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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 109, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/109\hfil Limit cycles bifurcated from a center]
{Limit cycles bifurcated from a center in a three dimensional
system}

\author[B. Sang, B. Fer\v{c}ec, Q.-L. Wang \hfil EJDE-2016/109\hfilneg]
{Bo Sang,  Brigita Fer\v{c}ec, Qin-Long Wang}

\address{Bo Sang (corresponding author) \newline
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China.
\newline 
Guangxi Education Department Key Laboratory of Symbolic Computation and 
Engineering Data Processing, Hezhou University, Hezhou 542899, China}
\email{sangbo\_76@163.com}

\address{Brigita Fer\v cec \newline
Faculty of Energy Technology, University of Maribor,
Ho\v{c}evarjev trg 1, 8270 Kr\v{s}ko, Slovenia}
\email{brigita.fercec@gmail.com}

\address{Qin-Long Wang \newline
School  of Science, Hezhou University, Hezhou 542800, China}
\email{wqinlong@163.com}

\thanks{Submitted September 25, 2015. Published April 26, 2016.}
\subjclass[2010]{34C05, 34C07}
\keywords{Focal value; limit cycle; Hopf bifurcation; center}

\begin{abstract}
 Based on the pseudo-division algorithm, we introduce a method for
 computing focal values of a class of 3-dimensional autonomous systems.
 Using the ${\epsilon}^{1}$-order focal values computation, we determine
 the number of limit cycles bifurcating from each component of the center
 variety (obtained by Mahdi et al). It is shown that at most four limit
 cycles can be bifurcated from the center with identical quadratic
 perturbations and that the bound is sharp.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks


\section{Introduction}

Many real world  phenomena  can be modeled  by autonomous systems of the
form
\begin{equation}\label{eq1}
\frac{d\mathbf{x}}{dt}=\mathbf{f}(\mathbf{x}), \quad
\mathbf{x}\in \mathbb{R}^{m},
\end{equation}
where $\mathbf{f}: D \to \mathbb{R}^{m} $ is a smooth function
and $D$ an open connected subset of $\mathbb{R}^{m}$. A limit
cycle of the system is a periodic orbit which is isolated among
periodic orbits. Limit cycles may be used to model the behavior
of many real-world oscillator systems
of great importance (see \cite{Fi06, mr, mul, Rt1989, s15}). 
The study of limit cycles was initiated by Poincar\'{e} \cite{p1886}.
Further research was perhaps motivated by Hilbert's 16th problem. 
A fundamental question in these studies
is the determination of upper bounds, $H_{n}$, for the number
of limit cycles in planar polynomial vector fields of degree $n$
and their relative position. Although the problem
was formulated more than a hundred years ago, it is not yet solved
even for planar quadratic systems. Moreover, it is unknown even
whether a uniform upper bound exists (see \cite{gine07,Rouss}).

An essential part of the problem, called the local 16th Hilbert problem 
\cite{Fran}, is the investigation of the number of limit cycles 
bifurcated from singular points, i.e., the cyclicity of singular points.
The concept of cyclicity was introduced by Bautin in \cite{Bautin},
 where he showed that in  antisaddles of quadratic systems
  at most three small-amplitude limit cycles
can bifurcate out of one equilibrium point. Bautin's work is
important not only because of the bound that it provides, but also
because of the approach it gives to the study of the problem of
cyclicity in any polynomial system. Specifically, Bautin showed
that the cyclicity problem in the case of a simple focus or center
could be reduced to the problem of finding a basis for the ideal
of focal values.  Bautin's approach  is described in detail and
further developed in \cite{IY,Rouss}. The cyclicity problem for
some families of polynomial systems was treated also in
\cite{ch05, Fran, han07, sib65, yak95, zol94}.

Higher-dimensional vector fields may not only exhibit limit
cycles, but also may co-exist with chaotic dynamics. For
results on limit cycles for higher-dimensional vector fields
see \cite{bb, han15, li08} and more references therein.
Llibre et al.\cite{li08} studied the limit cycles of polynomial
vector fields in $\mathbb{R}^3$ which bifurcate from three
different kinds of two dimensional centers (non-degenerate and
degenerate). Buzzi et al. \cite{bb} studied the maximal number of
limit cycles that can bifurcate from a periodic orbits of the
linear center in $\mathbb{R}^4$. Han and Yu \cite{han15} showed
that perturbing a simple quadratic system in $\mathbb{R}^3$ with
a center-type equilibrium point can yield at least 10
small-amplitude limit cycles around an equilibrium point.

The computation of focal values (focus values, Lyapunov constants)
plays an important role in the study of the center-focus problem and
small-amplitude limit cycles arising in degenerated Hopf
bifurcations (see \cite{bvl, ch05, gine12a, gine12b,
gy09,han15,k04,lv,mrs,s15,t14,yu13,w10} and references therein).
For the definition and computation of focal values in
3-dimensional systems, see \cite{yu98,w10} and the second part of
this paper.

We consider the general $n$-dimensional system
\begin{equation}\label{eq10}
{\frac{\mathbf{d{x}}}{d{t}}={\mathbf{f}_1(\mathbf{x},\mathbf{p}_{k_1})+\epsilon
\mathbf{f}_2(\mathbf{x}, \mathbf{p}_{k_2})}},
\end{equation}
(which is an integrable system for $\epsilon=0$) associated with a
Hopf bifurcation, where
$\mathbf{p}_{k_1}=(p_1,p_2,\dots,p_{k_1})$ is the system
parameter and $\mathbf{p}_{k_2}=(p_{k_1+1},\dots,p_{k-1})$ is the
perturbation parameter. Since the corresponding Hopf equilibrium
point is a center for the flow on the center manifold when
$\epsilon=0$, the $k$th focal values of system \eqref{eq10} can be
written in the form of
\begin{equation}
V_k=\tilde{v}_{k,1}(\mathbf{p})\epsilon+\tilde{v}_{k,2}(\mathbf{p}){\epsilon}^2
+\tilde{v}_{k,3}(\mathbf{p}){\epsilon}^3+\dots, \quad k=1,2,\dots,
\end{equation}
where $\tilde{v}_{k,m}$ are said to be ${\epsilon}^{m}$-order
focal values (see \cite{yu13}). For $|{\epsilon}|>0$
sufficiently small, we may use $\tilde{v}_{k,1}$ to determine
the number of small-amplitude limit cycles of system \eqref{eq10}
bifurcated from the equilibrium point.


\begin{lemma}[\cite{han15}] \label{lem1}
Assume that at $\mathbf{p}=\mathbf{p}_c=(p_{1c} , p_{2c} , \dots ,
p_{(k-1)c} )$, the $\epsilon^{1}$-order focal values of system
\eqref{eq10} satisfy
$$
\tilde{v}_{j,1}(\mathbf{p}_c)=0,\quad
j=1,2,\dots,k-1;\;\tilde{v}_{k,1}(\mathbf{p}_c)\neq {0}
$$ 
and
\begin{equation}\label{bigcond}
\operatorname{Rank} \Big[\frac {D(\tilde{v}_{1,1}, \tilde{v}_{2,1},
\dots,\tilde{v}_{k-1,1})} {D (p_1, p_2,  \dots,
p_{k-1})}\Big]_{\mathbf{p}=\mathbf{p}_c}=k-1. 
\end{equation} 
Then, proper perturbations can be made to the parameters $p_1, p_2,
\dots , p_{k-1}$ around the critical point $\mathbf{p}_c$ to
generate $k-1$ small-amplitude limit cycles in the vicinity of the
Hopf equilibrium point.
\end{lemma}

In fact, this Lemma was first presented in \cite{han15}  without
proof. For completeness and convenience, we  give a proof
using the  basic idea found in \cite{han04}.

\begin{proof}
Proving the existence of $k-1$ small-amplitude limit cycles
near the equilibrium point for system \eqref{eq10} is equivalent
to proving  that the amplitude equation of the normal form (expressed in
polar coordinates) of the system up to $k$th order, given by
\begin{align*}
\frac{dr}{dt}
&={r^3}(V_1+V_2r^2+\dots+V_kr^{2k-2})\\
&={r^3}\sum_{j=1}^{k}{\Big(\sum_{s=1}^{\infty}{\tilde{v}_{j,s}
 (\mathbf{p}){\epsilon}^{s}}  \Big){(r^2)}^{j-1}}\\
&= {\epsilon}{r^3}\sum_{j=1}^{k}{\tilde{v}_{j,1}(\mathbf{p})
 }{(r^2)}^{j-1}+o(\epsilon)\\
&= {\epsilon}{r^3}\Big(\sum_{j=1}^{k}{\tilde{v}_{j,1}(\mathbf{p})
 }{(r^2)}^{j-1}+o(1)\Big),\quad \epsilon\to{0},
\end{align*}
has $k-1$ small positive zeros for ${r^2}$, where
$\mathbf{p}=(\mathbf{p}_{k_1},\mathbf{p}_{k_2})=(p_1 , p_2 , \dots ,
p_{k-1} )$. Let ${\rho}=r^2$. By the implicit function theorem,
one can reduce the problem to the existence of $k-1$ small
positive simple zeros for $\rho$ in the algebraic
equation
\begin{equation}\label{alg}
\tilde{v}_{k,1}(\mathbf{p}){\rho}^{k-1}+\tilde{v}_{k-1,1}(\mathbf{p}){\rho}^{k-2}
+\dots+\tilde{v}_{1,1}(\mathbf{p})=0.
\end{equation}
By the conditions given in \eqref{bigcond}, there exists a
$\mathbf{p}=(p_1 , p_2 , \dots , p_{k-1} )$ in a neighborhood
of  $\mathbf{p}_c$, such that
\[
|\tilde{v}_{1,1}(\mathbf{p})|\ll |\tilde{v}_{2,1}(\mathbf{p})|\ll
|\tilde{v}_{3,1}(\mathbf{p})|\ll \dots \ll
|\tilde{v}_{k,1}(\mathbf{p})|\ll {1},\quad
\tilde{v}_{j,1}(\mathbf{p})\tilde{v}_{j+1,1}(\mathbf{p})<{0},
\]
for $j=1,2,\dots k-1$,
which ensures the existence of $k-1$ positive simple zeros for 
\eqref{alg}. Hence, $k-1$ small-amplitude limit cycles can
bifurcate from the equilibrium point. This completes the proof.
\end{proof}

Mulholland \cite{mul} studied the behavior of the solutions of the
third-order non-linear differential equation
\begin{equation}\label{new1}
{\frac {{d}^3x}{{d}{t}^3}}+F(r){\frac
{{d}^{2}x}{{d}{t}^{2}}}+F(r){\frac {{d}x}{{d}{t}}}+x=0,
\end{equation}
in which
\begin{equation}\label{new2}
F(r)=1-\epsilon f(r), \quad f(r)=1-r^2,\quad   r^2=x^2+({\frac
{{d}x}{{d}{t}}})^2+({\frac {{d}^{2}x}{{d}{t}^{2}}})^2.
\end{equation}
Here $F(r)$ represents a central restoring force, which has
important applications in modern control theory. For this equation
with small non-linearities, the existence of a limit cycle is
established by a fixed point technique, the approach to the limit
cycle is approximated by averaging methods, and the periodic
solution is harmonically represented by perturbation.

Mahdi et al \cite{mah13, mah} investigated the center-focus
problem of a third-order differential equation of the form
\begin{equation}\label{eq12}
{\frac {{d}^3u}{{d}{t}^3}}={\frac
{{d}^{2}u}{{d}{t}^{2}}}+{\frac {{d}u}{{d}{t}}}+u+f(u,{\frac
{{d}u}{{d}{t}}},{\frac {{d}^{2}u}{{d}{t}^{2}}}),
\end{equation}
where $f(u,{\frac {{d}u}{{d}{t}}},{\frac {{d}^{2}u}{{d}{t}^{2}}})$
is an analytic function starting with quadratic terms. It can be
reduced to a system of first order differential equations
\begin{equation}\label{eq13} 
\begin{gathered} 
\frac{{du}}{{dt}}=-v+h(u,v,w),\\
\frac{{dv}}{{dt}}=u+h(u,v,w),\\
\frac{{dw}}{{dt}}=-w+h(u,v,w),
\end{gathered} 
\end{equation}
where $h(u,v,w)=f(-u+w,v-w,u+w)/2$, which is equivalent to
\eqref{eq12}, see \cite{mah13}. The center conditions on the local
center manifold for system \eqref{eq13} with
\begin{equation}\label{eq14}
h(u,v,w)=a_1u^2+a_2v^2+a_3w^2+a_4uv+a_5uw+a_6vw
\end{equation}
were obtained in \cite{mah}. For center conditions of some other
polynomial differential systems in $\mathbb{R}^3$ we refer to
\cite{alg, gine16, cv15}.



\begin{lemma}[\cite{mah}] \label{lem2}
The system \eqref{eq13} with $h(u,v,w)$ as in \eqref{eq14} admits
a center on the local center manifold (for the equilibrium point
at the origin) if and only if one of the following holds:
\begin{itemize}
\item[(1)] $a_1=a_2=a_4=0$;
\item[(2)] $a_1-a_2=a_3=a_5=a_6=0$;
\item[(3)] $a_1+a_2=a_3=a_5=a_6=0$;
\item[(4)] $a_1+a_2=2a_2-a_3+a_6=a_3-a_4-2a_5=2a_4+3a_5+a_6=0$;
\item[(5)] $2a_1-a_6=2a_2+a_5=2a_3-a_5+a_6=a_4+a_5+a_6=0$;
\item[(6)] $a_1-a_2=2a_2+a_6=a_4=a_5+a_6=0$;
\item[(7)] $2a_1+a_2=2a_2+a_6=4a_3+5a_6=a_4=2a_5-a_6=0$.
\end{itemize}
\end{lemma}

For system \eqref{eq13}, we consider the perturbed system (with
identical nonlinearities for the three components) of the form
\begin{equation}\label{eq15} 
\begin{gathered} 
\frac{{du}}{{dt}}=-v+h(u,v,w)+\epsilon h_1(u,v,w),\\
\frac{{dv}}{{dt}}=u+h(u,v,w)+\epsilon h_1(u,v,w),\\
\frac{{dw}}{{dt}}=-w+h(u,v,w)+\epsilon h_1(u,v,w),
\end{gathered}
\end{equation}
where
\begin{equation}\label{eq16}
h_1(u,v,w)=b_1u^2+b_2v^2+b_3w^2+b_4uv+b_5uw+b_6vw.
\end{equation}
The purpose of this paper is prove that four is an  upper bound for
 small amplitude limit cycles that bifurcate from the
origin of the system \eqref{eq15} when one of conditions in 
Lemma \ref{lem2} holds.

The remainder of the paper is organized as follows.  Based on a
previously developed algorithm of Sang \cite{s14} for
2-dimensional systems, in Section 2, we introduced a new algorithm
for computing focal values of 3-dimensional systems. In Section 3,
we prove that at most four small-amplitude limit cycles can
bifurcate out of the center based on the analysis of
$\epsilon^{1}$-order focal values.


\section{Algorithm for computing focal values}

In this section, we present an algorithm for computing focal
values of a class of 3-dimensional differential systems
\begin{equation}\label{eq2} 
\begin{gathered} 
\frac{{du}}{{dt}}=-v+\sum_{j+k+s=2}^{\infty}\tilde{a}_{{jks}}{u}^{j}{v}^{k}w^{s},\\
\frac{{dv}}{{dt}}=u+\sum_{j+k+s=2}^{\infty}\tilde{b}_{{jks}}{u}^{j}{v}^{k}w^{s},\\
\frac{{dw}}{{dt}}=-dw+\sum_{j+k+s=2}^{\infty}\tilde{c}_{{jks}}{u}^{j}{v}^{k}w^{s},
\end{gathered} 
\end{equation}
which is analytic in the neighborhood of the origin and 
$u,v,w,t \in\mathbb{R}$, $d>0$, 
$\tilde{a}_{jks},\tilde{b}_{jks},\tilde{c}_{jks}\in
\mathbb{R}$, $j,k,s\in \mathbb{N}\cup\{0\}$.

By means of the transformation
\begin{equation}\label{eq3}
u=\frac{1}{2}(x+y),\quad  v=\frac{\texttt{i}}{2}(-x+y),\quad  w=z,  \quad
t={-\texttt{i}}t_1,
\end{equation}
where $\texttt{i}=\sqrt{-1}$, system \eqref{eq2} can be
transformed into the following complex system
\begin{equation}\label{eq4} 
\begin{gathered} 
\frac{{dx}}{{dt_1}}=x+\sum_{j+k+s=2}^{\infty}a_{{jks}}{x}^{j}{y}^{k}z^{s}=X(x,y,z),\\
\frac{{dy}}{{dt_1}}=-y+\sum_{j+k+s=2}^{\infty}b_{{jks}}{x}^{j}{y}^{k}z^{s}=Y(x,y,z),\\
\frac{{dz}}{{dt_1}}=\texttt{i}dz+\sum_{j+k+s=2}^{\infty}c_{{jks}}{x}^{j}{y}^{k}z^{s}=Z(x,y,z).
\end{gathered}
\end{equation}


\begin{lemma}[\cite{w10}] \label{lem3}
For system \eqref{eq4}, there exists a formal power series
\begin{equation}\label{eq5}
F(x,y,z)=xy+\sum_{s=3}^{\infty}\sum_{k=0}^{s}
\sum_{j=0}^{s-k}{B_{s,k,j}}{x}^{s-k-j}{y}^{k}z^{j},
\end{equation}
such that
\begin{equation}\label{eq6}
\frac{dF}{dt_1}\Big|_{\eqref{eq4}}=\frac{{\partial F}}{{\partial
x}}{X} + \frac{{\partial F}}{{\partial y}}{Y}+\frac{{\partial
F}}{{\partial z}}{Z}=\sum_{n=1}^{\infty} W_{n}(xy)^{n+1},
\end{equation}
where ${B_{s,k,j}}$ are determined by the recursive formula (see
\cite{w10}) with ${B_{2k,k,0}}=0$. The terms $W_{n}$ are called the
$n$th singular point values of system \eqref{eq4} at the origin.
\end{lemma}

\begin{lemma}[\cite{w10}] \label{lem4}
For any positive natural number $n$, the following assertion
holds:
\begin{equation}\label{eq7}
V_{n}=\texttt{i}\pi W_{n} \mod{\langle{W_1,W_2,\dots,W_{n-1}}\rangle },
\end{equation}
where $V_{n}$ is the $n$-th focal value of system \eqref{eq2}, and
$W_{j}$ is the $j$th singular point value of system \eqref{eq4},
$j=1,2,\dots,n$.  More precisely, when
$W_1=W_2=\dots=W_{n-1}=0$, the following assertion holds:
\begin{equation}\label{eq8}
V_{n}=\texttt{i}\pi W_{n}.
\end{equation}
\end{lemma}

Note that the equilibrium point of system \eqref{eq2} at the
origin is either a center or a fine focus for the flow on the
local center manifold (see \cite{w10}). The problem of
distinguishing between these two cases is called the center
problem. The origin is said to be a fine focus of order $k$ 
($k\in \mathbb{N}$) if $V_k$ is the first non-zero focal value. In this
case at most $k$ limit cycles can be bifurcated from the fine
focus; these limit cycles are called small-amplitude limit cycles.
The origin is a center when all the focal values are zero.

Grouping like terms in the second expression of \eqref{eq6}, we
obtain
\begin{equation}\label{eq9}
\begin{aligned}
&\frac{{\partial F}}{{\partial x}}{X} +
\frac{{\partial F}}{{\partial y}}{Y}+\frac{{\partial F}}{{\partial z}}{Z} \\
& = \sum_{s= 3}^{2n+1} \sum_{k = 0}^{s}
\sum_{j = 0}^{s-k} {{f_{s,k,j}}{x^{s
-k-j}}{y^k}{z^j}} \\
& \quad +\sum_{\substack{k,j\geq {0}\\k+j\leq
2n+2\\(k,j)\neq{(n+1,0)}}}{{f_{2n +
2,k,j}}{x^{2n+2-k-j}}{y^k}{z^j}} + {D_n}{(xy)^{n + 1}} +\dots,
\end{aligned}
\end{equation}
where $D_{{n}},f_{{s,k,j}},f_{2n+2,k,j}$ can be considered as
linear polynomials of variables $B_{s,k,j}$ with coefficients
formed from $a_{jks},b_{jks},c_{jks}$.


Suppose that $W_1=W_2=\dots=W_{n-1}=0$. In this case, we are
in a position to develop the algorithm for computing the $n$th
singular point value $W_{n}$ of system \eqref{eq4} based on
pseudo-divisions. It is remarkable that the idea behind it is
similar to the situation of 2-dimensional systems described by
Sang in \cite{s14}. When computing the $n$th singular point value
$W_{n}$, the coefficients $f_{{s,k,j}},f_{2n+2,k,j}$ have to be
zero. Thus in order to eliminate variables $B_{{s,k,j}}$ from
$D_{n}$, we use successive pseudo-divisions: first choosing an
adequate  variable order of $B_{s,k,j}$; then rearranging some
polynomials $f_{{s,k,j}},f_{2n+2,k,j}$ to get a triangular set
$TS_{n}$, next  performing successive pseudo-division of
$D_{n}+\xi$ by $TS_{n}$  to get the pseudo-remainder $R_{n}$,
and finally  expressing the $n$th singular point value ${W_n}$ as
$\frac{{{R_n}}}{{{\operatorname{coeff}}({R_n},\xi)}}-\xi$, where
${\operatorname{coeff}}({R_n},\xi)$ is the coefficient of $\xi$ in the polynomial
$R_{n}$, and $\xi$ is a dummy variable.  The termination of the
algorithm is trivial because the number of variables $B_{{s,k,j}}$
is finite when $n$ is fixed.

Recalling Lemma \ref{lem4}, once $W_{n}$ is returned, the $n$th
focal value $V_{n}$ of system \eqref{eq2} can be obtained from
relation \eqref{eq8}.  Thus, the algorithm can be modified for
computing the $n$th focal value $V_{n}$ of system \eqref{eq2}.

\section{Four limit cycles obtained
from $\epsilon$-order focal values}

Suppose that the condition (1) in Lemma \ref{lem2} holds.  It is easy to
obtain the $\epsilon^{1}$-order focal values of system
\eqref{eq15} (up to a positive constant multiple):
\begin{align*}
\tilde{v}_{1,1}
&= {\frac {13 a_5b_1}{20}}+{\frac {7 a_5b_2}{20}}-{\frac{1}{20}} a_5b_4+{\frac
{11 a_6b_1}{20}}+{\frac {9 a_6b_2}{20}}+{\frac
{3 a_6b_4}{20}},\\
\tilde{v}_{2,1}
&= {\frac
{473 {a_5}^3b_1}{1080}}+{\frac {371 {a_5}^3b_2}{1080}}-{\frac {2 {a_5}^3b_4}{27}}+{\frac
{409 a_{{6 }}{a_5}^{2}b_1}{1080}}+{\frac
{71 {a_5}^{2}a_6b_2 }{360}}\\
&\quad +{\frac {31 a_5{a_6}^{2}b_1}{360}}-{\frac {a_5
{a_6}^{2}b_2}{120}}+{\frac{{a_6}^3b_1}{120}}-{\frac{1}{40}}\, {a_6}^3b_2,\\
\tilde{v}_{3,1}
&= -{\frac {49 {a_5}^5b_1}{2720}}+{\frac {2661 a_6{a_{{5}
}}^4b_1}{19040}}-{\frac
{3981 {a_5}^3{a_6}^{2}b_1 }{3332}}-{\frac
{156509 {a_5}^{2}{a_6}^3b_1}{13328}}\\
&\quad -{ \frac{667381 {a_5}^{2}{a_6}^3b_2}{93296}}
 -{\frac{ 5379919 a_5{a_6}^4b_1}{133280}}
 -{\frac{142871 a_5 {a_6}^4b_2}{5831}}
 -{\frac {25080959 {a_6}^5b_1}{ 932960}}\\
&\quad -{\frac{2066875 {a_6}^5b_2}{93296}}
 -{\frac {342555 {a_6}^5b_4}{46648}},\\
\tilde{v}_{m,1}
&= 0, \quad  m\ge 4,
\end{align*}
where the quantity $\tilde{v}_{k,1}$ is reduced with respect to the
Gr\"{o}bner basis of $\{\tilde{v}_{j,1}: j<k\}$.

\begin{theorem} \label{thm1}
Based on the analysis of $\epsilon$-order focal values for the
case $(1)$,  the perturbed system \eqref{eq15} can have at most
two small-amplitude limit cycles bifurcated from the center, and
the bound is sharp.
\end{theorem}

\begin{proof}
For this case, an appropriate selection of $(a_5,b_1,b_2)$
for system \eqref{eq15} is:
\begin{equation}
a_5=0,\quad b_1=-{\frac{3}{14}}b_4,\quad
b_2=-{\frac{1}{14}}b_4, \quad b_4, a_6\neq {0},
\end{equation}
which implies 
$$
\tilde{v}_{1,1}=\tilde{v}_{2,1}=0, \quad
\tilde{v}_{3,1}=-{\frac {11 b_4{a_6}^5}{38080}}\neq{0},
$$ 
and the rank of Jacobian matrix (evaluated at the critical
point) of $\tilde{v}_{1,1}, \tilde{v}_{2,1}$ with respect to
$b_1,b_2$ is two, hence by Lemma \ref{lem1}, the perturbed system
\eqref{eq15} can have at most two small-amplitude limit cycles
bifurcated from the center, and the bound is sharp.
\end{proof}


Now, we assume that condition (2) in Lemma \ref{lem2} holds. It is easy to
obtain the $\epsilon^{1}$-order focal values of system
\eqref{eq15} (up to a positive constant multiple):
\begin{align*}
\tilde{v}_{1,1}
&= -a_2b_1+a_2b_2+b_5a_2+b_6a_2-\frac{1}{20} b_{
{5}}a_4+{\frac {3 b_6a_4}{20}},\\
\tilde{v}_{2,1}
&= {\frac {326 b_1{a_2}^3}{405}}-{\frac {326 b_2{a_2}
^3}{405}}+{\frac {86 b_5{a_2}^3}{81}}-{\frac
{704 b_{{6} }{a_2}^3}{405}}-4 b_{{3}}{a_2}^3+{\frac
{107 b_1{a_{ {2}}}^{2}a_4}{135}}\\
&\quad -{\frac{107 b_2{a_2}^{2}a_4}{135} }
 -{\frac{302 a_4{a_2}^{2}b_5}{405}}
 +{\frac{2}{5}} b_{{3}}a_4{a_2}^{2}
 -{\frac {4 b_1{a_4}^{2}a_2}{45}}
 +{\frac{4 b _2{a_4}^{2}a_2}{45}} \\
&\quad +{\frac{43 a_2{a_4}^{2}b_{{5 }}}{135}}
 -{\frac{53 b_{{3}}{a_4}^{2}a_2}{150}}
 +{\frac {{a_{{4}}}^3b_5}{180}}
 +{\frac {{a_4}^3b_{{3}}}{300}},\\
\tilde{v}_{3,1}
&= -{\frac {1376747429 {a_2}^5b_1}{1275}}
 +{\frac {1376747429{a_2}^5b_2}{1275}}
 -{\frac{1917866 {a_2}^5b_{{3}}}{85 }}\\
&\quad +{\frac {185356263 b_5{a_2}^5}{170}}
 +{\frac {913355709 b _6{a_2}^5}{850}}
 -{\frac {68663387 {a_2}^4a_4b_{{1 }}}{1275}}\\
&\quad +{\frac {68663387 {a_2}^4a_4b_2}{1275}} 
 +{ \frac{5211337 {a_2}^4a_4b_{{3}}}{2550}}
 +{\frac {33139013{a_2}^4a_4b_6}{150}}\\
&\quad -{\frac {55023 {a_2}^3{a_{{4} }}^{2}b_1}{34}}
 +{\frac{55023 {a_2}^3{a_4}^{2}b_2}{ 34}}
 -{\frac {50399083 {a_2}^3{a_4}^{2}b_{{3}}}{25500}}\\
&\quad +{ \frac {50559593 {a_2}^3{a_4}^{2}b_6}{5100}}
 -{\frac {24129 {a_2}^{2}{a_4}^3b_1}{1700}} \\
&\quad +{\frac{24129 {a_{{2 }}}^{2}{a_4}^3b_2}{1700}}
 +{\frac{1877003 {a_2}^{2}{a_{ {4}}}^3b_6}{10200}}
 -{\frac{269 a_2{a_4}^4b_1}{ 3400}} \\
&\quad +{\frac{269 a_2{a_4}^4b_2}{3400}}
 +{\frac {31057  a_2{a_4}^4b_6}{20400}}
 +{\frac {87 {a_4}^5b_{{6} }}{13600}},\\
\tilde{v}_{4,1}
&= -{\frac {153971816454597031246 {a_2}^{7}b_1}{491160925125}}
 +{\frac {153971816454597031246 {a_2}^{7}b_2}{491160925125}} \\
&\quad -{\frac {16610753188310104 {a_2}^{7}b_{{3}}}{2518773975}}
 +{\frac {691036797753634751 b_5{a_2}^{7}}{2182937445}}\\
&\quad +{\frac {1309536133953556187 b_6{a_2}^{7}}{4197956625}}
 -{\frac {264798540296377058 {a_2}^{6}a_4b_1}{16936583625}}\\
&\quad +{\frac{264798540296377058 {a_2}^{6}a_4b_2}{16936583625}}
 +{\frac {293602854116874307 {a_2}^{6}a_4b_{{3}}}{491160925125}}\\
&\quad +{\frac {31507609090246881862 {a_2}^{6}a_4b_6}{491160925125}}
 -{\frac {25504331681017132 {a_2}^5{a_4}^{2}b_{{1}}}{54573436125}}\\
&\quad +{\frac{25504331681017132 {a_2}^5{a_4}^{2}b_2}{54573436125}}
 -{\frac{218556152766434059 {a_2}^5{a_4}^{2}b_{{3}}}{377816096250}}\\
&\quad +{\frac{2825095303169559761 {a_{{2}}}^5{a_4}^{2}b_6}{982321850250}}
 -{\frac{27926323546531 {a_2}^4{a_4}^3b_1}{7276458150}}\\
&\quad +{\frac{27926323546531{a_2}^4{a_4}^3b_2}{7276458150}}
 +{\frac{103518014332944689 {a_2}^4{a_4}^3b_6}{1964643700500}}\\
&\quad -{\frac {1454958217433 {a_2}^3{a_4}^4b_1}{163720308375}}
 +{\frac {1454958217433 {a_2}^3{a_4}^4b_{{2}}}{163720308375}}\\
&\quad +{\frac {755866567299851 {a_2}^3{a_4}^4b_6}{1964643700500}}
 +{\frac{265910137 {a_2}^{2}{a_4}^5b_1}{2509123500}}\\
&\quad -{\frac {265910137 {a_2}^{2}{a_4}^5b_{{2}}}{2509123500}}
 -{\frac {529 a_2{a_4}^{6}b_1}{4806750}}
 + {\frac{529 a_2{a_4}^{6}b_2}{4806750}},\\
\tilde{v}_{5,1}
&= \tilde{v}_{5,1}(a_2,a_4,b_5,b_6),\\
\tilde{v}_{m,1}
&= 0, \quad m\ge 6,
\end{align*}
where the quantity $\tilde{v}_{k,1}$ is reduced with respect to the
Gr\"{o}bner basis of $\{\tilde{v}_{j,1}: j<k\}$ and
the expression of $\tilde{v}_{5,1}$ is too lengthy to be
presented here.

\begin{theorem} \label{thm2}
Based on the analysis of $\epsilon$-order focal values for the
case $(2)$, the perturbed system \eqref{eq15} can have at most
four small-amplitude limit cycles bifurcated from the center, and
the bound is sharp.
\end{theorem}

\begin{proof} For this case, an appropriate selection of
$(a_2,b_1,b_3,b_5)$ for system \eqref{eq15} is that
\begin{align*}
a_2&= - 0.124090617911090057250206646846125321120939094367768\text{(cont.)}\\
&\quad 2051487342750 a_4,\\
b_1&= 1.000000000000000000000000000000000000000000000000000000000000001 b_2 \\
&\quad +6.264551641463457593818886043698212586619541984593273\text{(cont.)}\\
&\quad 257643565588 b_6,\\
b_3 &= 1.309260028633156162644799671413719205654316212923457750123345163 b_6,\\
b_5&=  4.614157131800682898459705227984840546895131617055149564347672012 b_6
\end{align*}
with $a_4,b_6\neq {0}$, which implies
\begin{align*}
\tilde{v}_{1,1}&=  1.0\times 10^{-64} a_4b_2-{
1.00\times 10^{-63}} b_6a_4,\\
\tilde{v}_{2,1}&= { 2.0\times 10^{-65}} b_2{a_4}^3-{
3.0\times 10^{-64}} b_{ {6}}{a_4}^3,\\
\tilde{v}_{3,1}&= { 2.0\times 10^{-62}} {a_4}^5b_2-{
1.0\times 10^{-61}} {a_ {{4}}}^5b_6,\\
\tilde{v}_{4,1}&= { 1.00\times 10^{-61}} {a_4}^{7}b_2-{
1.00\times 10^{-60}} { a_4}^{7}b_6,\\
\tilde{v}_{5,1}
&= - 0.00025178105021258923344860574000162257838903488776\text{(cont.)}\\
&\quad 44746869970435709 {a_4}^{9}b_6\neq {0}.
\end{align*}
The errors on
$\tilde{v}_{1,1},\tilde{v}_{2,1},\tilde{v}_{3,1},\tilde{v}_{4,1}$
are due to numerical computation in the step of solving a
12th-degree polynomial. The rank of Jacobian matrix (evaluated at
the critical point) of $\tilde{v}_{1,1}, \tilde{v}_{2,1},
\tilde{v}_{3,1}, \tilde{v}_{4,1}$ with respect to
$a_2,b_1,b_3,b_5$ is four, hence by Lemma \ref{lem1}, the perturbed
system \eqref{eq15} can have at most four small-amplitude limit
cycles bifurcated from the center, and the bound is sharp.
\end{proof}

Imitating the arguments  in the proof of Theorem \ref{thm2}, we obtain
the following result.

\begin{theorem} \label{thm3}
Based on the analysis of $\epsilon$-order focal values for the
cases $(3)-(7)$, the perturbed system \eqref{eq15} can have at
most four small-amplitude limit cycles bifurcated from the center
respectively, and the bound is sharp.
\end{theorem}


\subsection*{Acknowledgements}
This work was supported by a grant from the National Natural
Science Foundation of China (No. 11461021), a grant from Guangxi
Colleges and Universities Key Laboratory of Symbolic Computation
and Engineering Data Processing (No. FH201505), a grant from the
Foundation for Research in Experimental Techniques of Liaocheng
University (No. LDSY2014110),  
a grant from the Scientic Research Foundation of Guangxi Education 
Department (No. ZD2014131), and PhD research startup foundation of 
Liaocheng University.

The authors are very grateful to the referee for the
helpful and valuable comments on this manuscript. The authors
should also be grateful to Dr. A. Mahdi for providing the citation
information of his preprint \cite{mah}.

\begin{thebibliography}{99}


\bibitem{alg} A. Algaba, F. Fern\'{a}ndez-S\'{a}nchez, M. Merino, A. J.
Rodr\'{i}guez-Luis; 
\emph{Centers on center manifolds in the
Lorenz, Chen and L\"{u} systems}, Commun. Nonlinear Sci. Numer.
Simul., {\textbf{19}} (2014), No. 4, 772--775.

\bibitem{bvl} L. Barreira,  C. Valls,  J. Llibre;  
\emph{Integrability and limit cycles of the Moon-Rand system},
Int. J. Nonlin. Mech., \textbf{69} (2015), 129--136.

\bibitem{Bautin} N. N. Bautin;
\emph{On the number of limit cycles which appear with the
variation of coefficients from an equilibrium position of focus or
center type}, Math. Sb.(N.S.), \textbf{30} (1952), 181--196;
 Amer. Math. Soc. Transl., \textbf{100} (1954), 181--196.

\bibitem{bb} C. A. Buzzi, J. Llibre, J. C. Medrado, J. Torregrosa; 
\emph{Bifurcation of limit cycles from a centre in $R^4$ in resonance $1:N$}, 
Dynam. Syst., \textbf{24} (2009), 123--137.

\bibitem{ch05} C. Christopher; 
\emph{Estimating limit cycle bifurcations from
centers}, in: D.M. Wang, Z.M. Zheng,  \emph {Differential
equations with symbolic computation, Trends Math.},
Birkh\"{a}user, Basel, 2005, 23--35.


\bibitem{Fi06} A. Fidlin; 
\emph{Nonlinear oscilations in mechanical engineering}, 
Springer-Verlag, Berlin Heidelberg, 2006.

\bibitem{Fran} J. P. Francoise, Y. Yomdin; 
\emph{Bernstein Inequalities and Applications to
Analytic Geometry and Differential Equations}, Journal of
Functional Analysis, \textbf{146} (1997), 185--205.

\bibitem{gine07} J. Gin\'{e}; 
\emph{On some open problems in planar differential systems
and Hilbert's 16th problem}, Chaos, Solitions and Fractals,
\textbf{31} (2007), 1118--1134.

\bibitem{gine12a} J. Gin\'{e}; 
\emph{Limit cycle bifurcations from a non-degenerate
center},
Appl. Math. Comput., \textbf{218} (2012), 4703--4709.

\bibitem{gine12b} J. Gin\'{e}; 
\emph{Higher order limit cycle bifurcations from non-degenerate
centers}, Appl. Math. Comput., \textbf{{218}}(2012), 8853--8860.

\bibitem{gine16} J. Gin\'{e}, C. Valls;
 \emph{Center problem in the center manifold for
quadratic differential systems in $\mathbb{R}^3$}, J. Symbolic
Comput., \textbf{73} (2016), 250--267.

\bibitem{gy09} M. Gyllenberg, P. Yan; 
\emph{Four limit cycles for a three-dimensional competitive Lotka-Volterra
system with a heteroclinic cycle}, Comput. Math. Appl., \textbf{58} (2009), 649--669.

\bibitem{han04}M. A. Han, Y. Lin, P. Yu; 
\emph{A study on the existence of limit cycles of a
planar system with third-degree polynomials}, Int. J. Bifurcation
and Chaos, \textbf{{14}}(2004), 41--60.

\bibitem{han07} M. A. Han, H. Zang, T. H. Zhang;
 \emph{A new proof to Bautin's theorem}, Chaos, Solitons and Fractals, 
\textbf{31} (2007), 218--223.

\bibitem{han15}M. A. Han, P. Yu; 
\emph{Ten limit cycles around a center-type singular point in a 3-d
quadratic system with quadratic perturbation}, Appl. Math. Lett.,
\textbf{44} (2015), 17--20.


\bibitem{IY} Yu. Il'yashenko, S. Yakovenko; 
\emph{Lectures on analytic differential equations}, Graduate Studies 
in Mathematics, Vol. \textbf{86}, American Mathematical Society, Providence, RI,
2008.

\bibitem{k04} Y. A. Kuznetsov; 
\emph{Elements of Applied Bifurcation Theory}, Springer Verlag, New York, 2004.


\bibitem{li08} J. Llibre, J. Yu, X. Zhang; 
\emph{Limit cycles coming from the perturbation of
2-dimensional centers of vector fields in $\mathbb{R}^3$},
Dynamic Systems and Applications,  \textbf{17} (2008), 625--636.

\bibitem{lv} J. Llibre,  C. Valls; 
\emph{Hopf bifurcation of a generalized Moon-Rand system}, 
Commun. Nonlinear Sci. Numer. Simulat., \textbf{20} (2015), 1070--1077.

\bibitem{mr} F. C. Moon, R. H. Rand; 
\emph{Parametric stiffness control of flexible structures}, in:
 Jet Propulsion Laboratory Publication 85-29, Vol.
\textbf{II}, California Institute of Technology, 1985, 329--342.

\bibitem{mrs} A. Mahdi, V. G. Romanovski, D. S. Shafer; 
\emph{Stability and periodic oscillations in the Moon-Rand systems}, 
Nonlinear Anal.-Real, \textbf{14} (2013), 294--313.

\bibitem{mah13} A. Mahdi; 
 \emph{Center problem for third-order ODEs}, Int. J. Bifurcation and Chaos, 
\textbf{{23}}(2013), 1350078 (11 pages).

\bibitem{mah} A. Mahdi, C. Pessoa, J. D. Hauenstein; 
\emph{A hybrid symbolic-numerical approach to the center-focus problem},  
arXiv preprint, arXiv:1509.00375(2015), 1--18.

\bibitem{mul} R. J. Mulholland; 
\emph{Non-linear oscillations of a third-order differential equation}, 
Int. J. Nonlin. Mech., \textbf{6} (1971), 279--294.

\bibitem{p1886} H. Poincar\'{e}; 
\emph{Sur les courbes d\'{e}finies par une \'{e}quation diff\'{e}rentielle}, Journal
de Math\'{e}ma\-tiques Pures et Appliqu\'{e}es, IV,
\textbf{2} (1886), 151--217.

\bibitem{Rt1989} M. I. Rabinovich, D. I. Trubetskov; 
\emph{Oscillations and waves in linear and nonlinear systems}, 
in: M. Hazewinkel et al (Eds.), Mathematics and its Applications 
(Soviet Series), Vol. \textbf{50}, Kluwer Academic Publishers, Netherlands, 1989.

 \bibitem{Rouss} R. Roussarie; 
\emph{Bifurcation of planar vector fields and Hilbert's sixteenth problem}, 
Progress in Mathematics, Vol. \textbf{{164}}, Birkh\"{a}user Verlag, Basel, 1998.

\bibitem{s14} B. Sang; 
\emph{Center problem for a class of degenerate quartic systems},
 Electron. J. Qual. Theory  Differ. Equ., Vol. \textbf{2014} (2014), No. 74,
 pp. 1--17.


\bibitem{s15} B. Sang,  Q. L. Wang, W. T. Huang;
 \emph{Computation of focal values and stability analysis of
  4-dimensional systems},  Electron. J. Diff. Equ., Vol. \textbf{2015}
(2015), No. 209,  pp. 1--11.

\bibitem{sib65} K. S. Sibirskii; 
\emph{On the number of limit cycles in the neighborhood of a singular point}, 
 Differ. Equ.,  \textbf{{1}} (1965), 36--47.

\bibitem{t14} Y. Tian, P. Yu; 
\emph{Seven limit cycles around a focus point in a simple
three-dimensional quadratic vector field}, Int. J. Bifurcation and
Chaos, \textbf{24} (2014),  1450083 (10 pages).

\bibitem{cv15} C. Valls;  
\emph{Center problem in the center manifold for
quadratic and cubic differential systems in $\mathbb{R}^3$},
Appl. Math. Comput., \textbf{251} (2015), 180--191.

\bibitem{yak95} S. Yakovenko; 
\emph{A geometric proof of the Bautin theorem. Concerning the Hilbert 
Sixteenth Problem}, Advances in Mathematical Sciences, Amer. Math. Soc. 
Transl., \textbf{165} (1995), 203--219.

\bibitem{yu98}P. Yu; 
\emph{Computation of normal forms via a perturbation technique}, 
J. Sound Vib., \textbf{{211}}(1998), 19--38.

\bibitem{yu13} P. Yu, M. A. Han; 
\emph{Eight limit cycles around a center in quadratic hamiltonian system 
with third-order perturbation},
Int. J. Bifurcation and Chaos, \textbf{23} (2013), 1350005 (18 pages).

\bibitem{w10} Q. L. Wang,  Y. R. Liu,  H. B. Chen; 
\emph{Hopf bifurcation for a class of three-dimensional nonlinear
    dynamic systems}, Bull. Sci. Math., \textbf{134} (2010), 786--798.

\bibitem{zol94} H. \.{Z}o{\l}\c{a}dek; 
\emph{On a certain generalization of Bautin's theorem}, Nonlinearity,
 \textbf{7} (1994), 273--279.

\end{thebibliography}



\end{document}