have studied the number of limit cycles which may
bifurcate from the periodic orbits of a center of a polynomial
differential system when it is perturbed up to first order in the
parameter of the perturbation. This problem is known as the
\emph{weak Hilbert's problem}. See for example
\cite{ALGM-biurcation, BloPer94}.

Among the many tools for studying the maximum number of limit
cycles that may bifurcate from the periodic annulus of a center
we have  the Poincar\'e return map, the Poincar\'e-Melnikov
integrals, the Abelian integrals, and the averaging theory. The last
three methods are equivalent at first order, see for instance
\cite{HRZ}. For studies on the  weak Hilbert's problem see, for
example, the second part of  \cite{Chr} and the hundreds of
references quoted therein.

Here we consider the \emph{polynomial differential systems} 
\begin{equation}\label{sisweight}
\begin{gathered}
\dot{x}=P(x,y),\\
\dot{y}=Q(x,y),
\end{gathered}
\end{equation}
where $P$ and $Q$ are polynomials with real coefficients. The
\emph{degree} of the system is the maximum of the degrees of the
polynomials $P$ and $Q$ .

We say that system \eqref{sisweight} is \emph{weight-homogeneous}
if there exist $(s_1,s_2)\in\mathbb{N}^2$ and $d\in\mathbb{N}$ such
that for any $\lambda\in\mathbb{R}^+ =\{\lambda\in\mathbb{R}:\lambda>0\}$ we have
\[P(\lambda^{s_1}x,\lambda^{s_2}y)=\lambda^{s_1-1+d}P(x,y),\quad
Q(\lambda^{s_1}x,\lambda^{s_2}y)=\lambda^{s_2-1+d}Q(x,y).\] The
vector $(s_1,s_2)$ is called the \emph{weight-exponent} of system
\eqref{sisweight} and $d$ is called \emph{weight-degree} with
respect to the weight-exponent $(s_1,s_2)$.

Our main goal is to solve the weak Hilbert's problem for the weight-homogeneous
polynomial differential systems of weight-degree 3.

The classification of all centers of a planar
\emph{weight-homogeneous} polynomial differential systems
up to weight-degree 4 is found in \cite{Pessoa}. In \cite{Pessoa}
two families of weight-homogeneous polynomial differential
systems having centers with weight-degree 3 are provided. The first family can
be written as
\begin{equation}\label{sist1}
\begin{gathered}
\dot{x}=ax^3+(b-3\alpha\mu)x^2y-axy^2-\alpha y^3,\\
\dot{y}=\alpha x^3+ax^2y+(b+3\alpha\mu)xy^2-ay^3,
\end{gathered}
\end{equation}
with $\alpha\in\{-1,1\}$, $a,b,\mu\in\mathbb{R}$ and $\mu>-1/3$, after doing
an affine change of variables and a rescaling of the time. The
weight-exponent of this family is $(s_1,s_2) = (1,1)$.

The second family is 
\begin{equation}\label{sis1}
\begin{gathered}
\dot{x}=ax^3+by=P(x,y),\\
\dot{y}=cx^5+dx^2y=Q(x,y),
\end{gathered}
\end{equation}
with $bc\neq0$, $3a+d=0$ and $12(bc-ad)<0$. The weight-exponent of
this family is $(s_1,s_2) = (1,3)$.


In \cite{BJ_Elec} we provide a polynomial whose real positive simple
zeros give exactly the number of limit cycles that bifurcate from
the periodic orbits of  \eqref{sist1} when perturbed
in the class of all polynomial differential systems of degree
$n$ and nonzero  first order average function is. In particular
the maximum number of limit cycles obtained is exactly $[(n-1)/2]$,
where $[x]$ denotes the integer part of $x$.

In this article we give two explicit polynomials whose real positive
simple zeros provide the number of limit cycles which bifurcate from
the periodic orbits of the center of the weight-homogeneous
polynomial differential system \eqref{sis1}, when the first order
average function is non-zero. The maximum number of limit cycles is
reached when the parameter $d\neq0$.

More precisely here we consider the polynomial differential system
\begin{equation}
\begin{gathered}
\dot{x}=-\frac{d}{3}x^3+by+ \varepsilon p(x,y),\vspace{0.2cm}\\
\dot{y}=cx^5+dx^2y+ \varepsilon q(x,y),
\end{gathered}\label{sis2}
\end{equation}
where
\begin{gather}\label{eq1}
p(x,y)=\sum_{k=0}^{5}p_k(x,y), \quad q(x,y)=\sum_{i=0}^{5}q_k(x,y), \\
p_k(x,y)=\sum_{i=0}^{k}c_{k-i\,i}\,\,\,x^{k-i}y^i,\quad
q_k(x,y)=\sum_{i=0}^{k}d_{k-i\,i}\,\,\,x^{k-i}y^i, \nonumber
\end{gather}
and $\varepsilon$ is a small parameter.

In \cite{Li1} the authors found an upper bound for the maximum
number of limit cycles of system \eqref{sis2}. Looking at statement
(c) of Theorem A of \cite{Li1} with $n=5$, $p=3$ and $q=1$ an upper
bound for the number of limit cycles of system \eqref{sis2} with
$\varepsilon$ sufficiently small coming from the periodic orbits of the
center \eqref{sis1} is 8. \emph{Here we prove that the maximum
number of limit cycles that system \eqref{sis2} can have is exactly
6 and it is reached.} See Corollary \ref{cor1}. See \cite[Theorem 2]{LTT}
for another  study on  the limit cycles that may  bifurcate from the periodic
orbits surrounding a non-Hamiltonian center using averaging theory.

In what follows we state our main results where the functions
$f_i(\theta)$, for $i=0,1,\dots,9$, $W(\theta)$, $g_1(\theta)$ and
$k(\theta)$ are given in Section \ref{sec3}  and the functions
$f_i^{*}(\theta)$, for $i=0,1,\dots,9$, $W^{*}(\theta)$,
$g_1^{*}(\theta)$ and $k^{*}(\theta)$ are given in Section
\ref{sec4}. We do not provide these functions here due to their
length.

\begin{theorem}\label{teo1}
Suppose that $d\neq0$. Let $ r_{0,s}$ be a positive simple root of
the polynomial
\begin{equation}\label{eqS01}
r_0\mathcal{F}(r_0)
=\frac{1}{2\pi}\sum_{k=1}^{7}r_0^{2k-2}
\int_0^{2\pi}A_{2k+1}(\theta)d\theta,
\end{equation}
where
$$
A_i(\theta)=\frac{W(\theta)f_i(\theta) k(\theta)^{i-5}}{g_1(\theta)^2}.
$$
Then for $|\varepsilon|>0$ sufficiently small parameter the perturbed
systems \eqref{sis2} have a limit cycle bifurcating from the
periodic orbit $r(\theta, r_{0,s})=k(\theta)r_{0,s}$ of the period
annulus of the center \eqref{sis1} if the first order average
function is non-zero. In particular, the polynomial \eqref{eqS01}
has at most $6$ positive simple real roots and they are reached.
\end{theorem}

\begin{theorem}\label{teor2}
Suppose that $d=0$. Let $ r_{0,s}$ be a positive simple root of the
polynomial
\begin{equation}\label{eqS02}
r_0\mathcal{F}^*(r_0)=
\frac{1}{2\pi}\Big(\sum_{k=1}^{5}r_0^{2k-2}
\int_0^{2\pi}A_{2k+1}^*(\theta)d\theta+r_0^{11}\int_0^{2\pi}
A_{15}^*(\theta)d\theta\Big),
\end{equation}
where
$$
A_i^*(\theta)=\frac{W^*(\theta)f_i^*(\theta)
k^*(\theta)^{i-5}}{g_1^*(\theta)^2}.
$$
Then for $|\varepsilon|>0$ sufficiently small the perturbed systems
\eqref{sis2} have a limit cycle bifurcating from the periodic orbit
$r(\theta, r_{0,s})=k^{*}(\theta)r_{0,s}$ of the period annulus of
the center \eqref{sis1} if the first order average function is
non-zero. In particular, the polynomial \eqref{eqS02} has at most
$5$ positive simple real roots and they are reached.
\end{theorem}

Theorem \ref{teo1} is proved in Section \ref{sec3} and Theorem
\ref{teor2} is proved in Section \ref{sec4}. In Section \ref{sec5}
we provide two examples that illustrate Theorems \ref{teo1} and
\ref{teor2} with the maximum number of limit cycles.

From Theorems \ref{teo1} and \ref{teor2} the next result follows.

\begin{corollary}\label{cor1}
Applying the averaging theory of first order to the perturbed system
\eqref{sis2} when  $|\varepsilon|$ is sufficiently small we can obtain at most
$6$ limit cycles bifurcating from the periodic orbits of the center
\eqref{sis1}, and we have systems where these $6$ limit cycles are reached.
\end{corollary}


\section{Preliminaries}\label{sec2}

In this section we give some well known results that we shall need
for proving Theorem \ref{teo1}.
Consider the system 
\begin{equation}\label{eq2}
\dot{\mathrm{x}}=F_0(t,\mathrm{x})+\varepsilon F_1(t,\mathrm{x})+\mathcal{O}(\varepsilon^2),
\end{equation}
where $\varepsilon\neq0$ is sufficiently small and the functions
$F_0,F_1:\mathbb{R}\times\Omega\to \mathbb{R}^n$ and $F_2:\mathbb{R}\times
\Omega\times (-\varepsilon_0,\varepsilon_0)\to \mathbb{R}^n$ are $\mathcal{C}^2$
functions, $T-$periodic in the first variable and $\Omega$ is an
open subset of $\mathbb{R}^n$. We suppose that the unperturbed system
\begin{equation}\label{eq3}
\dot{\mathrm{x}}=F_0(t,\mathrm{x})
\end{equation}
has a submanifold of periodic solutions of dimension $n$.

Let $x(t,z,\varepsilon)$ be the solution of system \eqref{eq3} such that
$\mathrm{x}(0,\mathrm{z},\varepsilon)=z$. The linearization of the unperturbed
system along a periodic solution $\mathrm{x}(t,\mathrm{z},0)$ is
\begin{equation}\label{eq4}
\dot{\mathrm{y}}=D_{\mathrm{x}}F_0(t,\mathrm{x}(t,\mathrm{z},0))\mathrm{y}.
\end{equation}

In what follows we denote by $M_{\mathrm{z}}(t)$ the fundamental matrix
solution
of the linearized system \eqref{eq4} such that $M_{\mathrm{z}}(0)$ is
the identity matrix.

We assume that there is an open set $U$ with
$\mathrm{Cl}(U)\subset\Omega$ such that for each
$z\in\mathrm{Cl}(U)$, $\mathrm{x}(t,\mathrm{z},0)$ is $T-$periodic, where
$\mathrm{x}(t,z,0)$ denotes the solution of the unperturbed system
\eqref{eq3}, and   $\mathrm{Cl}(U)$ the closure of $U$. The
set $\mathrm{Cl}(U)$ is \emph{isochronous} for system \eqref{eq3},
i.e. it is formed only by periodic orbits with period $T$.

The following result is the a version of averaging theorem for
studying the bifurcation of $T-$periodic solutions of system
\eqref{eq2} from the periodic solutions $\mathrm{x}(t,\mathrm{z},0)$
contained in $\mathrm{Cl}(U)$ of system \eqref{eq3} when $|\varepsilon|>0$
is sufficiently small. See \cite{Bui1} for a proof. For more details
on the averaging theory see \cite{BuiLli} and
\cite{SanVer}.

\begin{theorem}[Perturbations of an isochronous set]\label{teo2}
We assume that there exists an open and bounded set $U$ with
$\mathrm{Cl}(U)\subset\Omega$ such that for each $\mathrm{z}\in
\mathrm{Cl}(U)$, the solution $\mathrm{x}(r,\mathrm{z},0)$ is $T-$periodic.
Consider the function $\mathcal{F}:\mathrm{Cl}(U)\to \mathbb{R}^n$
\begin{equation}\label{eq5}
\mathcal{F}(\mathrm{z})=\frac{1}{T}\int_0^T M_{\mathrm{z}}^{-1}(t)
F_1(t,\mathrm{x}(t,\mathrm{z},0))dt.
\end{equation}
Then if there exists $\mathbf{a}\in U$ with
$\mathcal{F}(\mathbf{a})=0$ and $\det((\partial \mathcal{F}/\partial
\mathrm{z})(\mathbf{a}))\neq0$ then there exists a $T-$periodic solution
$\mathrm{x}(t,\varepsilon)$ of system \eqref{eq2} such that
$\mathrm{x}(0,\varepsilon)\to \mathbf{a}$ when $\varepsilon\to 0$.
\end{theorem}

In fact, if $\mathrm{x}(t,\mathrm{z},\varepsilon)$ denotes the solution of the
differential system \eqref{eq2} such that $\mathrm{x}(0,\mathrm{z},\varepsilon)=
\mathrm{z}$, then the average function satisfies that
$\mathrm{x}(T,\mathrm{z},\varepsilon)-\mathrm{z}= \varepsilon \mathcal{F}(\mathrm{z})+O(\varepsilon^2)$, see
for more details \cite{Bui1, HRZ}. Then, by the Implicit Function
Theorem it follows that if $\mathcal{F}(\mathrm{z})\ne 0$, then the
simple zeros of the function $\mathcal{F}(\mathrm{z})$ provide limit
cycles of the differential system \eqref{eq2}.

The following result is the generalized Descartes Theorem about the
number of zeros of a real polynomial. See \cite{Descartes} for a
proof.

\begin{theorem}\label{descartes}
Consider the real polynomial
$p(x)=a_{i_1}x^{i_1}+a_{i_2}x^{i_2}+\dots+a_{i_r}x^{i_r}$ with $0\leq
i_1<i_2<\dots<i_r$ and $a_{i_j}\neq0$ real constants for
$j\in\{1,2,\dots,r\}$. When $a_{i_j}a_{i_{j+1}}<0$, we say that
$a_{i_j}$ and $a_{i_{j+1}}$ have a variation of sign. If the number
of variations of signs is $m$, then $p(x)$ has at most $m$ positive
real roots. Moreover, it is always possible to choose the
coefficients of $p(x)$ in such a way that $p(x)$ has exactly $r-1$
positive real roots.
\end{theorem}

\section{Proof of Theorem \ref{teo1}}\label{sec3}

Suppose that $d\neq0$. We apply the affine change of variables
\[
\tilde{x}=\alpha x, \quad \tilde{y}=\frac{\alpha^3 b}{d}y,\quad
\tilde{t}=\frac{d}{\alpha^2}t,
\]
with $\alpha\neq0$ and system \eqref{sis1} becomes
\begin{gather*}
\dot{x}=P(x,y)=-\frac{1}{3}x^3+y,\\
\dot{y}=Q(x,y)= a_1x^5+x^2y,
\end{gather*}
where $a_1=-(4+b^2)/12$ and $b\neq0$. In the case $b=0$ working in a
similar way we also can reach the previous differential system. The
perturbed system corresponding to the previous system is 
\begin{equation}\label{siste1}
\begin{gathered}
\dot{x}=-\frac{1}{3}x^3+y+\varepsilon p(x,y),\\
\dot{y}= a_1x^5+x^2y+\varepsilon q(x,y)\,.
\end{gathered}
\end{equation}
We write system \eqref{siste1} in the generalized polar coordinates
$x=r \cos\theta$, $y=r^3 \sin\theta$, and we obtain the differential
equation
\begin{equation}\label{eq100}
\frac{dr}{d\theta}= F_0(r,\theta)+\varepsilon F_1(r,\theta)+
\mathcal{O}(\varepsilon^2),
\end{equation}
in the standard form for applying the averaging theory of first
order described in Section \ref{sec2}, where
\begin{gather*}
F_0(r,\theta) = \frac{h_1(\theta)}{g_1(\theta)}r, \\
\begin{aligned}
F_1(r,\theta)&=\frac{144(\cos^2\theta+3\sin^2\theta)}
{r^7 g_1(\theta)^2}\Big(Q(r\cos\theta,r^3 \sin\theta)
p(r\cos\theta,r^3 \sin\theta) \\
&\quad - P(r\cos\theta,r^3 \sin\theta)q(r\cos\theta,r^3 \sin\theta)\Big),
\end{aligned}\\
h_1({\theta})=\cos \theta \left(\left(b^2+4\right) \sin \theta
\cos ^4\theta-6 \sin \theta (\sin (2 \theta )+2)+4 \cos
^3\theta\right),\\
g_1(\theta)=(4+b^2)\cos^6\theta-24\cos^3\theta\sin\theta+36\sin^2\theta.
\end{gather*}

Note that the differential equation \eqref{eq100} satisfies the
assumptions of Theorem \ref{teo2}. Consider $r(\theta,r_0)$ the
periodic solution of the differential equation
\begin{equation*}\label{rt}
\frac{dr}{d\theta}=  r \,\frac{h_1(\theta)}{g_1(\theta)},
\end{equation*}
such that $r(0,r_0)= r_0$.
By solving the previous differential equation we get
\[
r(\theta,r_0)= k(\theta)r_0,
\]
where
\[
k(\theta)=\frac{2^{5/6}(4+b^2)^{1/6}}{B(\theta)},
\]
with
\begin{align*}
B(\theta)&=\big(3 \left(5 b^2-172\right) \cos (2 \theta )+6
\left(b^2+4\right) \cos (4 \theta )+\left(b^2+4\right) \cos (6
\theta )+10 b^2 \\
&\quad -192 \sin (2 \theta )-96 \sin (4 \theta )+616  \big)^{1/6}.
\end{align*}

Solving the variational equation \eqref{eq4} for the differential
equation \eqref{eq100} we see that the fundamental matrix solution
$M(\theta)$ is $k(\theta)$. Using the polynomials $p$ and $q$
given in \eqref{eq1} and system \eqref{sis1} we have that the
integrant of the integral \eqref{eq5} for the differential equation
\eqref{eq100} is
\begin{align*}
M^{-1}(\theta)F_1(\theta ,r(\theta,r_0))
&= \sum_{i=0}^{17}\frac{W(\theta)f_i(\theta)}{ g_1(\theta)^2M(\theta)}
r(\theta,r_0)^{i-4} \\
&=\sum_{i=0}^{17} r_0^{i-4} \frac{W(\theta)f_i(\theta)
k(\theta)^{i-5}}{ g_1(\theta)^2}\\
&= \sum_{i=0}^{17} r_0^{i-4}A_i(\theta),
\end{align*}
where
\begin{align*}
f_0(\theta)&= 4 d_{00} \cos ^3\theta-12 d_{00} \sin \theta ,\\
f_1(\theta)&= 4 d_{10} \cos ^4\theta-12 d_{10} \sin \theta \cos \theta ,\\
f_2(\theta)&= \cos ^5\theta \left(-b^2 c_{00}-4 c_{00}+4
   d_{20}\right)+12 (c_{00}-d_{20}) \sin \theta \cos
   ^2\theta ,\\
f_3(\theta)&= \cos ^6\theta \left(-b^2c_{10}-4 c_{10}+4
   d_{30}\right)+4 \sin \theta \cos ^3\theta (3
   c_{10}+d_{01}-3 d_{30})\\
&\quad-12 d_{01} \sin ^2\theta ,\\
f_4(\theta)&= \cos ^7\theta \left(-b^2c_{20}-4 c_{20}+4
   d_{40}\right)+4 \sin \theta \cos ^4\theta (3
   c_{20}+d_{11}-3 d_{40})\\
&\quad -12 d_{11} \sin ^2\theta
   \cos \theta,\\
f_5(\theta)&= \sin \theta \cos ^5\theta \left(-b^2c_{01}-4 c_{01}+12
   c_{30}+4 d_{21}-12 d_{50}\right)\\
&\quad +\cos ^8\theta
   \left(-b^2c_{30}-4 c_{30}+4 d_{50}\right)+12
   (c_{01}-d_{21}) \sin ^2\theta \cos ^2\theta,\\
f_6(\theta)&= \sin \theta \cos ^6\theta \left(-b^2c_{11}-4 c_{11}+12
   c_{40}+4 d_{31}\right)-\left(b^2+4\right) c_{40} \cos
   ^9\theta\\
&\quad +4 \sin ^2\theta \cos ^3\theta (3
   c_{11}+d_{02}-3 d_{31})-12 d_{02} \sin ^3\theta ,\\
f_7(\theta)&= \sin \theta \cos ^7\theta \left(-b^2c_{21}-4 c_{21}+12
   c_{50}+4 d_{41}\right)-\left(b^2+4\right) c_{50} \cos
   ^{10}\theta\\
&\quad +4 \sin ^2\theta \cos ^4\theta (3
   c_{21}+d_{12}-3 d_{41})-12 d_{12} \sin ^3\theta
   \cos \theta ,\\
f_8(\theta)&= \sin ^2\theta \cos ^5\theta \left(-b^2c_{02}-4
   c_{02}+12 c_{31}+4 d_{22}\right)\\
&\quad -\left(b^2+4\right)
   c_{31} \sin \theta \cos ^8\theta+12
   (c_{02}-d_{22}) \sin ^3\theta \cos ^2\theta ,\\
 f_9(\theta)&= \sin ^2\theta \cos ^6\theta \left(-b^2c_{12}-4
   c_{12}+12 c_{41}+4 d_{32}\right)\\
&\quad -\left(b^2+4\right)
   c_{41} \sin \theta \cos ^9\theta+4 \sin ^3\theta \cos
   ^3\theta (3 c_{12}+d_{03}-3 d_{32})\\
&\quad -12 d_{03}
   \sin ^4\theta ,\\
f_{10}(\theta)&= -\left(b^2+4\right) c_{22} \sin ^2\theta \cos ^7\theta+4 (3
   c_{22}+d_{13}) \sin ^3\theta \cos ^4\theta\\
&\quad -12   d_{13} \sin ^4\theta \cos \theta ,\\
f_{11}(\theta)&= \sin ^3\theta \cos ^5\theta \left(-b^2c_{03}-4
   c_{03}+12 c_{32}+4 d_{23}\right)\\
&\quad -\left(b^2+4\right)
   c_{32} \sin ^2\theta \cos ^8\theta+12
   (c_{03}-d_{23}) \sin ^4\theta \cos ^2\theta ,\\
f_{12}(\theta)&= -\left(b^2+4\right) c_{13} \sin ^3\theta \cos ^6\theta+4 (3
   c_{13}+d_{04}) \sin ^4\theta \cos ^3\theta\\
&\quad -12    d_{04} \sin ^5\theta ,\\
f_{13}(\theta)&= -\left(b^2+4\right) c_{23} \sin ^3\theta \cos ^7\theta+4 (3
   c_{23}+d_{14}) \sin ^4\theta \cos ^4\theta\\
&\quad -12   d_{14} \sin ^5\theta \cos \theta ,\\
f_{14}(\theta)&= 12 c_{04} \sin ^5\theta \cos ^2\theta-\left(b^2+4\right)
   c_{04} \sin ^4\theta \cos ^5\theta ,\\
f_{15}(\theta)&= -\left(b^2+4\right) c_{14} \sin ^4\theta \cos ^6\theta+4 (3
   c_{14}+d_{05}) \sin ^5\theta \cos ^3\theta \\
&\quad -12   d_{05} \sin ^6\theta ,\\
f_{16}(\theta)&= 0 ,\\
f_{17}(\theta)&= 12 c_{05} \sin ^6\theta \cos ^2\theta-\left(b^2+4\right)
   c_{05} \sin ^5\theta \cos ^5\theta ,\\
W(\theta)&= 12 \left(3 \sin ^2\theta +\cos ^2\theta \right).
\end{align*}

Computing  integral \eqref{eq5} we obtain
\[
\mathcal{F}(r_0)= \frac{1}{2\pi}\int_0^{2\pi} M^{-1}
(\theta)F_1(\theta ,r(\theta,r_0)) d\theta =   \frac{1}{2\pi}
\sum_{i=0}^{17}r_0^{i-4}\int_0^{2\pi}A_i(\theta)d\theta,
\]
where the function $A_i(\theta)$ is defined in the statement of
Theorem \ref{teo1}.


If $i$ is even then it is easy to check that $f_i(\theta )=
-f_i(\theta+\pi)$, for $i=0,\dots,17,$ and
$\theta\in[\pi,3\pi/2]\cup[3\pi/2,\pi]$. Since that
$k(\theta)=k(\theta+\pi)$, $g_1(\theta)=g_1(\theta+\pi)$ and
$W(\theta)=W(\theta+\pi)$, for
$\theta\in[\pi,3\pi/2]\cup[3\pi/2,\pi]$ we can easily show that
\begin{align*}
\int_\pi^{\frac{3\pi}{2}} A_i(\theta)d\theta 
&= \int_\pi^{\frac{3\pi}{2}} \frac{f_i(\theta)W(\theta) k(\theta)^{i-5}}
 {g_1(\theta)^2 }d\theta\\
&= \int_0^{\frac{\pi}{2}} \frac{f_i(\theta+\pi)W(\theta+\pi) k(\theta+\pi)^{i-5}}
 {g_1(\theta+\pi)^2 }d\theta\\
&= \int_0^{\frac{\pi}{2}} -\frac{f_i(\theta)W(\theta) k(\theta)^{i-5}}
 {g_1(\theta)^2 }d\theta\\
&=  - \int_0^{\frac{\pi}{2}} A_i(\theta)d\theta,\\
\end{align*}
\begin{align*}
\int_{\frac{3\pi}{2}}^{2\pi} A_i(\theta)d\theta 
&= \int_{\frac{3\pi}{2}}^{2\pi} \frac{f_i(\theta)W(\theta) k(\theta)^{i-4}}
 {g_1(\theta)^2 M(\theta)}d\theta\\
&= \int_{\frac{\pi}{2}}^{\pi} \frac{f_i(\theta+\pi) W(\theta+\pi)k(\theta+\pi)^{i-4}}
 {g_1(\theta+\pi)^2 M(\theta+\pi)}d\theta\\
&= \int_{\frac{\pi}{2}}^{\pi} - \frac{f_i(\theta)W(\theta) k(\theta)^{i-4}}
 {g_1(\theta)^2 M(\theta)}d\theta\\
&=  - \int_{\frac{\pi}{2}}^{\pi} A_i(\theta)d\theta.
\end{align*}
Thus if $i$ is even we conclude that
\[
\int_0^{2\pi} A_i(\theta)d\theta=0.
\]

The coefficients $A_1$ and $A_{17}$ are
\begin{gather*}
A_1= [768\ 2^{2/3} d_{10} \cos \theta  (\cos (2 \theta )-2) (-12 \sin
   \theta +3 \cos \theta +\cos (3 \theta ))]/L,\\
A_{17}= [3\times2^{22} \left(b^2+4\right)^2 c_{05} \sin ^5\theta \cos
   ^2\theta  (\cos (2 \theta )-2) M]/N,
\end{gather*}
where
\begin{align*}
L&= \left(b^2+4\right)^{2/3} \Big(3 \left(5 b^2-172\right) \cos (2 \theta
   )+6 \left(b^2+4\right) \cos (4 \theta )\\
&\quad +\left(b^2+4\right) \cos (6
   \theta )+10 b^2-192 \sin (2 \theta )-96 \sin (4 \theta
   )+616\Big)^{4/3},\\
M&= \left(b^2+4\right) \cos ^3\theta -12 \sin \theta,\\
N&= \Big(3 \left(5 b^2-172\right) \cos (2 \theta )+6 \left(b^2+4\right)
   \cos (4 \theta )+\left(b^2+4\right) \cos (6 \theta )\\
&\quad +10 b^2-192 \sin
   (2 \theta )-96 \sin (4 \theta )+616\Big)^4.
\end{align*}

Computing the integrals of the coefficients $A_1$ and $A_{17}$ in the variable 
$\theta$, in the interval $[0,2\pi]$ we obtain that both are zero.



\noindent\text\bf{Claim:}  For $i=3,5,7,9,11,13$ or $15$ we can choose
the parameters that appear in $A_i$ such that 
$\int_0^{2\pi} A_i(\theta)d\theta\neq0$.

The proof of this claim follows from Example \ref{ex1}.
In summary the function $\mathcal{F}$ defined in \eqref{eq5} can be
written as
\begin{equation}\label{eqqS0}
\mathcal{F}(r_0)= \frac{1}{2\pi}\sum_{k=1}^{7}r_0^{2k-3}
\int_0^{2\pi}A_{2k+1}(\theta)d\theta.
\end{equation}

Note that the coefficients $A_{2k+1}(\theta )$ in \eqref{eqqS0} are
linearly independent for $k=1,..,7$. Thus by the generalized
Descartes Theorem, the average function $\mathcal{F}$ has at most
$6$ positive simple zeros which provide limit cycles of system
\eqref{sis2}, when the average function is non-zero.


\section{Proof of Theorem \ref{teor2}}\label{sec4}

Suppose that $d=0$. We take the affine change of coordinates
\[
\tilde{x}=x , \quad \tilde{y}=y\sqrt{-b/c} , \quad
\tilde{t}=t\sqrt{-bc},
\]
and system \eqref{sis1} becomes $\dot{x}=-y$, $\dot{y}=x^5$. We
write system \eqref{sis2} in the generalized polar coordinates $x=r
\cos\theta$, $y=r^3 \sin\theta$, and we obtain the differential
equation
\begin{equation}\label{equ100}
\frac{dr}{d\theta}= F_0(r,\theta)+\varepsilon F_1(r,\theta)+
\mathcal{O}(\varepsilon^2),
\end{equation}
in the standard form for applying the averaging theory of first
order described in Section \ref{sec2}, where
\begin{align*}
F_0(r,\theta) 
&=  \frac{r \left(\sin \theta  \cos ^5\theta -\sin
   \theta  \cos \theta \right)}{\cos ^6\theta
   -3 \cos ^2\theta +3}, \\
F_1(r,\theta)
&= -\frac{\cos (2 \theta )-2}{r^4 \left(3 \sin ^2\theta
   +\cos ^6\theta \right)^2}\Big(r^2 \cos ^5\theta \;
    p\left(r \cos\theta ,r^3 \sin
   \theta \right)\\
&\quad +\sin \theta \; q\left(r
   \cos \theta ,r^3 \sin \theta
   \right)\Big).
\end{align*}
Denote by $g_1^{*}(\theta)=\cos ^6\theta -3 \cos ^2\theta +3$. Note
that the differential equation \eqref{equ100} satisfies the
assumptions of Theorem \ref{teo2}. Consider $r(\theta,r_0)$ the
periodic solution of the differential equation $ \dot{r}=
F_0(r,\theta)$ such that $r(0,r_0)= r_0$. For  solving this
differential equation we take $z=\cos^2 \theta $ in
$g_1^{*}(\theta)$, and we obtain a polynomial of degree 3 in $z$
which can be factorized in the form
\[
g_1^{*}(z)=(z-z_1)(z-z_2)(z-z_3),
\]
where the coefficients of $g_2^*(z)=(z-z_1)$ and
$g_3^*(z)=(z-z_2)(z-z_3)$ are reals, and $z_i$ are the roots of
$g_1^{*}$, for $i=1,\dots,3$ given by
\begin{gather*}
z_1=  {-\frac{2+\sqrt[3]{2}
   \left(3-\sqrt{5}\right)^{2/3}}{2^{2/3}
   \sqrt[3]{3-\sqrt{5}}}},\\
z_{2,3}=  {\frac{2 \sqrt[3]{2} \left(1\mp i
   \sqrt{3}\right)+\left(1\pm i \sqrt{3}\right)
   \left(6-2 \sqrt{5}\right)^{2/3}}{4
   \sqrt[3]{3-\sqrt{5}}}}.
\end{gather*}
Thus the differential equation \eqref{equ100} with $\varepsilon = 0$ can be
rewritten in the form
\begin{equation}\label{rt2}
\frac{dr}{d\theta}=r
\Big(C_1\frac{\cos\theta\sin\theta}{-z_3+\cos^2\theta}+C_2
\frac{\cos\theta\sin\theta}{-z_1+\cos^2\theta}+C_3\frac{\cos\theta
\sin\theta}{-z_2+\cos^2\theta}\Big),
\end{equation}
where
\[
C_1=\frac{z_3^2-1}{(z_3-z_1)(z_3-z_2)},\quad
C_2=\frac{z_1^2-1}{(z_1-z_2)(z_1-z_3)},\quad
C_3=\frac{z_2^2-1}{(z_2-z_1)(z_2-z_3)}.
\]

The solution of differential equation \eqref{rt2} with initial
condition $r(0,r_0)=r_0$ is
\[
r(\theta,r_0)= r_0k^*(\theta),
\]
where
\begin{align*}
k^*(\theta)
&= r_0 (1-z_3)^{C_1/2}
   (1-z_1)^{C_2/2}
   (1-z_2)^{C_3/2} \left(\cos ^2\theta
   -z_3\right)^{-C_1/2}\\
& \quad\times \left(\cos
   ^2\theta -z_1\right)^{-C_2/2}
   \left(\cos ^2\theta
   -z_2\right)^{-C_3/2}.
\end{align*}

Solving the variational equation \eqref{eq4} for our differential
equation \eqref{equ100} we get that the fundamental matrix is the
function $M^*(\theta)=k^*(\theta)$. Note that $M^*(\theta)$ does
not depend on $r_0$. Using the polynomials $p$ and $q$ given in
\eqref{eq1} and system \eqref{sis1} we have that the integrant of
the integral \eqref{eq5} for the differential equation
\eqref{equ100} is
\begin{align*}
{M^{*}}^{-1}(\theta)F_1(\theta ,r(\theta,r_0))
&=
\sum_{i=0}^{17}\frac{W^*(\theta)f_i^*(\theta)}{ g_1^*(\theta)^2M^*(\theta)}
r(\theta,r_0)^{i-4} \\
&= \sum_{i=0}^{17} r_0^{i-4}
\frac{W^*(\theta)f_i^*(\theta)k^*(\theta)^{i-5}}{ g_1^*(\theta)^2}\\
&= \sum_{i=0}^{17} r_0^{i-4}A_i^*(\theta),
\end{align*}
where
\begin{align*}
f_0^*(\theta)&= d_{00} \sin \theta,\\
f_1^*(\theta)&= d_{10} \sin \theta \cos \theta,\\
f_2^*(\theta)&= c_{00} \cos ^5\theta+d_{20} \sin \theta
   \cos ^2\theta,\\
f_3^*(\theta)&= c_{10} \cos ^6\theta+d_{01} \sin ^2\theta
   +d_{30} \sin \theta \cos ^3\theta,\\
f_4^*(\theta)&= c_{20} \cos ^7\theta+d_{11} \sin ^2\theta
   \cos \theta+d_{40} \sin \theta \cos
   ^4\theta,\\
f_5^*(\theta)&= (c_{01}+d_{50}) \sin \theta \cos ^5\theta
   +c_{30} \cos ^8\theta+d_{21} \sin
   ^2\theta \cos ^2\theta,\\
f_6^*(\theta)&= c_{11} \sin \theta \cos ^6\theta+c_{40}
   \cos ^9\theta+d_{02} \sin ^3\theta
   +d_{31} \sin ^2\theta \cos ^3\theta,\\
f_7^*(\theta)&= c_{21} \sin \theta \cos ^7\theta+c_{50}
   \cos ^{10}\theta+d_{12} \sin ^3\theta \cos
   \theta+d_{41} \sin ^2\theta \cos ^4\theta,
   \\
f_8^*(\theta)&= c_{02} \sin ^2\theta \cos ^5\theta+c_{31}
   \sin \theta \cos ^8\theta+d_{22} \sin
   ^3\theta \cos ^2\theta,\\
f_9^*(\theta)&= c_{12} \sin ^2\theta \cos ^6\theta+c_{41}
   \sin \theta \cos ^9\theta+d_{03} \sin
   ^4\theta+d_{32} \sin ^3\theta \cos
   ^3\theta,\\
f_{10}^*(\theta)&= c_{22} \sin ^2\theta \cos ^7\theta+d_{13}
   \sin ^4\theta \cos \theta,\\
f_{11}^*(\theta)&= c_{03} \sin ^3\theta \cos ^5\theta+c_{32}
   \sin ^2\theta \cos ^8\theta+d_{23} \sin
   ^4\theta \cos ^2\theta,\\
f_{12}^*(\theta)&= c_{13} \sin ^3\theta \cos ^6\theta+d_{04}
   \sin ^5\theta,\\
f_{13}^*(\theta)&= c_{23} \sin ^3\theta \cos ^7\theta+d_{14}
   \sin ^5\theta \cos \theta,\\
f_{14}^*(\theta)&= c_{04} \sin ^4\theta \cos ^5\theta,\\
f_{15}^*(\theta)&= c_{14} \sin ^4\theta \cos ^6\theta+d_{05}
   \sin ^6\theta,\\
f_{16}^*(\theta)&= 0,\\
f_{17}^*(\theta)&= c_{05} \sin ^5\theta \cos ^5\theta,\\
W^*(\theta)&= 3 \sin ^2\theta+\cos ^2\theta.
\end{align*}

Computing the integral \eqref{eq5} we obtain
\[
\mathcal{F}^*(r_0)= \frac{1}{2\pi}\int_0^{2\pi} (M^*)^{-1}
(\theta)F_1(\theta ,r(\theta,r_0)) d\theta =   \frac{1}{2\pi}
\sum_{i=0}^{17}r_0^{i-4}\int_0^{2\pi}A_i^*(\theta)d\theta,
\]
where the function $A_i^{*}(\theta)$ is defined in the statement of
Theorem \ref{teo2}.

Analogously as in the proof of Theorem \ref{teo1} we can
show that if $i$ is even then
$$
\int_{0}^{2\pi} A_i^*(\theta)d\theta = 0.
$$

The coefficients $A_1^*$, $A_{13}^*$ and $A_{17}^*$ are given by
\begin{align*}
A_1^*&= -d_{10}\frac{2^{\frac{55}{9}} \left(7-3
   \sqrt{5}\right)^{4/9} \sin (2 \theta
   ) (\cos (2 \theta
   )-2)}{\left(3-\sqrt{5}\right)^{8/9}
   (-33 \cos (2 \theta )+6 \cos (4
   \theta )+\cos (6 \theta )+58)^{4/3}},\\
A_{13}^*&= -\frac{2^\frac{4}{3}
   \left(2+\sqrt{5}\right)^{4/9} \sin
   ^3\theta  \cos \theta  (\cos (2
   \theta )-2) \left(c_{23} \cos
   ^6\theta +d_{14} \sin ^2\theta
   \right)}{\left(1+\sqrt{5}\right)^{4/
   3} \left(\cos ^6\theta -3 \cos
   ^2\theta +3\right)^{4/3} \left(3
   \sin ^2\theta +\cos ^6\theta
   \right)^2},\\
A_{17}^*&= -c_{05}\frac{32 \sin ^5\theta \cos ^5\theta  (\cos
   (2 \theta )-2)}{R(\theta)},
\end{align*}
where
\begin{align*}
R(\theta)&= \sqrt[3]{47+21 \sqrt{5}} \bigg(2 \sqrt[3]{2} \cos
   ^4\theta+4 \sqrt[3]{3 \Big(2+2^{2/3}
   \sqrt[3]{3-\sqrt{5}}+2^{2/3}
   \sqrt[3]{3+\sqrt{5}}\Big)} \\
&\quad\times \cos ^2\theta+\left(2   (3+\sqrt{5})\right)^{2/3}+(6-2 \sqrt{5})^{2/3}
 +4 \sqrt[3]{2}\bigg) \Big(-2 (3-\sqrt{5})^{2/3} \\
&\quad\times  \cos ^4\theta
   +\left(2 \sqrt[3]{6-2 \sqrt{5}}-2^{2/3}
   (\sqrt{5}-3)\right) \cos ^2\theta
   +\sqrt[3]{30 \sqrt{5}-50} \\
&\quad +2
   (3-\sqrt{5})^{2/3}-3 \sqrt[3]{6-2
   \sqrt{5}}-2\ 2^{2/3}\Big)^2 \left(3 \sin ^2\theta
   +\cos ^6\theta\right)^2.
\end{align*}

The integrals of the coefficients $A_1^*$, $A_{13}^*$ and $A_{17}^*$
in the variable $\theta$, in the interval $[0,2\pi]$ are zero
because $A_1^*$, $A_{13}^*$ and $A_{17}^*$ are odd functions.


\noindent{Claim:} 
For $i=3,5,7,9,11$ or $15$ we can
choose the parameters that appear in $A_i^{*}$ such that
$\int_0^{2\pi} A_i^*(\theta)d\theta\neq0$.

The proof of this claim follows from Example \ref{ex2}.


In short the function $\mathcal{F}$ defined in \eqref{eq5} can be
written as
\begin{equation}\label{eqS0}
\mathcal{F}^*(r_0)=
\frac{1}{2\pi}\Big(\sum_{k=1}^{5}r_0^{2k-3}
\int_0^{2\pi}A_{2k+1}^*(\theta)d\theta+r_0^{11}\int_0^{2\pi}
A_{15}^*(\theta)d\theta\Big).
\end{equation}

Note that the coefficients $A_{2k+1}^*(\theta )$ in \eqref{eqS0} are
linearly independent for $k=1,2,3,4,5,7$. Thus by the generalized
Descartes Theorem, the average function $\mathcal{F}^*$ has at most
$5$ positive simple zeros which provide limit cycles of system
\eqref{sis2}, when the average function is non-zero.


\section{Examples}\label{sec5}

\begin{example}\label{ex1}\rm
 Consider the quintic polynomial differential system with a
center at the origin
\[\dot{x}=-\frac{1}{3}x^3+y,\quad \dot{y}=-\frac{5}{12}x^5+x^2y,\]
with the perturbation
\begin{equation}\label{eq6}
\dot{x}=-\frac{1}{3}x^3+y, \quad
\dot{y}=-\frac{5}{12}x^5+x^2y+\varepsilon q(x,y),
\end{equation}
where
\[
q(x,y)=d_{01}y+d_{21}x^2y+d_{12}xy^2+d_{03}y^3+d_{23}x^2y^3+d_{14}xy^4+d_{05}y^5.
\]
Writing system \eqref{eq6} in the coordinates $x=r\cos\theta$ and
$y=r^3\sin\theta$ and taking the quotient $\dot{r}/\dot{\theta}$ we
get the following system in the standard form of Theorem \ref{teo2}
for applying the averaging theory
\begin{equation}\label{abc}
\frac{dr}{d\theta}=F_0(r,\theta)+\varepsilon
F_1(r,\theta)+\mathcal{O}(\varepsilon^2),
\end{equation}
where
\begin{align*}
F_0(r,\theta)
&= \frac{r \cos \theta  \left(-6 \sin \theta  (\sin (2 \theta
   )+2)+4 \cos ^3\theta +5 \sin \theta  \cos ^4\theta
   \right)}{36 \sin ^2\theta +5 \cos ^6\theta -24 \sin
   \theta  \cos ^3\theta },\\
F_1(r,\theta)
&= -48\,C(\theta)E(\theta)\frac{q\left(r \cos \theta ,r^3 \sin
   \theta \right)}{r^4 \left(36 \sin ^2\theta +5 \cos
   ^6\theta -24 \sin \theta  \cos ^3\theta\right)^2},
\end{align*}
with $C(\theta)= \cos (2 \theta )-2$, and $E(\theta)= \cos ^3\theta
-3 \sin \theta$. Thus for system \eqref{abc} we have $M(\theta)=
k(\theta)= (160/G(\theta))^{1/6}$, where
\[
G(\theta)=-192 \sin (2 \theta )-96 \sin (4 \theta )-501 \cos (2
\theta )+30 \cos (4 \theta )+5 \cos (6 \theta )+626,
\]
and the integrant of the integral \eqref{eq5} of system \eqref{abc}
is
\[\sum_1^7r_0^{2k-3}A_{2k+1}(\theta),\]
with
\begin{align*}
A_3(\theta)&= -d_{01}\frac{3072 \sqrt[3]{\frac{2}{5}}  \sin \theta\, C(\theta) (-12 \sin \theta +3 \cos \theta+\cos (3 \theta
   ))}{T(\theta)^{5/3}},\\
A_5(\theta)&= -d_{21}\frac{48  \sin \theta \cos ^2\theta\, C(\theta) E(\theta)}{\left(36 \sin
   ^2\theta+5 \cos ^6\theta-24 \sin \theta \cos ^3\theta
   \right)^2},\\
A_7(\theta)&= -d_{12}\frac{98304\ 2^{2/3} \sqrt[3]{5}  \sin ^2\theta \cos
   \theta \, C(\theta) E(\theta)}{T(\theta)^{7/3}},\\
A_9(\theta)&= -d_{03}\frac{393216 \sqrt[3]{2}\, 5^{2/3}  \sin ^3\theta \, C(\theta) E(\theta)}{T(\theta)^{8/3}},\\
A_{11}(\theta)&= -d_{23} \frac{7864320 \sin ^3\theta \cos ^2\theta \, C(\theta) E(\theta)}{T(\theta)^3},\\
A_{13}(\theta)&= -d_{14}\frac{15728640\ 2^{2/3} \sqrt[3]{5}  \sin ^4\theta \cos
   \theta \, C(\theta) E(\theta)}{T(\theta)^{10/3}},\\
A_{15}(\theta)&= -d_{05}\frac{62914560 \sqrt[3]{2} 5^{2/3}  \sin ^5\theta \, C(\theta) E(\theta)}{T(\theta)^{11/3}},
\end{align*}
and
\[
T(\theta)=-192 \sin (2 \theta )-96 \sin (4 \theta )-501 \cos (2 \theta
   )+30 \cos (4 \theta )+5 \cos (6 \theta )+626.
\]

Computing numerically the integral \eqref{eq5} for system
\eqref{abc} we obtain
\begin{align*}
\mathcal{F}(r_0)
&= \frac{1}{r_0}\Big( -4.2608..\,d_{01}- 2.0944..
\,d_{21} r_0^2
 -1.2770..\,d_{12} r_0^4 -1.2427..\, d_{03} r_0^6 \\
&\quad  -1.0908..\, d_{23} r_0^8
-0.7348..\, d_{14} r_0^{10}-0.5419..\, d_{05} r_0^{12} \Big).
\end{align*}
Taking
\begin{gather*}
d_{01}=-\frac{720}{ 4.2608..},\quad
d_{21}=\frac{1764}{ 2.0944..}, \quad 
d_{12}=-\frac{1624}{ 1.2770..}, \\
d_{03}=\frac{735}{1.2427..},\quad 
d_{23}=-\frac{175}{1.0908..},\quad
d_{14}=\frac{21}{0.7348..}, \quad 
d_{05}=-\frac{1}{0.5419..}.
\end{gather*}
The function $\mathcal{F}$ becomes
\[
\mathcal{F}(r_0)=\frac{r_0^{12}-21 r_0^{10}+175 r_0^8-735 r_0^6+1624
r_0^4-1764 r_0^2+720}{r_0}=\frac{1}{r_0}\prod_{i=1}^{6}(r_0^2-i).
\]
Thus we have that $\mathcal{F}$ has 6 positive simple zeros given by
$r_{0,i}=\sqrt{i}$, for $i=1,\ldots,6$ which by Theorem \ref{teo2},
provide 6 limit cycles of the perturbed system \eqref{eq6} for
$\varepsilon\neq0$ sufficiently small.
\end{example}

\begin{example}\label{ex2}
\rm Consider the quintic polynomial differential system with a
center at the origin
\[
\dot{x}=-y,\quad \dot{y}=x^5,
\]
with the perturbation
\begin{equation}\label{eq66}
\dot{x}=-y+\varepsilon p(x,y), \quad \dot{y}=x^5+\varepsilon q(x,y),
\end{equation}
where
\begin{gather*}
p(x,y)=c_{30}x^3+c_{50}x^5+c_{14}xy^4,\\
q(x,y)=d_{01}y+d_{03}y^3+d_{23}x^2y^3.
\end{gather*}

Writing system \eqref{eq66} in the coordinates $x=r\cos\theta$ and
$y=r^3\sin\theta$ and taking the quotient $\dot{r}/\dot{\theta}$ we
obtain the following system in the standard form of Theorem
\ref{teo2} for applying the averaging theory
\begin{equation}\label{abcd}
\frac{dr}{d\theta}
=F_0(r,\theta)+\varepsilon
F_1(r,\theta)+\mathcal{O}(\varepsilon^2),
\end{equation}
with $F_0(r,\theta)$ given in the proof
of Theorem \ref{teo1} and
\begin{align*}
F_1(r,\theta)
&= -\frac{\cos(2\theta)-2} {r \left(3 \sin
   ^2\theta+\cos ^6\theta
   \right)^2}\Big[r^6
   \sin ^4\theta \left(c_{14}
   r^6 \cos ^6\theta
   +d_{03}+d_{23} r^2 \cos
   ^2\theta\right) \\
&\quad   +r^2 \cos ^8\theta
   \left(c_{30}+c_{50} r^2 \cos
   ^2\theta\right)+d_{01} \sin
   ^2\theta \Big].
\end{align*}

The functions $k^*(\theta)$ and $M^*(\theta)$ for system
\eqref{abcd} are given also in the proof of Theorem \ref{teo1} and
the integrant of the integral \eqref{eq5} of system \eqref{abcd} is
\[
\sum_{k=1}^{5}r_0^{2k-3} A_{2k+1}^*(\theta)+r_0^{11}A_{15}^*(\theta),
\]
where
\begin{align*}
A_3^*(\theta)
&= -d_{01}\frac{256 \left(6-2
   \sqrt{5}\right)^{2/9} \sin ^2\theta
 (\cos (2 \theta )-2)}{\sqrt[9]{7-3
   \sqrt{5}} (-33 \cos (2 \theta )+6
   \cos (4 \theta )+\cos (6 \theta
   )+58)^{5/3}},\\
A_5^*(\theta)
&= -c_{30}\frac{\cos ^8\theta (\cos (2 \theta )-2)}{\left(3 \sin ^2\theta+\cos ^6\theta
   \right)^2},\\
A_7^*(\theta)&= -c_{50}\frac{\cos ^{10}\theta (\cos (2 \theta )-2)}{\sqrt[3]{\cos
   ^6\theta-3 \cos ^2\theta+3} \left(3 \sin ^2\theta+\cos
   ^6\theta\right)^2},\\
A_{9}^*(\theta)&= -d_{03}\frac{2^{\frac{40}{3}} \sin ^4\theta
   (\cos (2 \theta )-2)}{(-33 \cos (2
   \theta )+6 \cos (4 \theta )+\cos (6
   \theta )+58)^{8/3}},\\
A_{11}^*(\theta)&= -d_{23}\frac{2^\frac{43}{3} \left(\sqrt{5}-1\right)^{4/3}
   \sin ^4\theta \cos ^2\theta (\cos (2 \theta
   )-2)}{\left(3-\sqrt{5}\right)^{2/3} (-33 \cos (2
   \theta )+6 \cos (4 \theta )+\cos (6 \theta )+58)^3},\\
A_{15}^*(\theta)&= -c_{14}\frac{2^\frac{55}{3} \sin ^4(\theta
   ) \cos ^6\theta (\cos (2 \theta
   )-2)}{(-33 \cos (2 \theta )+6 \cos (4
   \theta )+\cos (6 \theta )+58)^{11/3}},
\end{align*}

Computing numerically the integral \eqref{eq5} for system
\eqref{abcd} we obtain
\begin{align*}
\mathcal{F}^*(r_0)
&= \frac{1}{r_0}\Big( 2.1033..\,d_{01}+1.8138..
\,c_{30} r_0^2
 +1.6169..\,c_{50} r_0^4 +0.6310..\, d_{03} r_0^6  \\
&\quad  +0.1512..\, d_{23} r_0^8
+0.0394..\, c_{14} r_0^{12} \Big).
\end{align*}
Taking
\begin{gather*}
d_{01}=-\frac{1800}{2.1033..},\quad 
c_{30}=\frac{3990}{ 1.8138..},\quad 
c_{50}=-\frac{3101}{ 1.6169..}, \\
d_{03}=\frac{1050}{0.6310..},\quad
d_{23}=-\frac{140}{0.1512..},\quad 
c_{14}=\frac{1}{0.0394..}.
\end{gather*}
The function $\mathcal{F}^*$ is now given by
\begin{align*}
\mathcal{F^*}(r_0)
&=\frac{r_0^{12}-140 r_0^8+1050 r_0^6-3101
r_0^4+3990 r_0^2-1800}{r_0}\\
&=\frac{r_0^2+15}{r_0}\prod_{i=1}^{5}(r_0^2-i).
\end{align*}
Thus we have that $\mathcal{F^*}$ has 5 positive simple zeros given
by $r_{0,i}=\sqrt{i}$, for $i=1,\ldots,5$ which by Theorem
\ref{teo2}, provide 5 limit cycles of the perturbed system
\eqref{eq66} with $\varepsilon\neq0$ sufficiently small.
\end{example}


\subsection*{Acknowledgements}
J. LLibre was supported by MINECO grants
MTM2016-77278-P and MTM2013-40998-P, by  AGAUR grant number
2014SGR-568, by grants FP7-PEOPLE-2012-IRSES 318999, and by CAPES
grant  88881.030454/2013-01 from the program CSF-PVE.
B. D. Lopes was supported by PNPD/CAPES. 
J. R. de Moraes was supported by FUNDECT-219/2016.


\begin{thebibliography}{99}

\bibitem{ALGM-biurcation} {A. A. Andronov, E. A. Leontovich, I. I. Gordon, 
 A. G. Ma{\u\i}er};
\textit{Theory of bifurcations of dynamic systems on a plane}, Halsted Press [A division of John Wiley \&
Sons], New York--Toronto, Ont., 1973, Translated from the Russian.

\bibitem{Descartes}{I.S. Berezin, N.P. Zhidkov};
\textit{Computing methods}, vol. II, Pergamon Press, Oxford,
1964.

\bibitem{BloPer94}{T.R. Blows, L.M. Perko};
\textit{Bifurcation of limit cycles from centers and separatrix
cycles of planar analytic systems}, SIAM Rev. \textbf{36} (1994), 341--376.

\bibitem{Bui1}{A. Buic{\u{a}}, J. P. Fran\c{c}oise, J. Llibre};
\textit{Periodic solutions of nonlinear periodic
differential systems with a small parameter}, Commun. Pure Appl. Anal. \textbf{6} (2007), 103--111.

\bibitem{BuiLli} {A. Buic{\u{a}}, J. Llibre};
\textit{Averaging methods for finding periodic orbits via Brouwer
degree}, Bull. Sci. Math. \textbf{128} (2004), 7--22.

\bibitem{Chr} {C. Christopher, C. Li};
\textit{Limit cycles in differential equations}, Birkhauser, Boston,
2007.

\bibitem{HRZ} {M. Han, V. G. Romanovski, X. Zhang};
\textit{Equivalence of the melnikov function and
the averaging method}, Qual. Th. Dyn. Sys. (2016).

\bibitem{HI} {D. Hilbert};
\textit{Mathematische probleme}, Lecture, Second Internat. Congr. Math. (Paris,
1900), \textit{Nachr. Ges. Wiss. G''ttingen Math. Phys. KL.} (1900), 253--297; English transl.,
\textit{Bull. Amer. Math. Soc.} \textbf{8} (1902), 437--479; \textit{Bull. (New Series) Amer. Math. Soc.} \textbf{37}
(2000), 407--436.

\bibitem{Li1}{W. Li, J. Llibre, J. Yang, Z. Zhang};
\textit{Limit cycles bifurcating from the period annulus of quasi--homogeneous centers},
J. Dynam. Differential Equations
\textbf{21} (2009), 133--152.

\bibitem{BJ_Elec}{J. Llibre, B. D. Lopes, J. R. de Moraes};
\textit{Limit cycles bifurcating from the periodic
annulus of cubic homogeneous polynomial centers}, Electronic Journal of Differential
Equations \textbf{271} (2015), 1--8.

\bibitem{Pessoa}{J. Llibre, C. Pessoa};
\textit{On the centers of the weight-homogeneous polynomiL vector fields on the plane},
J. Math. Anal. Appl. \textbf{359} (2009), 722--730.


\bibitem{LTT}{J. Llibre, M.A. Teixeira, J. Torregrosa};
\textit{Limit cycles bifurcating from a k--dimensional isochronous center contained in $r^n$ with $k\leq n$},
Mathematical Physics, Analysis and Geometry \textbf{10} (2007), 237--249.

\bibitem{SanVer}{J.A. Sanders, F. Verhulst};
\textit{Averaging methods in nonlinear dynamical systems},
Applied Mathematical Sciences, vol. 59, Springer, New York, 2007.

\bibitem{SS}{S. Smale};
\textit{Mathematical problems for next century},
Math. Intelligencer \textbf{20} (1998), 7--15.

\end{thebibliography}

\end{document}