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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 246, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/246\hfil Limit cycles bifurcating]
{Limit cycles bifurcating from the period annulus of a uniform isochronous
center in a quartic polynomial differential system}

\author[J. Itikawa, J. Llibre \hfil EJDE-2015/246\hfilneg]
{Jackson Itikawa, Jaume Llibre}

\address{Jackson Itikawa \newline
Departament de Matem\`atiques,
Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain.
 Fax +34 935812790. Phone +34 93 5811303}
\email{itikawa@mat.uab.cat}

\address{Jaume Llibre \newline
Departament de Matem\`atiques,
Universitat Aut\`onoma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain.
 Fax +34 935812790. Phone +34 93 5811303}
\email{jllibre@mat.uab.cat}

\thanks{Submitted October 15, 2014. Published September 22, 2015.}
\subjclass[2010]{34A36, 34C07, 34C25, 37G15}
\keywords{Polynomial vector field; limit cycle; averaging method;
\hfill\break\indent  periodic orbit; uniform isochronous center}

\begin{abstract}
 We study the number of limit cycles that bifurcate from the periodic
 solutions surrounding a uniform isochronous center located at the
 origin of the quartic polynomial differential system
 $$
 \dot{x}=-y+xy(x^2+y^2),\quad \dot{y}=x+y^2(x^2+y^2),
 $$
 when perturbed in the class of all quartic polynomial differential
 systems. Using the averaging theory of first order we show that at
 least 8 limit cycles bifurcate from the period annulus of the center.
 Recently this problem was studied by Peng and Feng \cite{Peng},
 where the authors found 3 limit  cycles.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction and statement of main results}\label{sectionintroduction}

One of the main open problems in the qualitative theory of
 polynomial differential systems in $\mathbb{R}^2$ is the
determination of their limit cycles, see for instance
\cite{Hilbert}. A classical method to produce limit cycles is by
perturbing a system which has a center. In this case the perturbed
system displays limit cycles that bifurcate either from the
center (having the so-called Hopf bifurcation); or from some of the
periodic orbits around the center, (see
\cite{Christopher,Pontrjagin}
and  references therein); or from the graph in
the boundary of the period annulus of the center.

Isochronous differential systems constitute a large class of
polynomial systems with interesting properties. They
arise in many applications. For recent studies on the bifurcation of
limit cycles of planar polynomial differential systems having a
uniform isochronous center  see for
instance \cite{Dias,IL1,IL2}. In this paper we shall perturb the
uniform isochronous center of the quartic polynomial differential
system
\begin{equation}\label{peng01}
\dot{x}=-y+xy(x^2+y^2), \quad \dot{y}=x+y^2(x^2+y^2),
\end{equation}
inside the class of all quartic polynomial differential systems.

Peng and Feng \cite{Peng} studied the differential system
\eqref{peng01}, showing that under any quartic homogeneous
polynomial perturbations, at most 2 limit cycles bifurcate from the
period annulus of such system using averaging theory of first order,
and that this upper bound can be reached. In addition these authors
proved that for the family of perturbed quartic polynomial
differential systems
\begin{equation}\label{pengparticular}
\begin{gathered}
\begin{aligned}
\dot{x}&=-y+xy(x^2+y^2)+\varepsilon(a_{10}x+a_{01}y+a_{11}xy+a_{21}x^2y+a_{03}y^3\\
&\quad +a_{40}x^4+a_{31}x^3y+a_{22}x^2y^2+a_{13}xy^3+a_{04}y^4),
\end{aligned}\\
\begin{aligned}
\dot{y}&=x+y^2(x^2+y^2)+\varepsilon(b_{10}x+b_{01}y+b_{20}x^2+b_{02}y^2+b_{30}x^3\\
&\quad+b_{12}xy^2+b_{40}x^4+b_{31}x^3y+b_{22}x^2y^2+b_{13}xy^3+b_{04}y^4),
\end{aligned}
\end{gathered}
\end{equation}
there are at most 3 limit cycles bifurcating from the period annulus
of \eqref{peng01} using averaging theory of first order, and that  this
upper bound is sharp. We remark that the perturbed system
\eqref{pengparticular} studied by Peng and Feng do not cover all
possible perturbed quartic polynomial differential systems because
the authors do not consider the coefficients $a_{00}$, $a_{20}$,
$a_{02}$, $a_{30}$, $a_{12},$ $b_{00}$, $b_{11}$, $b_{21}$, $b_{03}$ in their
analysis.

We study the limit cycles which bifurcate from the periodic
solutions of the uniform isochronous center located at the origin of
system \eqref{peng01} when it is pertur\-bed inside the whole class of
quartic polynomial diffe\-rential systems. More precisely we consider
the following differential systems
\begin{equation}\label{peng02}
\begin{gathered}
\dot{x}= -y+xy(x^2+y^2)+\varepsilon\sum_{i=0}^4 p_i(x,y),\\
\dot{y}=\;\;x+y^2(x^2+y^2)+\varepsilon\sum_{i=0}^4 q_i(x,y),
\end{gathered}
\end{equation}
where 
\[
p_i=\sum_{j+k=i}a_{jk}x^jy^k, \quad
q_i=\sum_{j+k=i}b_{jk}x^jy^k
\]
 are homogeneous   polynomials of degree $i$, and
$a_{jk},b_{jk}\in\mathbb{R}$.
In what follows we state our main result.

\begin{theorem}\label{mainteo}
For  $|\varepsilon|\neq0$ sufficiently small there are quartic
polynomial differential systems \eqref{peng02} having at least 8
limit cycles bifurcating from the periodic orbits of the period
annulus of the uniform isochronous center located at the origin of
system \eqref{peng01}.
\end{theorem}

Note that Theorem \ref{mainteo} improves the result of Peng and Feng
by providing five additional limit cycles.
All our calculations were performed with the assistance of the software
\emph{Mathematica}.

\section{Preliminary results}\label{sectionpreliminaries}
In this section we introduce some preliminary results on uniform
isochronous centers and on averaging theory that we shall use in our
study.

Let $p\in\mathbb{R}^2$ be a center of a differential polynomial
system in $\mathbb{R}^2$. Without loss of generality we can assume
that $p$ is the origin of   coordinates. We say that $p$ is
an  \emph{isochronous center} if it is a center having a
neighborhood such that all the periodic orbits in this
neighborhood have the same period. We say that $p$ is a
\emph{uniform isochronous center} if the system, in polar
coordinates $(r,\theta)$ where $ x=r\cos\theta$,
$ y=r\sin\theta$, takes the form $\dot{r}=G(\theta,r)$,
$\dot{\theta}=k, \; k\in\mathbb{R}\setminus\{0\}$, for more details
see Conti \cite{Conti}. The \emph{period annulus} of a center is the
biggest connected set of periodic solutions surrounding a center and
having in its inner boundary the center itself. The next result is
well-known.

\begin{proposition}\label{charactuniformisochronous}
Assume that a planar differential polynomial system 
$\dot{x}=P(x,y)$, $\dot{y}= Q(x,y)$ of degree $n$ has a center at the origin 
of coordinates. Then, this center is uniform isochronous if and only if 
through a linear change of variables and time rescaling it can be written under 
the form
\begin{equation}\label{07}
\dot{x}=-y+x\;f(x,y), \quad \dot{y}=x+y\;f(x,y),
\end{equation}
where $f(x,y)$ is a polynomial in x and y of degree $n-1$, and $f(0,0)=0$.
\end{proposition}

In 1994, Conti \cite{Conti} proved the following theorem.

\begin{theorem}\label{contihomogeneous}
Let $f(x,y)=\sum_{i+j=n-1}p_{i,j}x^iy^j$ be a homogeneous polynomial of degree 
$n-1$. Then system  \eqref{07} has a uniform isochronous center at the origin 
if either $n$ is even, or if $n$ is odd and
\[
\sum_{\nu=0}^{n-1}\Big[p_{n-1-\nu,\nu}\int_0^{2\pi}{\cos^{n-1-\nu}
\theta\sin^{\nu}\theta\ d\theta}\Big]=0.
\]
\end{theorem}

The next result is the first-order averaging theory developed for continuous 
differential systems.
Consider the differential system
\begin{equation}\label{pert}
\dot{x}=\varepsilon F_1(t,x)+\varepsilon^2 F_2(t,x,\varepsilon), \quad x(0)=x_0
\end{equation}
with $x\in D$, where $D$ is an open subset of $\mathbb{R}^n$, $t\geq 0$.
Furthermore we suppose that the functions $F_1(t,x)$ and
$F_2(t,x,\varepsilon)$ are $T-$periodic in $t$. We define in $D$ the
averaged differential system
\begin{equation}\label{eqb2}
\dot{y}=\varepsilon f_1(y),\quad y(0)=x_0,
\end{equation}
where 
\[
f_1(y)=\frac{1}{T}\int_0^TF_1(t,y)dt.
\]
As we shall see under convenient assumptions, the equilibria
solutions of the averaged system will provide $T-$periodic solutions
of system \eqref{pert}.

\begin{theorem}\label{teobif04}
Consider the two initial value problems \eqref{pert} and
\eqref{eqb2}. Assume that
\begin{itemize}
\item[(i)] the functions $F_1$, $\partial F_1/\partial x$, $\partial ^2
F_1/\partial x^2$,
$F_2$ and $\partial F_2/\partial x$ are defined, continuous and
bounded by a constant independent of $\varepsilon$ in $[0,\infty)\times D$
and $\varepsilon\in(0,\varepsilon_0]$;

\item[(ii)] the functions $F_1$ and $F_2$ are $T-$periodic in $t$ ($T$
independent of
$\varepsilon$).
\end{itemize}
Then the following statements hold.

(a) If $p$ is an equilibrium point of the averaged system \eqref{eqb2}
satisfying
\[
\det\Big(\frac{\partial f_1}{\partial
y}\Big)\Big|_{y=p}\neq 0,
\]
then there is a $T-$periodic solution $\varphi(t,\varepsilon)$ of system
\eqref{pert} such that $\varphi(0,\varepsilon)\to p$ as
$\varepsilon\to 0$.

(b) The kind of stability or instability of the periodic solution
$\varphi(t,\varepsilon)$
coincides with the kind of stability or instability of the
equilibrium point $p$ of the averaged system \eqref{eqb2}. The
equilibrium point $p$ has the kind of stability behavior of the
Poincar\'{e} map associated to the periodic solution $\varphi(t,\varepsilon)$.
\end{theorem}

For a proof of the above, see \cite[sec. 6.3, 11.8]{Ve}.
The next theorem provides a method to write a perturbed differential system 
under the form \eqref{pert}.

\begin{theorem}\label{teobifenergy}
Consider the unperturbed system $\dot{x}=P(x,y),\ \dot{y}=Q(x,y)$, where 
$P,Q:\mathbb{R}^2\to\mathbb{R}$ are continuous functions, and assume that 
this system has a continuous family of period solutions 
$\{\Gamma_h\}\subset\{(x,y) : \mathcal{H}(x,y)=h, h_1<h<h_2\}$, 
where $\mathcal{H}$ is a first integral of the system. For a given first 
integral $H$ assume that $xQ(x,y)-yP(x,y)\neq0$ for all $(x,y)$ in the period 
annulus formed by the ovals $\{\Gamma_h\}$. 
Let $\rho:(\sqrt{h_1},\sqrt{h_2})\times[0,2\pi)\to[0,\infty)$ 
be a continuous function such that
\[
H(\rho(R,\theta)\cos\theta,\rho(R,\theta)\sin\theta)=R^2
\]
for all $R\in(\sqrt{h_1},\sqrt{h_2})$ and all $\theta\in[0,2\pi)$. 
Then the differential equation which describes the dependence between 
the square root of the energy $R=\sqrt{h}$ and the angle $\theta$ 
for the perturbed system $\dot{x}=P(x,y)+\varepsilon p(x,y)$,
$\dot{y}=Q(x,y)+\varepsilon q(x,y)$,  where 
$p,q:\mathbb{R}^2\to\mathbb{R}$ are continuous functions is
\begin{equation}\label{eqbifenergy}
\frac{dR}{d\theta}=\varepsilon\frac{\mu(x^2+y^2)(Qp-Pq)}{2R(Qx-Py)}
+\mathcal{O}(\varepsilon^2)
\end{equation}
where $\mu=\mu(x,y)$ is the integrating factor corresponding to the first 
integral H of the unperturbed system and $x=\rho(R,\theta)\cos\theta$, 
$y=\rho(R,\theta)$ $\sin\theta$.
\end{theorem}

For more details see \cite{Buica}.
We also need the next result, which can be found in 
\cite[Prop. 1]{Llibreswirszcz}.

\begin{proposition}\label{propdif01}
Let $f_0, \ldots , f_n$ be analytic functions defined on an open interval 
$I\subset\mathbb{R}$. If $f_0, \ldots , f_n$ are linearly independent then 
there exists $s_1, \ldots , s_n\in I$ and 
$\lambda_0, \ldots , \lambda_n\in\mathbb{R}$ such that for every 
$j\in\{1, \ldots , n\}$ we have $\sum_{i=0}^n\lambda_if_i(s_j)=0$.
\end{proposition}


\section{Proof of Theorem \ref{mainteo}}\label{sectionproofmainteo}

By Theorem \ref{contihomogeneous} it follows that system \eqref{peng01} has 
a uniform isochronous center at the origin. A first integral $H$ and its 
corresponding integrating factor $\mu$ for system \eqref{peng01} are
$$
H(x,y)=\frac{1}{3(x^2+y^2)^{3/2}}-\frac{x}{(x^2+y^2)^{1/2}},\quad
\mu(x,y)=\frac{1}{(x^2+y^2)^{5/2}},
$$
respectively. If $h\in(1,+\infty)$ then $H(x,y)=h$ are periodic solutions 
around the center $(0,0)$. For proving Theorem \ref{mainteo} we shall use 
Theorem \ref{teobifenergy}. We choose
$$
\rho(R,\theta)=\frac{1}{(R^2+3\cos\theta)^{1/3}},
$$
then $H(\rho\cos\theta,\rho\sin\theta)=R^2/3$ for all $R>\sqrt{3}$ and 
$\theta\in[0,2\pi)$. Therefore all the hypotheses of Theorem \ref{teobifenergy} 
are satisfied for system \eqref{peng01}. Using Theorem \ref{teobifenergy} 
we transform the perturbed differential system \eqref{peng02} into the form
\begin{equation}\label{eqproof01}
\frac{dR}{d\theta}=\varepsilon\Big(\frac{3}{2R}\frac{Qp-Pq}{\rho^5}\Big)
\Big|_{x=\rho\cos\theta,y=\rho\sin\theta}+O(\varepsilon^2),
\end{equation}
where
\[
Qp-Pq=A+B,
\]
with
\begin{align*}
A&=a_{00}x+b_{00}y+(a_{02}+b_{11})xy^2+a_{20}x^3+(a_{00}+b_{03})y^4\\
&\quad-b_{00}xy^3+(a_{00}+a_{12}+b_{21})x^2y^2-b_{00}x^3y+a_{30}x^4+a_{02}y^6\\
&\quad +(a_{02}+a_{20}-b_{11})x^2y^4+(a_{20}-b_{11})x^4y^2+(a_{12}-b_{03})xy^6\\
&(a_{12}+a_{30}-b_{03}-b_{21})x^3y^4+(a_{30}-b_{21})x^5y^2,
\end{align*}
\begin{align*}
B&= a_{10}x^2+(a_{01}+b_{10})xy+b_{01}y^2+(a_{11}+b_{20})x^2y+b_{02}y^3\\
&\quad +(a_{21}+b_{30})x^3y+(a_{03}+b_{12})xy^3+a_{40}x^5+(a_{31}+b_{40}-b_{10})x^4y\\
&\quad +(a_{22}+a_{10}+b_{31}-b_{01})x^3y^2+(a_{13}+a_{01}+b_{22}-b_{10})x^2y^3\\
&\quad +(a_{04}+a_{10}+b_{13}-b_{01})xy^4+(a_{01}+b_{04})y^5-b_{20}x^5y\\
&\quad +(a_{11}-b_{20}-b_{02})x^3y^3+(a_{11}-b_{02})xy^5-b_{30}x^6y\\
&\quad +(a_{21}-b_{30}-b_{12})x^4y^3+(a_{21}+a_{03}-b_{12})x^2y^5+a_{03}y^7\\
&\quad -b_{40}x^7y+(a_{40}-b_{31})x^6y^2+(a_{31}-b_{40}-b_{22})x^5y^3\\
&\quad +(a_{40}+a_{22}-b_{31}-b_{13})x^4y^4+(a_{31}+a_{13}-b_{22}-b_{04})x^3y^5.
\end{align*}
We remark that the coefficients $\{a_{ij},b_{ij}\}_{i,j\in\{0,\ldots,4\}}$ 
which appear in $A$ and $B$ are different. The expression $B$ corresponds 
to the perturbed system \eqref{pengparticular} studied in \cite{Peng}. 
The authors of \cite{Peng} obtained for this system the following averaging function
\begin{equation}\label{eqproof03}
\begin{aligned}
g_B(R)&=\frac{3}{4R}\Big[\Big(M_4-\frac{3M_1+4M_2+8M_3}{36}\Big)R^2
 -\frac{M_1+2M_2}{82}R^6-\frac{2M_1}{729}R^{10} \\
&\quad +\Big(\frac{2M_1}{729}R^{12}+\frac{2M_2}{81}R^8
 +\frac{2M_3}{9}R^4-2(M_1+M_2+M_3)\Big)\frac{1}{\sqrt{R^4-9}}\Big],
\end{aligned}
\end{equation}
where
\begin{equation}\label{eqproofM}
\begin{gathered}
M_1= a_{22}-a_{40}-a_{04}+b_{31}-b_{13},\\
M_2= -2a_{22}+a_{40}+3a_{04}-b_{31}+2b_{13},\\
M_3= a_{22}-3a_{04}-b_{13},\quad
M_4= a_{10}+b_{01}.
\end{gathered}
\end{equation}
Peng and Feng proved that the function $g_B(R)$ has at most 3 zeros 
in $R\in(\sqrt{3},+\infty)$, and using the averaging theory of first order 
they showed that the maximum number of limit cycles of system \eqref{pengparticular} 
emerging from the period annulus of the unperturbed system \eqref{peng01} is 3.

In this work we extend the results presented in \cite{Peng} by calculating 
the part of the averaging function of system \eqref{peng02} which corresponds 
to the expression $A$. In order to do that, we perturb the center of 
system \eqref{peng01} inside the whole class of quartic polynomial differential 
systems. We note that \eqref{eqproof01} is continuous and bounded 
for $\theta\in(0,2\pi)$ and $R\in(\sqrt{3},+\infty)$ therefore the 
integral of \eqref{eqproof01} is the sum of the integrals of its parts $A$ and $B$. 
Then from the expression \eqref{eqproof01} we have
\[
\frac{dR}{d\theta}=\varepsilon\Big(\frac{3}{2R}\frac{A}{\rho^5}\Big)
\Big|_{\substack{x=\rho\cos\theta \\ y=\rho\sin\theta}}
+\varepsilon\Big(\frac{3}{2R}\frac{B}{\rho^5}\Big)
\Big|_{\substack{x=\rho\cos\theta \\
y=\rho\sin\theta}}+O(\varepsilon^2).
\]
We obtain the averaging function $f(R)=g_A(R)+g_B^{\ast}(R)$ where
\begin{gather*}
\begin{aligned}
g_A(R)&= a_{00}g_0(R)+a_{02}g_1(R)+a_{12}g_2(R)+a_{20}g_3(R)+a_{30}g_4(R)\\
&\quad + b_{03}g_5(R)+b_{11}g_6(R)+b_{21}g_7(R),
\end{aligned}\\
g_B^{\ast}(R)= \sum_{i=1}^4M_i\,g_{Mi}(R),
\end{gather*}
and $g_B^{\ast}(R)$ is the function \eqref{eqproof03} rearranged in a 
convenient way, with $M_i$, $i\in\{1,\ldots,4\}$ given in \eqref{eqproofM}. 
The expressions of $g_i(R)$, $i\in\{0,\ldots,7\}$ are shown in the Appendix 
\ref{appendix1}, and the functions $g_{Mi}(R)$, $i\in\{1,\ldots,4\}$ are presented 
in the Appendix \ref{appendix2}.

Out of the 12 functions $G_i=g_i:(\sqrt{3},+\infty)\to\mathbb{R}$, 
$i\in\{0,\ldots,7\}$, $G_{i+7}=g_{Mi}:(\sqrt{3},+\infty)\to\mathbb{R}$, 
$i\in\{1,\ldots,4\}$ we have that 9 are linearly independent. Indeed, 
using the software \emph{Mathematica} to calculate the Taylor expansions 
for those 12 functions in the variable $R$ until its $15^{th}$ power around 
$R=2$, which are too long and therefore they are not presented here, 
we construct a $12\times16$ matrix, where in the $k$ row we place the 16 
coefficients of $R^0, R^1,\ldots, R^{15}$ of the Taylor expansion of 
$G_k$, $k\in\{0,\ldots, 11\}$, and we conclude that the rank of such matrix is 9.

By Proposition \ref{propdif01} since there are 9 linearly independent functions 
among the 12 previously described, then there exists a linear combination 
of them with at least 8 zeros, because all the coefficients of these functions 
are linearly independent, as it is easy to check. Thus there exist 
$R_1,R_2,\ldots,R_8\in(\sqrt{3},+\infty)$ and coefficients 
$a_{ij}, b_{ij}\in\mathbb{R}$, $i,j\in\{0,\ldots,4\}$ such that 
$f(R_k)=0$, $k\in\{1,\ldots,8\}$.

In summary, applying Theorem \ref{teobif04} we conclude that there are 
planar quartic polynomial differential systems \eqref{peng02} having at 
least 8 limit cycles bifurcating from the period orbits of the period annulus 
of the uniform isochronous center located at the origin of the unperturbed 
differential system \eqref{peng01}.

\section{Appendix: Averaging functions $g_i(R)$, $i\in\{0,\ldots,7\}$}\label{appendix1}

\begin{align*}
g_0&= -3 \pi  \bigg((R^2+3) (-6 R^{10} (R^2+3)^{2/3}+59 R^6 (R^2+3)^{2/3}
 -1440 R^2(R^2+3)^{2/3}\\
&\quad +6 R^8 ((R^4-9)^{2/3} \sqrt[3]{R^2-3}+3 (R^2+3)^{2/3})
 -R^4 \big(709(R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +177 (R^2+3)^{2/3}\big)  +360 (12 (R^2+3)^{2/3}
  -61 \sqrt[3]{R^2-3} (R^4-9)^{2/3}))\\
&\quad\times {}_2F_1(-\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3}) \sqrt[3]{R^2-3}
 +\Big(7320 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad -1440 (R^2+3)^{2/3}+R^2 \Big(1346 (R^4-9)^{2/3} \sqrt[3]{R^2-3}
+6 R^8 (R^2+3)^{2/3}\\
&\quad +618 (R^2+3)^{2/3}-6 R^6
\big((R^4-9)^{2/3} \sqrt[3]{R^2-3}+(R^2+3)^{2/3}\big)\\
&\quad -R^4 (12 (R^4-9)^{2/3}
 \sqrt[3]{R^2-3}+71 (R^2+3)^{2/3})+R^2 (685 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +59(R^2+3)^{2/3})\Big)\Big) 
 \, _2F_1(\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3}) (R^2-3)^{4/3}\\
&\quad +(R^2+3)^{4/3}
\Big(\Big(-1346 (R^4-9)^{2/3} \sqrt[3]{R^2+3}+618 (R^2-3)^{2/3}\\
&\quad +R^2 \Big(685 (R^4-9)^{2/3} \sqrt[3]{R^2+3}-59 (R^2-3)^{2/3}\\
&\quad +R^2 \Big(12 (R^4-9)^{2/3} \sqrt[3]{R^2+3}
 -71(R^2-3)^{2/3}+6 R^2 \big(-(R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad +R^2 (R^2-3)^{2/3}
 +(R^2-3)^{2/3}\big)\Big)\Big)\Big)R^2+120 (61 (R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad +12 (R^2-3)^{2/3})\Big)
\, _2F_1(\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})-(R^2-3) 
 \sqrt[3]{R^2+3} \\
&\quad\times \Big(6 R^{10}(R^2-3)^{2/3}-59 R^6 (R^2-3)^{2/3}
 +1440 R^2 (R^2-3)^{2/3}\\
&\quad +360 (61(R^4-9)^{2/3} \sqrt[3]{R^2+3}+12 (R^2-3)^{2/3}) \\
&\quad +R^4 (709 \sqrt[3]{R^2+3} (R^4-9)^{2/3}
 -177 (R^2-3)^{2/3})+6 R^8 (3 (R^2-3)^{2/3}\\
&\quad -\sqrt[3]{R^2+3} (R^4-9)^{2/3})\Big)\;
{}_2F_1(-\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\bigg)\\
&\quad \div 14560 R (R^2-3)^{2/3} (R^2+3)^{2/3} (R^4-9)^{2/3};
\end{align*}
\begin{align*}
g_1&= \pi  \Big(-(2 R^4-39) (3 R^2 \sqrt[3]{R^2-3}+9 \sqrt[3]{R^2-3}
 +2 (R^2+3)^{2/3} \sqrt[3]{R^4-9}) R^2\\
&\quad{}_2F_1(-\frac{2}{3},\frac{1}{2};1;\frac{6}{R^2+3})
 +2 (R^4-12) ((R^2+3) (3 R^2 \sqrt[3]{R^2+3}\\
&\quad -9 \sqrt[3]{R^2+3}+2 (R^2-3)^{2/3} \sqrt[3]{R^4-9}) 
 \, _2F_1(\frac{1}{3},\frac{1}{2};1;-\frac{6}{R^2-3})\\
&\quad +(R^2-3) (3 R^2 \sqrt[3]{R^2-3}+9 \sqrt[3]{R^2-3}+2 (R^2+3)^{2/3}
  \sqrt[3]{R^4-9})\\
&\quad\times{}_2F_1(\frac{1}{3},\frac{1}{2};1;\frac{6}{R^2+3}))
 -R^2 (2 R^4-39) (3 R^2 \sqrt[3]{R^2+3}-9 \sqrt[3]{R^2+3}\\
&\quad +2 (R^2-3)^{2/3} \sqrt[3]{R^4-9}) 
 \, _2F_1(-\frac{2}{3},\frac{1}{2};1;-\frac{6}{R^2-3})\Big)\\
&\div 880 R \sqrt[3]{R^4-9};
\end{align*}
\begin{align*}
g_2&= 3 \pi  \bigg((R^2+3) \Big(72 R^{14} (R^2+3)^{2/3}-840 R^{10} (R^2+3)^{2/3}+391 R^6 
(R^2+3)^{2/3}\\
&\quad +50688 R^2 (R^2+3)^{2/3}-72 R^{12} ((R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +3 (R^2+3)^{2/3})+24 R^8 (73 (R^4-9)^{2/3} \sqrt[3]{R^2-3}
 +105 (R^2+3)^{2/3})\\
&\quad -51 R^4 (261 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+23 (R^2+3)^{2/3})\\
&\quad -1728 (88 (R^2+3)^{2/3}-15 \sqrt[3]{R^2-3} (R^4-9)^{2/3})\Big) \\
&\quad\times{}  _2F_1(-\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3})
 \sqrt[3]{R^2-3}+((12294 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad -24702 (R^2+3)^{2/3}+R^2(8535 (R^4-9)^{2/3} \sqrt[3]{R^2-3}
 +391 (R^2+3)^{2/3}\\
&\quad +R^2 (-2640 (R^4-9)^{2/3}\sqrt[3]{R^2-3}-1711 (R^2+3)^{2/3}\\
&\quad -24 R^2 (61 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+3 R^6 (R^2+3)^{2/3}
  +35 (R^2+3)^{2/3}\\
&\quad -3 R^4 ((R^4-9)^{2/3}  \sqrt[3]{R^2-3}+(R^2+3)^{2/3})
 -R^2 (6 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +41 (R^2+3)^{2/3}))))) R^2+576 (88 (R^2+3)^{2/3}
 -15 \sqrt[3]{R^2-3} (R^4-9)^{2/3})) \\
&\quad\times{}_2F_1(\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3}) (R^2-3)^{4/3}
 +\sqrt[3]{R^2+3} Big((50688 (R^2-3)^{2/3}\\
&\quad +R^2 ((391 (R^2-3)^{2/3}+24 R^2 (73(R^4-9)^{2/3}\sqrt[3]{R^2+3}
 +3 R^6 (R^2-3)^{2/3}\\
&\quad -35 R^2 (R^2-3)^{2/3}-105(R^2-3)^{2/3}+R^4 (9 (R^2-3)^{2/3}\\
&\quad -3 \sqrt[3]{R^2+3} (R^4-9)^{2/3}))) R^2+51 (23(R^2-3)^{2/3}\\
&\quad -261\sqrt[3]{R^2+3} (R^4-9)^{2/3}))) R^2+1728 (15 (R^4-9)^{2/3} 
\sqrt[3]{R^2+3} \\
&\quad +88 (R^2-3)^{2/3})\Big) 
 \,_2F_1(-\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})(R^2-3)\\
&\quad -(R^2+3)^{4/3} (((-8535 (R^4-9)^{2/3} \sqrt[3]{R^2+3}+391 (R^2-3)^{2/3}\\
&\quad +R^2 (-2640 (R^4-9)^{2/3}\sqrt[3]{R^2+3}+1711 (R^2-3)^{2/3}\\
&\quad +24 R^2 (61 (R^4-9)^{2/3} \sqrt[3]{R^2+3}-35 (R^2-3)^{2/3}+R^2 (6 (R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad -41 (R^2-3)^{2/3}+3 R^2 (-(R^4-9)^{2/3} \sqrt[3]{R^2+3}+R^2 (R^2-3)^{2/3}\\
&\quad +(R^2-3)^{2/3}))))) R^2+6 (2049 (R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad +4117 (R^2-3)^{2/3})) R^2+576 (15 (R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad +88 (R^2-3)^{2/3})) \,_2F_1(\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\bigg)\\
&\div 442624 R (R^2-3)^{2/3} (R^2+3)^{2/3} (R^4-9)^{2/3};
\end{align*}
\begin{align*}
g_3
&= \pi  \big((-29 R^2 \sqrt[3]{R^2+3}+87 \sqrt[3]{R^2+3}+6 R^6 \sqrt[3]{R^2+3}\\
&\quad +98 (R^2-3)^{2/3} \sqrt[3]{R^4-9}+R^4 (4 (R^2-3)^{2/3}\sqrt[3]{R^4-9}\\
&\quad -18 \sqrt[3]{R^2+3})) R^2 \, _2F_1(-\frac{2}{3},\frac{1}{2};1;-\frac{6}{R^2-3})\\
&\quad +(-29 R^2 \sqrt[3]{R^2-3}-87 \sqrt[3]{R^2-3}+6 R^6 \sqrt[3]{R^2-3}\\
&\quad +98 (R^2+3)^{2/3} \sqrt[3]{R^4-9}+2 R^4 (9 \sqrt[3]{R^2-3}\\
&\quad +2 (R^2+3)^{2/3} \sqrt[3]{R^4-9})) R^2 \, _2F_1(-\frac{2}{3},\frac{1}{2};1;\frac{6}{R^2+3})\\
&\quad -2 (R^2+3) (8 R^2 \sqrt[3]{R^2+3}-24 \sqrt[3]{R^2+3}+3 R^6 \sqrt[3]{R^2+3}\\
&\quad +64 (R^2-3)^{2/3} \sqrt[3]{R^4-9}+R^4 (2 (R^2-3)^{2/3} \sqrt[3]{R^4-9}\\
&\quad -9 \sqrt[3]{R^2+3})) \, _2F_1(\frac{1}{3},\frac{1}{2};1;-\frac{6}{R^2-3})-2 (R^2-3) (8 R^2 \sqrt[3]{R^2-3}\\
&\quad +3 R^6 \sqrt[3]{R^2-3}+R^4 (9 \sqrt[3]{R^2-3}+2 (R^2+3)^{2/3} \sqrt[3]{R^4-9})\\
&\quad +8 (3 \sqrt[3]{R^2-3}+8 (R^2+3)^{2/3} \sqrt[3]{R^4-9})) 
\, _2F_1(\frac{1}{3},\frac{1}{2};1;\frac{6}{R^2+3})\Big)\\
&\div 880 R \sqrt[3]{R^4-9};
\end{align*}
\begin{align*}
g_4&= 3 \pi  \Big((R^2+3) (6 R^8-7 R^4-1056) (R^2-3)^{2/3} 
 \, _2F_1(-\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3})\\
&\quad +(R^2+3)^{2/3} (6 R^8-7 R^4-1056) (R^2-3) \, _2F_1(-\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\\
&\quad -(6 R^8+12 R^6+17 R^4+58 R^2-352) (R^2-3)^{5/3} \, _2F_1(\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3})\\
&\quad +(R^2+3)^{2/3} (-6 R^{10}-6 R^8+19 R^6+7 R^4+526 R^2+1056)\\
&\quad\times{} _2F_1(\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\Big)
\Big/2912 R (R^4-9)^{2/3};
\end{align*}
\begin{align*}
g_5&= -9 \pi  \bigg(-(R^2+3) (-120 R^{14} (R^2+3)^{2/3}+1704 R^{10} (R^2+3)^{2/3}\\
&\quad -3641 R^6 (R^2+3)^{2/3}-11520 R^2 (R^2+3)^{2/3}+120 R^{12} ((R^4-9)^{2/3}\\
&\quad\times \sqrt[3]{R^2-3}+3 (R^2+3)^{2/3})+17280 (7 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +2 (R^2+3)^{2/3})-8 R^8 (403 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+639 (R^2+3)^{2/3})\\
&\quad +3 R^4 (10587 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+3641 (R^2+3)^{2/3})) \\
&\quad\times{}_2F_1(-\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3}) \sqrt[3]{R^2-3}
+\Big(35994 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad -9858 (R^2+3)^{2/3}+R^2 \Big(22585 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+3641 (R^2+3)^{2/3}\\
&\quad +R^2 \big(-5008 (R^4-9)^{2/3} \sqrt[3]{R^2-3}-6449 (R^2+3)^{2/3}\\
&\quad -8 R^2 (343 (R^4-9)^{2/3} \sqrt[3]{R^2-3}+213 (R^2+3)^{2/3}\\
&\quad +3 R^2 (-10 (R^4-9)^{2/3} \sqrt[3]{R^2-3}
 +5 R^4 (R^2+3)^{2/3}-81 (R^2+3)^{2/3}\\
&\quad\times \sqrt[3]{R^2-3}+(R^2+3)^{2/3}))\big)\Big)\Big) R^2 
 +5760 (7 (R^4-9)^{2/3} \sqrt[3]{R^2-3}\\
&\quad +2 (R^2+3)^{2/3})) \, _2F_1(\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3}) (R^2-3)^{4/3}\\
&\quad +\sqrt[3]{R^2+3} ((11520 (R^2-3)^{2/3}+R^2 (-31761 (R^4-9)^{2/3}
  \sqrt[3]{R^2+3}\\
&\quad +10923 (R^2-3)^{2/3}+R^2 (3641 (R^2-3)^{2/3}+8 R^2 (403 (R^4-9)^{2/3} \\
&\quad\times \sqrt[3]{R^2+3}+15 R^6 (R^2-3)^{2/3}-213 R^2 
 (R^2-3)^{2/3}-639 (R^2-3)^{2/3}\\
&\quad +15 R^4 (3 (R^2-3)^{2/3}-\sqrt[3]{R^2+3} (R^4-9)^{2/3}))))) R^2 \\
&\quad +17280 (2 (R^2-3)^{2/3}-7 \sqrt[3]{R^2+3} (R^4-9)^{2/3})) \\
&_2F_1(-\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3}) 
 (R^2-3)-(R^2+3)^{4/3} ((35994 (R^4-9)^{2/3} \\
&\quad\times \sqrt[3]{R^2+3}+9858 (R^2-3)^{2/3}
 +R^2 (-22585 (R^4-9)^{2/3} \sqrt[3]{R^2+3}\\
&\quad +3641 (R^2-3)^{2/3}+R^2 (-5008 (R^4-9)^{2/3} 
 \sqrt[3]{R^2+3}+6449 (R^2-3)^{2/3}\\
&\quad +8 R^2 (343 (R^4-9)^{2/3} \sqrt[3]{R^2+3}-213 (R^2-3)^{2/3}\\
&\quad +3 R^2 (10 (R^4-9)^{2/3} \sqrt[3]{R^2+3}-81 (R^2-3)^{2/3}
 +5 R^2 (-(R^4-9)^{2/3} \\
&\quad\times \sqrt[3]{R^2+3}+R^2 (R^2-3)^{2/3}+(R^2-3)^{2/3})))))) R^2
 +5760 (2 (R^2-3)^{2/3}\\
&\quad -7 \sqrt[3]{R^2+3} (R^4-9)^{2/3})) 
 \, _2F_1(\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\bigg)\\
&\div 2213120 R (R^2-3)^{2/3} (R^2+3)^{2/3} (R^4-9)^{2/3}; 
\end{align*}
\begin{align*}
g_6&= \pi  \Big(2 (29 R^2 \sqrt[3]{R^2+3}-87 \sqrt[3]{R^2+3}
 -6 R^6 \sqrt[3]{R^2+3}+78 (R^2-3)^{2/3}\\
&\quad\times \sqrt[3]{R^4-9}+2 R^4 (9 \sqrt[3]{R^2+3}-2 (R^2-3)^{2/3} \sqrt[3]{R^4-9})) R^2\\
&\quad\times {}_2F_1(-\frac{2}{3},\frac{1}{2};1;-\frac{6}{R^2-3})
 +2 (29 R^2 \sqrt[3]{R^2-3}+87 \sqrt[3]{R^2-3}\\
&\quad -6 R^6 \sqrt[3]{R^2-3}+78 (R^2+3)^{2/3} \sqrt[3]{R^4-9}-2 R^4 (9 \sqrt[3]{R^2-3}\\
&\quad +2 (R^2+3)^{2/3} \sqrt[3]{R^4-9})) R^2 \, _2F_1(-\frac{2}{3},\frac{1}{2};1;\frac{6}{R^2+3})\\
&\quad +4 (R^2+3) (8 R^2 \sqrt[3]{R^2+3}+3 R^6 \sqrt[3]{R^2+3}-24 (\sqrt[3]{R^2+3} \\
&\quad +(R^2-3)^{2/3}\sqrt[3]{R^4-9})+R^4 (2 (R^2-3)^{2/3} \sqrt[3]{R^4-9}-9 \sqrt[3]{R^2+3})) \\
&\quad\times{} _2F_1(\frac{1}{3},\frac{1}{2};1;-\frac{6}{R^2-3})+4 (R^2-3) (8 R^2 \sqrt[3]{R^2-3}\\
&\quad +3 R^6 \sqrt[3]{R^2-3}+R^4 (9 \sqrt[3]{R^2-3}+2 (R^2+3)^{2/3} \sqrt[3]{R^4-9})\\
&\quad +24 (\sqrt[3]{R^2-3}-(R^2+3)^{2/3} \sqrt[3]{R^4-9})) 
 \, _2F_1(\frac{1}{3},\frac{1}{2};1;\frac{6}{R^2+3})\Big)\\
&\div 1760 R \sqrt[3]{R^4-9};
\end{align*}
\begin{align*}
g_7&= 3 \pi  \Big(3 (R^2+3) (2 R^8-11 R^4+64) (R^2-3)^{2/3} 
 \, _2F_1(-\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3})\\
&\quad +3 (R^2+3)^{2/3} (2 R^8-11 R^4+64) (R^2-3) 
 \, _2F_1(-\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\\
&\quad -(6 R^8+12 R^6-9 R^4+6 R^2+64) (R^2-3)^{5/3} 
 \, _2F_1(\frac{1}{2},\frac{2}{3};1;\frac{6}{R^2+3})\\
&\quad -(R^2+3)^{5/3} (6 R^8-12 R^6-9 R^4-6 R^2+64) 
 \, _2F_1(\frac{1}{2},\frac{2}{3};1;-\frac{6}{R^2-3})\Big)\\
&\div 2912 R (R^4-9)^{2/3};
\end{align*}
where $_2F_1(a,b,c,z)$ is the hypergeometric function which has the series expansion 
$$
\sum_{k=0}^{+\infty}\frac{(a)_k(b)_k}{(c)_k}\,\frac{z^k}{k!},
$$
with
\[
(a)_k =  \begin{cases}
1 & \text{if } k=0; \\
a(a+1)(a+2)\cdots(a+k-1) & \text{if } k> 0.
\end{cases}
\]

\section{Averaging functions $g_{Mi}(R), \;i\in\{1,\ldots,4\}$}
\label{appendix2}

\begin{gather*}
g_{M1}= -\frac{R}{16}-\frac{R^5}{108}-\frac{R^9}{486}
 -\frac{3}{2 R \sqrt{R^4-9}}+\frac{R^{11}}{486 \sqrt{R^4-9}};\\
g_{M2}= -\frac{R}{12}-\frac{R^5}{54}+\frac{\sqrt{R^4-9}}{6 R}
 +\frac{1}{54} \sqrt{R^4-9} R^3;\\ 
g_{M3}= -\frac{R}{6}+\frac{\sqrt{R^4-9}}{6 R};\quad
g_{M4}= \frac{3 R}{4}.\\
\end{gather*}

\subsection*{Acknowledgements}
The first author is supported by a Ci\^{e}ncia sem Fronteiras-CNPq grant
number 201002/ 2012-4. The second author is partially supported by a
MINECO grant MTM2013-40998-P, an AGAUR grant
number 2014SGR-568, the grants FP7-PEOPLE-2012-IRSES 318999 and
316338, the MINECO/fEDER grant UNAB13-4E-1604, and a CAPES grant
number 88881.030454/2013-01 from the program CSF-PVE.

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\end{document}