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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 66, pp. 1--11.\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-2018/66\hfil Bifurcation of critical periods of a quintic system]
{Bifurcation of critical periods of a \\ quintic system}

\author[V. G. Romanovski, M. Han, W. Huang \hfil EJDE-2018/66\hfilneg]
{Valery G. Romanovski, Maoan Han, Wentao Huang}

\address{Valery G. Romanovski \newline
Department of Mathematics,
Shanghai Normal University,
Shanghai 200234, China.\newline
Faculty of Electrical Engineering and Computer Science,
University of Maribor,
Smetanova 17, Maribor, SI-2000 Maribor, Slovenia. \newline
Center for Applied Mathematics and Theoretical Physics,
 University of Maribor,
Mladinska 3, SI-2000 Maribor, Slovenia.\newline
Faculty of Natural Science and Mathematics,
University of Maribor,
Koro\v{s}ka cesta 160, SI-2000 Maribor, Slovenia}
\email{valery.romanovsky@uni-mb.si}

\address{Maoan Han (corresponding author) \newline
Department of Mathematics,
Shanghai Normal University,
Shanghai 200234, China. \newline
School of Mathematics Sciences,
Qufu Normal University, Qufu, 273165, China}
\email{mahan@shnu.edu.cn}

\address{Wentao Huang \newline
School of Science,
Guilin University of Aerospace Technology,
Guilin, 541004, China}
\email{huangwentao@163.com}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted March 17, 2017. Published March 13, 2018.}
\subjclass[2010]{34C23, 34C25, 37G15}
\keywords{Critical period; bifurcation; isochronicity; polynomial systems}

\begin{abstract}
 We investigate the critical period bifurcations of the system
 $$
 \dot x = ix + x \bar x ( a x^3 + b x^2 \bar x + \bar x \bar x^2+d \bar x^3)
 $$
 studied in \cite{GLM}. We prove that at most three critical periods
 can bifurcate from any nonlinear center of the system.
\end{abstract}

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

\section{Introduction}\label{s:intro}

Consider a system of ordinary differential equations on $\mathbb{R}^{2}$
of the form
\begin{equation}\label{1}
\begin{gathered}
\dot u = -v + P(u,v), \\
 \dot v = u + Q(u,v),
\end{gathered}
\end{equation}
where $u$ and $v$ are real unknown functions
 and $P$ and $Q$ are polynomials without constant and linear terms. The
singularity at the origin of system \eqref{1} is either a center or a focus.
In a neighborhood of a center the so-called {\it period function} $T(r)$
gives the least period of the periodic solution passing through the point
with coordinates $(u, v) =(r, 0)$ inside the period annulus of the center.

If $T(r) $ is constant in a neighbourhood of the origin, then
 the center at the origin is called isochronous.
For a center that is not isochronous any value $r>0$ for which $T'(r) =0$
is called a critical period.
The problem of critical period bifurcations is aimed on estimation
of the number of critical periods that can arise near the center under small
perturbations. It was investigated for the first time by Chicone and
Jacobs \cite{CJ} in 1989 for quadratic systems and some Hamiltonian systems.
After that, many studies were devoted to the problem
(see, e.g. \cite{CRZ,FLRS,GLY,GV,LLH,RT,RT2,V,WCZ,XW,Yu,ZCZ}
and references given there).
One of difficulties in investigations of this problem is that before studying
the critical periods bifurcation for
a polynomial system one should resolve the center problem for the system,
that is, find all systems in the family with a center at the origin.

Studies of the center problem are usually simpler if one considers
the problem in the complex setting. To perform a complexification
we can make the substitution $x = u+iv$ obtaining from \eqref{1}
the complex differential equation
\begin{equation} \label{7}
\dot x = i x -\sum_{j+k=1}^{n-1} a_{jk} x^{j+1} \bar x^k.
\end{equation}
 Adjoining
 to \eqref{7} its complex conjugate and considering $\bar a_{jk}$ as
a new parameter $b_{kj}$ and $\bar x$ as a distinct unknown function
 $y$ we obtain the system
\begin{equation}\label{8}
\begin{gathered}
\dot x= i x - \sum_{j+k=1}^{n-1} a_{jk} x^{j+1} y^k
= i x + \widetilde P(x, y), \\
\dot y= - i y + \sum_{j+k=1}^{n-1} b_{kj} x^k y^{j+1}
= - i y + \widetilde Q(x, y).
\end{gathered}
\end{equation}
This system is called the \emph{complexification} of \eqref{1} and
it is equivalent to \eqref{7} when $y = \bar x$
and $b_{kj} = \bar a_{jk}$.

By Poincar\'e-Lyapunov theorem system \eqref{1} has a center at the origin if
and only if it admits in a neighbourhood of the origin an analytic
first integral of the form
$$
\Phi(u,v)=u^2+v^2+\text{h.o.t.},
$$
which is equivalent to the existence of a first integral of the form
$$
\Psi(x,\bar x) =x \bar x +\text{h.o.t.}
$$
for system \eqref{7}.

Thus, extending the notion of center
from real systems to systems \eqref{8} it is said that
complex system \eqref{8} has a center at the origin if in a neighbourhood of the
origin it admits an analytic first integral of the form
\begin{equation} \label{9}
\Psi(x, y) = x y + \sum_{j+k=3}^{\infty} \Psi_{jk} x^j y^k.
\end{equation}

Since to each $a_{jk}$ in the first equation of \eqref{8} corresponds
the parameter $b_{kj}$ in the second equation of \eqref{8}, system \eqref{8}
 has $2 \ell$ parameters. We denote the ordered $2\ell$-tuple of the
parameters of \eqref{8} by $(a,b)$; that is,
\begin{equation} \label{e:coeff_order}
(a,b)=( a_{p_1 q_1}, \ldots, a_{p_\ell q_\ell}, b_{q_\ell, p_\ell},
\ldots, b_{q_1 p_1} ),
\end{equation}
and we use the notation $\mathbb{C}[a, b]$ for the ring of polynomials
 in the variables
$a_{p_1 q_1}$, $ a_{p_2 q_2}$, $ \ldots, b_{q_1 p_1}$ over $\mathbb{C}$.

Recently, Garc\'ia, Llibre and Maza \cite{GLM}
 studied limit cycle bifurcations near a center or a focus at the origin
 of the quintic system written in the complex form as the equation
$$
\dot x = ix + x \bar x ( a x^3 + b x^2 \bar x + c \bar x x^2+d \bar x^3),
$$
which, in order to use the notation similar to the one in \eqref{7},
 we write as the complex equation
\begin{equation} \label{cfr}
\dot x = i(x - a_{31} x^4 \bar x - a_{22} x^3 \bar x^2
- a_{13} x^2 \bar x^3- a_{04} x \bar x^4 ).
\end{equation}

In this paper we study critical period bifurcations from the center
at the origin of system \eqref{cfr}. We first describe a way to compute
the period function of system \eqref{7} using the normal form of its
complexification \eqref{8}. Then we prove that at most three
critical periods can bifurcate from any nonlinear center of the system.

\section{Preliminaries}\label{s:prelim}

To study critical period bifurcations of system \eqref{cfr} we have to compute
a series expansion of the period function $T(r)$
of the system. One possibility is to pass to polar coordinates. This way is
geometrically and theoretically straightforward,
however it is not computationally efficient since one needs to compute
integrals of trigonometric polynomials, and this is a difficult task in the case of
 polynomials of high degree.

Another possible computational approach relies on calculations of
Poincar\'e-Dulac normal form of the complexification
\eqref{8}. We briefly remind it following to \cite{RomSh} and \cite{FLRS}.

As it is well-known after a change of coordinates
\begin{equation} \label{chnf(4)}
\begin{gathered}
x = y_1 + \sum_{j + k \ge 2} h_1^{(j,k)} y_1^j y_2^k\,,\\
y = y_2 + \sum_{j + k \ge 2} h_2^{(j,k)} y_1^j y_2^k\,,
\end{gathered}
\end{equation}
system \eqref{8} can be brought to the Poincar\'e-Dulac normal form
\begin{equation}\label{inf}
\begin{gathered}
\dot y_1 = y_1 ( i + \sum_{j=1}^\infty Y_1^{(j+1,j)} (y_1 y_2)^j )
= y_1 (i + Y_1(y_1 y_2) ), \\
\dot y_2 = y_2 (-i + \sum_{j=1}^\infty Y_2^{(j,j+1)} (y_1 y_2)^j )
 = y_2 (-i + Y_2(y_1 y_2) ).
\end{gathered}
\end{equation}

The normal form \eqref{inf} is not uniquely defined since the so-called
resonant coefficient $h_1^{(j+1,j)}$ and $h_2^{(j,j+1)}$ in \eqref{chnf(4)}
can be chosen arbitrary.
 We will chose for all $j$ ($j=1,2,\dots $) $h_1^{(j+1,j)}=h_2^{(j,j+1)}=0$
(in such case the transformation \eqref{chnf(4)} is called distinguished).

The coefficients $Y_1^{(j+1,j)}$ and $Y_2^{(j,j+1)}$ in \eqref{inf} are
 polynomials of the ring $\mathbb{C}[a,b]$. Denote by ${\mathcal{Y}}$ the ideal generated by
all coefficients of the normal form
\begin{equation} \label{Ygen}
{\mathcal{Y}}:= \langle Y_1^{(j+1, j)},\ Y_2^{(j,j+1)} : j \in \mathbb{N} \rangle \subset \mathbb{C}[a,b],
\end{equation}
and by $\mathcal{Y}_K$ the ideal generated by the first $K$ pairs of the
 coefficients,
$$
\mathcal{Y}_K: = \langle Y_1^{(j+1, j)}, Y_2^{(j,j+1)}
: j = 1, \ldots, K \, \rangle.
$$

The normal form of a particular system $(a^*,b^*)$ with the fixed parameters
is linear when all the coefficients of the normal form evaluated at $(a^*,b^*)$
are equal to zero,
\[
 Y_1^{(j+1,j)}(a^*,b^*)= Y_2^{(j,j+1)}(a^*,b^*)=0 \quad\text{for all }
 j \in \mathbb{N},
\]
 that is, when the point $(a^*,b^*)$ belongs to the variety of the
ideal $\mathcal{Y}$.
The variety $\mathbf{V}(I)$ of a
 polynomial ideal $I$ is the set of common zeros of all polynomials of the ideal.
The variety $V_{\mathscr{L}} := \mathbf{V}({\mathcal{Y}})$ is called the
\emph{linearizability variety} of system \eqref{8}.
As it is well known system \eqref{1} has an isochronous center
 at the origin if and only if the
system is linearizable. Thus, the real systems \eqref{1},
which parameters after the complexification
are in $V_{\mathscr{L}}$, have isochronous centers at the origin.

For system \eqref{8} one can find a function \eqref{9} such that
\[
 [ i x + \widetilde P(x,y) ] \Psi_x(x, y)
+ [ -i y + \widetilde Q(x, y)] \Psi_y (x, y)
= g_{11} (xy)^2 + g_{22} (xy)^3 + \cdots,
\]
where $g_{kk}$ is a polynomial in the coefficients of system \eqref{8}.
The polynomial $g_{kk}$ is called the $k$-th \emph{focus
quantity}. Clearly, system \eqref{8} with fixed coefficients $(a^*,b^*)$
has a center at the origin if
and only if $g_{kk} \equiv 0 $ for all $k\in \mathbb{N}$. We call the ideal
\[
{\mathscr{B}} := \langle g_{kk} : k \in \mathbb{N} \rangle \subset \mathbb{C}[a,b]
\]
 the \emph{Bautin ideal} of system \eqref{8}.
The variety of ${\mathscr{B}}$, $V_{{\mathscr{C}}} = \mathbf{V}({\mathscr{B}})$, is called the \emph{center variety}.
We will also use the ideal generated by the first $K$ focus quantities,
which we denote
\[
{\mathscr{B}}_K:= \langle g_{kk} : k = 1, \dots, K \rangle \subset \mathbb{C}[a,b].
\]
Let us denote
\begin{gather*}
G = Y_1 + Y_2, \\ H = Y_1 - Y_2.
\end{gather*}
It is easy to see that the origin is a center for \eqref{8} if and only
if $G\equiv 0$, in which case $H$ has purely imaginary
coefficients and the distinguished normalizing transformation converges.
We also define
\[
\widetilde H(w) = -\frac{1}{2}iH(w).
\]

When \eqref{8} is the complexification of a real system one can recover
the real system by replacing every occurrence of $y_2$ by $\bar y_1$
in each equation of \eqref{inf}. In such case, performing the transformation
$y_1 = r e^{i{\varphi}}$ we obtain from \eqref{inf} the equations for
$\dot r$ and $\dot {\varphi}$ as follows:
\begin{equation} \label{eq7}
\begin{gathered}
\dot r = \frac{1}{2 r}(\dot y_1 \bar y_1 + y_1 {\dot{\bar y}}_1)
 = 0,\\
\dot {\varphi} = \frac{i}{2 r^2}
 (y_1 {\dot{\bar y}}_1 - \dot y_1 \bar y_1)
 = 1 + \widetilde H(r^2) \,.
\end{gathered}
\end{equation}
We write the function $\widetilde H$ as
$$
\widetilde H(w) = \sum_{k=1}^\infty \widetilde H_{2k+1 }w^k.
$$
The integration of the second equation in \eqref{eq7} gives the least
period of the periodic solution of \eqref passing through the point
with coordinates $(r, 0)$ as
\begin{equation} \label{pf}
T(r) = \frac{2\pi}{1 + \widetilde H(r^2)}
 = 2 \pi \Big( 1 + \sum_{k=1}^\infty p_{2k}(a,\bar a) r^{2k} \Big)
\end{equation}
for some coefficients $p_{2k}$.
The center at the origin of system \eqref{cfr} corresponding to a parameter
$a^*$ is isochronous if and only if $p_{2k}(a^*,\bar a^*) = 0$ for
$k \ge 1$.

It is easy to see that $p_{2k}$ are polynomials in the parameters
$a$, $ \bar a $ of system \eqref{7}. We can extend the polynomial
functions $p_{2k}(a,\bar a)$ to the set of parameters $(a,b)$ setting
in \eqref{eq7} $y_2$ instead of $\bar y_1$. Then instead of \eqref{pf}
we obtain the function
\begin{equation} \label{pfc}
T(r, a, b) = 2 \pi \Big( 1 + \sum_{k=1}^\infty p_{2k}(a,b) r^{2k} \Big),
\end{equation}
which coincides with the period function \eqref{pf} when $b = \bar a$.

We call the polynomial $p_{2k}(a,b)$ in \eqref{pfc} the
\emph{$k$-th isochronicity quantity}.
Using \eqref{pf} and the formula for the inversion of series the first three
polynomials $p_{2k}$ are computed as:
\begin{equation} \label{p2k.formulas}
\begin{gathered}
p_2 = - \widetilde H_3
 = \frac{i}{2} \big( Y_1^{(2,1)} - Y_2^{(1,2)} \big) \\
p_4 = - \widetilde H_5 + (\widetilde H_3)^2
 = \frac{i}{2} \big( Y_1^{(3,2)} - Y_2^{(2,3)} \big)
 - \frac14 \big( Y_1^{(2,1)} - Y_2^{(1,2)} \big)^2, \\
\begin{aligned}
p_6 &= - \widetilde H_7 + 2 \widetilde H_3 \widetilde H_5 - (\widetilde H_3)^3 \\
 &= \frac{i}{2} \big( Y_1^{(4,3)} - Y_2^{(3,4)} \big)
 - \frac12 \big( Y_1^{(2,1)} - Y_2^{(1,2)} \big)
 \big( Y_1^{(3,2)} - Y_2^{(2,3)} \big) \\
 &\quad - \frac{i}{8} \big( Y_1^{(2,1)} - Y_2^{(1,2)} \big)^3.
\end{aligned}
\end{gathered}
\end{equation}


Since values of the isochronicity quantity $p_{2k}$ are of interest
only on the center variety, we should work with the equivalence
 class $[p_{2k}]$ of $p_{2k}$ in the coordinate
ring $\mathbb{C}[V_{\mathscr{C}}]$ of the center variety,
which can be viewed as the set of equivalence classes of polynomials $\mathbb{C}[a,b]$
 by $V_{\mathscr{C}}$. That is, for polynomials $f,g \in \mathbb{C}[a,b]$,
$$
[f] = [g] \quad \text{in } \mathbb{C}[V_{\mathscr{C}}]
$$
 if and only if
$$
f -g \equiv 0 \quad \text{on} \quad V_{\mathscr{C}}.
$$
We denote
\[
P = \langle p_{2k} : k \in \mathbb{N} \rangle \subset \mathbb{C}[a,b]
\quad\text{and}\quad
\widetilde P = \langle [p_{2k}] : k \in \mathbb{N} \rangle \subset \mathbb{C}[V_{\mathscr{C}}],
\]
and for $K \in \mathbb{N}$,
\[
P_K = \langle p_2, \ldots, p_{2K} \rangle
\quad \text{and}\quad
\widetilde P_K = \langle [p_2], \ldots, [p_{2K}] \rangle\,.
\]
The ideal $P$ is called the \emph{isochronicity ideal}.

Finally, we remind that given a Noetherian ring $R$ and an ordered set
$$
B=\{b_1,b_2,\ldots\} \subset R,
$$
 we construct a basis $M_I$ of the ideal
$I=\langle b_1,b_2,\ldots\rangle$ as follows:
\begin{itemize}
\item[(a)] initially set $M_I=\{b_p\}$, where $b_p$ is the first
non-zero element of $B$;

\item[(b)] sequentially check successive elements $b_j$,
starting with $j=p+1$, adding $b_j$ to $M_I$ if and only if $b_j
\notin \langle M_I\rangle$
\end{itemize}
The cardinality of $M_I$ is called the \emph{Bautin depth} of $I$.

\section{An upper bound for critical periods bifurcating from centers
 of system \eqref{cfr}}\label{s:main}

Along with system
\eqref{cfr} we consider its complexification
\begin{equation} \label{com1}
\begin{gathered}
\dot x = i x (1 - a_{31} x^3 y - a_{22} x^2 y^2 - a_{13} x y^3- a_{04} y^4), \\
\dot y = - i y (1 - b_{40 } x^4 - b_{31} x^3 y - b_{22} x^2 y^2-b_{13} x y^3).
\end{gathered}
\end{equation}
Our study is based on the following theorem which is an immediate corollary of
\cite[Theorem 5.2 and Remark 5.3]{FLRS}.

\begin{theorem} \label{r:ext_thm}
Suppose that for the complexification \eqref{8} of the family \eqref{7}:
\begin{itemize}
\item[(a)] $V_{\mathscr{L}} = \mathbf{V}(P_K) \cap V_{{\mathscr{C}}}$,
\item[(b)] the Bautin depth (i.e., the cardinality of the minimal basis)
 of $\widetilde P_K$ in $\mathbb{C}[V_{\mathscr{C}}]$ is $m$, and
\item[(c)] a primary decomposition of $P_K + \sqrt{{\mathscr{B}}}$ can be written
$R \cap N$ where $R$ is the intersection of the ideals
 in the decomposition that are prime and $N$ is the intersection of the
remaining ideals in the decomposition.
\end{itemize}
Then for any system of family \eqref{7} corresponding to
$(a^*, \bar a^*) \in V_{\mathscr{C}} \setminus \mathbf{V}(N)$, at most
$m - 1$ critical periods bifurcate from a center at the origin.
\end{theorem}

Thus, to estimate the number of bifurcating critical periods of
system \eqref{com1} we have to know the center and linearizability varieties
 of the system.


First we note that it follows from
Corollary 3.4.6 in \cite{RomSh} that for system \eqref{com1}
the focus quantities $g_{2k+1,2k+1}$ are zero polynomials.
Using the results of \cite{GR} we can easily prove the following statement.

\begin{proposition} \label{pro1}
The center variety of system \eqref{com1} is defined by the seven
first non-zero focus quantities,
\begin{equation} \label{bv12}
 \mathbf{V}({\mathscr{B}}) = \mathbf{V}({\mathscr{B}}_{14}),
\end{equation}
 where ${\mathscr{B}}_{14}=\langle g_{2,2}, g_{4,4}, g_{6,6}, g_{8,8},g_{10,10},g_{12,12}, g_{14,14}
\rangle$,
 and it consists of four components defined by the following prime ideals:
\begin{gather*}
\begin{aligned}
I_1=&\langle
a_{22}-b_{22},
 a_{31} a_{13}-b_{13} b_{31},
 b_{31}^2 a_{04}-a_{13}^2 b_{40},
 \\
 & a_{31} b_{31} a_{04}-b_{13} a_{13} b_{40},
 a_{31}^2 a_{04}-b_{13}^2 b_{40}\rangle,
\end{aligned}\\
 I_ 2=\langle
 b_{40}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}\rangle, \\
I_3=\langle
 a_{04}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , a_{31}\rangle,\\
I_ 4=\langle
 a_{22}-b_{22}
 , 3 b_{13}-a_{13}
 , 3 a_{31}-b_{31}\rangle.
\end{gather*}
\end{proposition}

\begin{proof}
Using the algorithm in \cite[Chapter 3]{RomSh}
and a \emph{Mathematica} code similar to the one given in
\cite[Fig. 6.1 of Appendix]{RomSh}
we computed the focus quantities $ g_{2,2}, g_{4,4},\dots, g_{14,14}$
(since the expressions are long, we do not present them here, but one can easily
compute them using any available computer algebra system).
Then,
 using the routine \texttt{minAssGTZ}, which is based on the
algorithm of \cite{GTZ}, of the computer algebra system {\sc Singular}
\cite{Sin} we found that
the minimal associate primes of ${\mathscr{B}}_{14}$ are the prime
ideals $I_1,\dots, I_4$ in the statement of the theorem.

By the results of \cite{GR} if the parameters $(a,b)$ of system \eqref{com1}
are from one of the varieties $\mathbf{V}(I_1),\dots, \mathbf{V}(I_4)$,
 then the corresponding systems have a center.
This means that \eqref{bv12} holds.
\end{proof}

Note, that taking into account that
 $\mathbf{V}({\mathscr{B}})$ is a complex variety, from \eqref{bv12} we obtain
 that the radical of ${\mathscr{B}}$ coincides with the radical of $ {{\mathscr{B}}_{14}}$, that is,
$$
\sqrt{{\mathscr{B}}} = \sqrt{{\mathscr{B}}_{14}}.
$$

To find the linearizability variety of system \eqref{com1}
and the isochronicity quantities $p_{2k}$
we have computed the normal
form of system \eqref{com1} up to the order
17 and found four first non-zero pairs of the resonant coefficients
$Y_1^{(2 k+1,2 k )}, Y_2^{(2k ,2k +1)}$ as follows:
\begin{gather*}
Y_1^{(3,2)}=- i a_{22};\quad Y_2^{(2,3)}= i b_{22};\\
Y_1^{(5,4)}= i (-2 a_{13} a_{31} + a_{31} b_{13} - 3 a_{13} b_{31}
 - 2 a_{04} b_{40})/2; \\
Y_2^{(4,5)}=- i (a_{31} b_{13})/2 + (3 a_{13} b_{31})/2 + b_{13} b_{31}
 + a_{04} b_{40};\\ 
\begin{aligned}
Y_1^{(7,6)}&=(4 a_{13} a_{22} a_{31} + a_{22} a_{31} b_{13}
 - 6 a_{13} a_{31} b_{22} + a_{31} b_{13} b_{22} -
 11 a_{13} a_{22} b_{31} \\
&\quad - 6 a_{04} a_{31} b_{31} + 2 a_{22} b_{13} b_{31}
 - 3 a_{13} b_{22} b_{31} - 10 a_{04} b_{31}^2 - 12 a_{13}^2 b_{40} \\
&\quad - 5 a_{04} a_{22} b_{40} - 2 a_{04} b_{22} b_{40})/4;
\end{aligned}\\
\begin{aligned}
 Y_2^{(6,7)}&=i (-a_{22} a_{31} b_{13} - 2 a_{13} a_{31} b_{22}
 - a_{31} b_{13} b_{22} + 3 a_{13} a_{22} b_{31} +
 6 a_{22} b_{13} b_{31} \\
&\quad + 11 a_{13} b_{22} b_{31} - 4 b_{13} b_{22} b_{31}
 + 12 a_{04} b_{31}^2 + 10 a_{13}^2 b_{40} + 2 a_{04} a_{22} b_{40} \\
&\quad + 6 a_{13} b_{13} b_{40} + 5 a_{04} b_{22} b_{40})/4;
\end{aligned}\\
\begin{aligned}
 Y_1^{(9,8)}&=i (132 a_{13} a_{22}^2 a_{31} + 36 a_{13}^2 a_{31}^2
 + 108 a_{04} a_{22} a_{31}^2 - 150 a_{22}^2 a_{31} b_{13} -
 6 a_{13} a_{31}^2 b_{13} \\
&\quad + 6 a_{31}^2 b_{13}^2 + 192 a_{13} a_{22} a_{31} b_{22}
 - 72 a_{04} a_{31}^2 b_{22} + 204 a_{22} a_{31} b_{13} b_{22}\\
&\quad - 324 a_{13} a_{31} b_{22}^2 + 18 a_{31} b_{13} b_{22}^2
 - 198 a_{13} a_{22}^2 b_{31}  - 126 a_{13}^2 a_{31} b_{31} \\
&\quad + 192 a_{04} a_{22} a_{31} b_{31}
 + 24 a_{22}^2 b_{13} b_{31} - 24 a_{13} a_{31} b_{13} b_{31}
 + 6 a_{31} b_{13}^2 b_{31} \\
&\quad - 132 a_{13} a_{22} b_{22} b_{31}
 - 267 a_{04} a_{31} b_{22} b_{31} + 24 a_{22} b_{13} b_{22} b_{31}
 + 18 a_{13} b_{22}^2 b_{31} \\
&\quad - 162 a_{13}^2 b_{31}^2
 - 384 a_{04} a_{22} b_{31}^2 - 18 a_{13} b_{13} b_{31}^2
 - 132 a_{04} b_{22} b_{31}^2 - 396 a_{13}^2 a_{22} b_{40}\\
&\quad - 78 a_{04} a_{22}^2 b_{40} - 200 a_{04} a_{13} a_{31} b_{40}
 + 27 a_{13} a_{22} b_{13} b_{40} - 180 a_{13}^2 b_{22} b_{40} \\
&\quad - 48 a_{04} a_{22} b_{22} b_{40} + 90 a_{13} b_{13} b_{22} b_{40} 
- 800 a_{04} a_{13} b_{31} b_{40} - 56 a_{04} b_{13} b_{31} b_{40}\\
&\quad - 48 a_{04}^2 b_{40}^2)/48;
\end{aligned} \\
\begin{aligned}
 Y_2^{(8,9)}
&=i (-18 a_{22}^2 a_{31} b_{13} - 6 a_{13} a_{31}^2 b_{13}
 - 6 a_{31}^2 b_{13}^2 - 24 a_{13} a_{22} a_{31} b_{22} \\
&\quad - 204 a_{22} a_{31} b_{13} b_{22} - 24 a_{13} a_{31} b_{22}^2
 + 150 a_{31} b_{13} b_{22}^2 - 18 a_{13} a_{22}^2 b_{31} \\
&\quad + 18 a_{13}^2 a_{31} b_{31} - 90 a_{04} a_{22} a_{31} b_{31}
 + 324 a_{22}^2 b_{13} b_{31}  + 24 a_{13} a_{31} b_{13} b_{31} \\
&\quad + 6 a_{31} b_{13}^2 b_{31} + 132 a_{13} a_{22} b_{22} b_{31}
 - 27 a_{04} a_{31} b_{22} b_{31} - 192 a_{22} b_{13} b_{22} b_{31} \\
&\quad + 198 a_{13} b_{22}^2 b_{31} - 132 b_{13} b_{22}^2 b_{31}
 + 162 a_{13}^2 b_{31}^2 + 180 a_{04} a_{22} b_{31}^2
 + 126 a_{13} b_{13} b_{31}^2 \\
&\quad - 36 b_{13}^2 b_{31}^2
 + 396 a_{04} b_{22} b_{31}^2 + 132 a_{13}^2 a_{22} b_{40}
 + 56 a_{04} a_{13} a_{31} b_{40} \\
&\quad + 267 a_{13} a_{22} b_{13} b_{40}
 + 72 a_{22} b_{13}^2 b_{40} + 384 a_{13}^2 b_{22} b_{40}
 + 48 a_{04} a_{22} b_{22} b_{40} \\
&\quad - 192 a_{13} b_{13} b_{22} b_{40}
 - 108 b_{13}^2 b_{22} b_{40} + 78 a_{04} b_{22}^2 b_{40}
 + 800 a_{04} a_{13} b_{31} b_{40} \\
&\quad + 200 a_{04} b_{13} b_{31} b_{40}
 + 48 a_{04}^2 b_{40}^2)/48.
\end{aligned}
\end{gather*}

Then, using \eqref{p2k.formulas} for the calculation of $p_4$ and
 computing the series expansions \eqref{pfc} in order
 to find $p_8, p_{12} $ and $p_{16}$
 we obtain the first four non-zero reduced isochronicity quantities
 (by the reduced quantities we mean the polynomials
 obtained in such way that in formulas \eqref{p2k.formulas} and their
 extensions to any $p_{2k}$ only terms containing the highest order coefficients
 of the normal form are taking into account; it is sufficient to work with
 the reduced quantities since the other terms of $p_{2k}$ are in the ideal
 $\langle p_2,\dots, p_{2k-2} \rangle $) of system \eqref{com1} as follows:
\begin{gather*}
p_4 =\frac 12 (a_{22}+b_{22});\quad
p_8= -\frac 12 ( a_{31} b_{13}-a_{31} a_{13}-b_{13} b_{31}-3 a_{13}
b_{31}-2 a_{04} b_{40});\\
\begin{aligned}
p_{12}&= \frac 18 (-4 a_{13} a_{22} a_{31} - 2 a_{22} a_{31} b_{13}
 + 4 a_{13} a_{31} b_{22} - 2 a_{31} b_{13} b_{22}
 + 14 a_{13} a_{22} b_{31} \\
&\quad + 6 a_{04} a_{31} b_{31} + 4 a_{22} b_{13} b_{31}
 + 14 a_{13} b_{22} b_{31} - 4 b_{13} b_{22} b_{31} + 22 a_{04} b_{31}^2
 + 22 a_{13}^2 b_{40} \\
&\quad + 7 a_{04} a_{22} b_{40} + 6 a_{13} b_{13} b_{40}
 + 7 a_{04} b_{22} b_{40});
\end{aligned}\\
\begin{aligned}
p_{16}&= \frac 1{48} (-66 a_{13} a_{22}^2 a_{31} - 18 a_{13}^2 a_{31}^2
 - 54 a_{04} a_{22} a_{31}^2 + 66 a_{22}^2 a_{31} b_{13} - 6 a_{31}^2 b_{13}^2\\
&\quad - 108 a_{13} a_{22}  a_{31} b_{22} + 36 a_{04} a_{31}^2 b_{22}
 - 204 a_{22} a_{31} b_{13} b_{22} + 150 a_{13} a_{31} b_{22}^2\\
&\quad + 66 a_{31} b_{13} b_{22}^2 + 90 a_{13} a_{22}^2 b_{31}
 + 72 a_{13}^2 a_{31} b_{31} - 141 a_{04}  a_{22} a_{31} b_{31} \\
&\quad + 150 a_{22}^2 b_{13} b_{31} + 24 a_{13} a_{31} b_{13} b_{31}
 + 132 a_{13} a_{22} b_{22} b_{31} + 120 a_{04} a_{31} b_{22} b_{31}\\
&\quad - 108 a_{22} b_{13} b_{22} b_{31} + 90 a_{13} b_{22}^2 b_{31}
 - 66 b_{13} b_{22}^2 b_{31} + 162 a_{13}^2 b_{31}^2
 + 282 a_{04} a_{22} b_{31}^2 \\
&\quad + 72 a_{13} b_{13} b_{31}^2 - 18 b_{13}^2 b_{31}^2
 + 264 a_{04} b_{22} b_{31}^2 + 264 a_{13}^2 a_{22} b_{40}
 + 39 a_{04} a_{22}^2 b_{40} \\
&\quad + 128 a_{04} a_{13} a_{31} b_{40}
 + 120 a_{13} a_{22} b_{13} b_{40} + 36 a_{22} b_{13}^2 b_{40}
 + 282 a_{13}^2 b_{22} b_{40} \\
&\quad + 48 a_{04} a_{22} b_{22} b_{40}
 - 141 a_{13}b_{13} b_{22} b_{40} - 54 b_{13}^2 b_{22} b_{40}
 + 39 a_{04} b_{22}^2 b_{40} \\
&\quad + 800 a_{04} a_{13} b_{31} b_{40}
 + 128 a_{04} b_{13} b_{31} b_{40} + 48 a_{04}^2 b_{40}^2).
\end{aligned}
\end{gather*}
We now look for the linearizability variety of system \eqref{com1}.

\begin{proposition} \label{prop2}
For system \eqref{com1},
\begin{equation} \label{iso1}
V_{\mathscr{L}} = \mathbf{V}({\mathcal{Y}}_8) = V_{{\mathscr{C}}} \cap \mathbf{V}(P_8).
\end{equation}
\end{proposition}

\begin{proof}
Using the routine \texttt{minAssGTZ} of
{\sc Singular} we found that the minimal associate primes
of ideals $\langle {\mathcal{Y}}_8\rangle $ and $\langle {\mathscr{B}}_{14}, P_8 \rangle$ are the same.
Namely, they are the ideals:
\begin{gather*}
Q_1=\langle
 b_{40}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}\rangle,\quad
Q_2=\langle
 b_{40}
 , b_{31}
 , b_{22}
 , a_{22}
 , a_{31}\rangle, \\
Q_3=\langle
 b_{40}
 , a_{04}
 , b_{22}
 , a_{22}
 , b_{13}-3 a_{13}
 , a_{31}-3 b_{31}\rangle,\\
Q_4=\langle
 b_{40}
 , a_{04}
 , b_{22}
 , a_{22}
 , b_{13}+a_{13}
 , a_{31}+b_{31},\rangle \\
Q_5=\langle
 a_{04}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}\rangle, \quad
Q_6=\langle
 a_{04}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , a_{31}\rangle.
\end{gather*}

By the results of \cite{RCH} systems with the coefficients from the
 varieties of these ideals
are linearizable. This proves \eqref{iso1}.
\end{proof}


We now can estimate the number of critical periods near a center at the origin
of system \eqref{cfr}.


\begin{theorem}\label{thm3.4}
At most 3 critical periods bifurcate from nonlinear centers of system \eqref{cfr}.
\end{theorem}

\begin{proof}
By Proposition \ref{prop2}, part (a) of Theorem \ref{r:ext_thm} holds with $K=8$.
We then check that in $\mathbb{C}[V_{\mathcal{C}}]$:
\begin{equation} \label{pbd}
[p_8] \not \in  \langle [p_4] \rangle, \quad
[ p_{12}] \not \in  \langle [p_4], [p_8] \rangle, \quad
 [p_{16}] \not \in  \langle [p_4], [p_8], [p_{12}] \rangle.
\end{equation}
To this end, with the routine \texttt{radical} of the computer algebra
 system {\sc Singular} we compute the radical of the Bautin ideal
${\mathscr{B}}={\mathscr{B}}_{14}$ denoted $\mathcal{R}_{14}, $ that is,
 $$
\mathcal{R}_{14}= \sqrt{{\mathscr{B}}_{14}}
$$
 (one can also compute $\mathcal{R}_{14}$ using the routine \texttt{intersect}
of {\sc Singular} and the ideals $I_1-I_4$ given in the statement of
Proposition \ref{pro1} since it is follows from the proof of Proposition \ref{pro1}
 that $\mathcal{R}_{14}=\cap_{k=1}^4 I_k$).
Then with the \texttt{reduce} of {\sc Singular} we check that for $k=2,3,4$
the remainder of the division of the polynomial $p_{4k}$ by a Groebner
basis of the ideal
 $$
\langle p_4, \dots, p_{4(k-1)}, \mathcal{R}_{14}\rangle
$$
is nonzero. That means, that \eqref{pbd} holds, which, in turn,
yields that the Bautin depth of $\widetilde P_8$
in $\mathbb{C}[V_{\mathcal{C}}]$ is 4.

Then, with the routine \texttt{primdecGTZ} \cite{primdec,GTZ} of
{\sc Singular} we have computed the primary decomposition
of the ideal
$$
Q = \langle P_8, \mathcal{R}_{14} \rangle
$$
 and found that
$$
Q=\cap_{k=1}^{13} Q_k,
$$
where
$Q_1, \dots, $ $ Q_6$ are prime ideals given in the statement of
Proposition \ref{pro1},
$Q_7, \dots, Q_{13}$ are some ideals defined by many polynomials
(for these reason we do not present them here, however
the interested reader can easily compute $Q$ and the primary decomposition
$Q=\cap_{k=1}^{13} Q_k$ with an appropriate
computer algebra system using the ideals $P_8$ and $I_1-I_4$
presented above) whose associate primes
are:
\begin{gather*}
\sqrt{ Q_7}=  \langle
 b_{40}
 , b_{31}
 , b_{22}
 , a_{22}
 , b_{13}-3 a_{13}
 , a_{31}\rangle, \\
\sqrt {Q_8}= \langle
 b_{40}
 , b_{31}
 , b_{22}
 , a_{22}
 , b_{13}+a_{13}
 , a_{31}\rangle, \\
\sqrt{Q_9}= \langle
 a_{04}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}
 , a_{31}-3 b_{31} \rangle,\\
\sqrt {Q_{10}}= \langle
 a_{04}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}
 , a_{31}+b_{31}\rangle, \\
\sqrt {Q_{11} }
 = \langle b_{40}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}
 , a_{31} \rangle, \\
\sqrt{ Q_{12}}= \langle
 a_{04}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}
 , a_{31}\rangle, \\
\sqrt {Q_{13}}=
\langle b_{40}
 , a_{04}
 , b_{31}
 , a_{13}
 , b_{22}
 , a_{22}
 , b_{13}
 , a_{31}\rangle.
\end{gather*}

Thus, $\sqrt{Q_k}= Q_k$ for $k=1,\dots, 6$ and $\sqrt{Q_k}\ne Q_k$ for
$k=7,\dots, 13$; that is, the ideals $R$ and $N$ from the statement of
Theorem \ref{r:ext_thm} are
$$
R=\cap_{k=1}^6Q_k\quad \text{and} \quad N=\cap_{k=7}^{13}Q_k.
$$

To find systems \eqref{cfr} whose coefficients are in the variety of
the ideal $N$ we perform as follows. Let $T_s=\sqrt{Q_{s+6}}$ for $s=1,\dots, 7$.
 Using the \texttt{intersect} of
{\sc Singular} we compute the ideal
$T=\cap_{k=1}^7 T_k $
and find that
\begin{align*}
T&= \langle a_{22}, b_{22}, a_{04} b_{40}, a_{13} b_{40}, b_{13} b_{40},
 a_{04} b_{31}, a_{04} a_{31}, a_{13} b_{31}, \\
&\quad b_{13} b_{31}, a_{13} a_{31}, -3 a_{13}^2 - 2 a_{13} b_{13} + b_{13}^2, a_{31} b_{13},
 a_{31}^2 - 2 a_{31} b_{31} - 3 b_{31}^2\rangle.
\end{align*}
Clearly, $\mathbf{V}(N)=\mathbf{V}(T)$ in $\mathbb{C}^8$.

Since in the case when \eqref{com1} is a complexification of the real system
 the parameters $a_{ks}$ and $b_{sk}$ are complex conjugate we perform the
change of variables
\begin{gather*}
a_{31} = A_{31} + i B_{31},\quad
b_{13} = A_{31} - i B_{31},\\
a_{22} = A_{22} + i B_{22},\quad
b_{22} = A_{22} - i B_{22}, \\
a_{13} = A_{13} + i B_{13},\quad
b_{31} = A_{13} - i B_{13},\\
a_{04} = A_{04} + i B_{04},\quad
b_{40} = A_{04} - i B_{04},
\end{gather*}
where $A_{ks}, B_{ks}$ are real parameters. Substituting these values
into the ideal $T$ and computing a Groebner bases of the obtained ideal
in the ring
$$
\mathbb{Q} [A_{04}, A_{13}, A_{22}, A_{31}, B_{04}, B_{13}, B_{22}, B_{31}]
$$
we find the ideal
\begin{equation}\label{TR}
\begin{aligned}
T_{\mathbb{R}} &=\langle A_{22}, B_{22}, (B_{13} - B_{31}) (3 B_{13} + B_{31}), A_{31}^2
 + B_{31}^2, \\
&\quad A_{31} B_{13} + A_{13} B_{31},
 3 A_{13} B_{13} + 2 A_{31} B_{13} + A_{31} B_{31}, \\
&\quad A_{13} A_{31} - B_{13} B_{31},
 3 A_{13}^2 + 2 B_{13} B_{31} + B_{31}^2, A_{31} B_{04} + A_{04} B_{31},\\
&\quad -A_{13} B_{04} + A_{04} B_{13},
 A_{04} A_{31} - B_{04} B_{31}, A_{04} A_{13} + B_{04} B_{13}, A_{04}^2
 + B_{04}^2 \rangle.
\end{aligned}
\end{equation}
The basis of $T_{\mathbb{R}} $ contains the polynomials
$$
A_{22}, B_{22}, A_{31}^2 + B_{31}^2, A_{04}^2 + B_{04}^2.
$$
Since $A_{31},B_{31}, A_{04},B_{04}$ are real parameters we conclude
that
\begin{equation} \label{eqABO}
A_{22}= B_{22}=A_{31}=B_{31}=A_{04}=B_{04}=0.
\end{equation}
Substituting the values from \eqref{eqABO} into polynomials of the ideal $T_\mathbb{R}$
given in \eqref{TR} we find that also
$$
A_{13}=B_{13}=0.
$$

It means that the only system of the form \eqref{cfr} whose
parameters are in the variety of the ideal $N$ is the linear system \eqref{7},
that is, the system $\dot x=i x$.
Thus, by Theorem \ref{r:ext_thm} at most 3 critical periods bifurcate from
non-linear isochro\-nous centers of system \eqref{cfr}.
\end{proof}

\section*{Acknowledgments}

The first author is supported by the Slovenian Research Agency
(research core funding No. P1-0306), the second author is supported
by the National Natural Science Foundation of China (No.11431008 and No.
11771296) and the third one by the National Natural Science Foundation
of China (No.11261013, No. 11361017) and the Natural Science Foundation
 of Guangxi, China (No. 2016GXNSFDA380031).

\begin{thebibliography}{99}


\bibitem{CRZ} X.~Chen, V.~G.~Romanovski, W.~Zhang;
 \emph{Critical periods of perturbations of reversible rigidly isochronous centers,} {J.~Differential Equations} {\bf 251} (2011), 1505--1525.



\bibitem{CJ} C.~Chicone, M.~Jacobs; \emph{
 Bifurcation of critical periods for plane vector fields,
 } {Trans.~Amer.~Math.~Soc.} {\bf 312} (1989), 433--486.

\bibitem{Sin} W.~Decker, G.-M. Greuel, G.~Pfister, H. Sh\"onemann; \emph{
 {\sc Singular} 3-1-6---A Computer Algebra System for Polynomial Computations.}
 http://www.singular.uni-kl.de , 2012.

\bibitem{primdec} W.~Decker, S.~Laplagne, G.~Pfister, H.~A.~Sch\"onemann;
 \emph{ \texttt{primdec.lib} {\sc Singular} 3-1-6 library for computing the primary decomposition and
 radical of ideals}, 2010.


\bibitem{FLRS} B.~Fer\v cec, V.~Levandovskyy, V.~G.~Romanovski, D.~S.~Shafer;
 \emph{ Bifurcation of critical periods of polynomial systems.
 } {J.~Differential Equations} {\bf 259} (2015), 3825--3853.

\bibitem{GLM}
I.A. Garc\'ia, J. Llibre, S. Maza; \emph{
The Hopf cyclicity of the centers of a class of quintic polynomial vector fields},
 \emph{J. Differential Equations} {\bf 258} (2015), 1990-2009.



\bibitem{GLY}
A.~Gasull, C.~Liu, J.~Yang; \emph{ On the number of critical periods for planar polynomial systems of arbitrary degree,} {J. Differential Equations} {\bf 249}
 (2010), 684-692.

\bibitem{GTZ} P.~Gianni, B.~Trager, G.~Zacharias; \emph{
 Gr\"obner bases and primary decomposition of polynomials,}
 {J.~Symbolic Comput.} {\bf 6} (1988), 146--167.

\bibitem{GR} J. Gine, V. G. Romanovski; \emph{ Integrability conditions for Lotka-Volterra planar complex quintic systems, } { Nonlinear Anal. Real World Appl.} {\bf 11} (2010), 2100-2105.


\bibitem{GV} M.~Grau, J.~Villadelprat; \emph{
 Bifurcation of critical periods from Pleshkan's isochrones,
 } {J.~Lond. Math. Soc.~(2)} {\bf 81} (2010), 142--160.

\bibitem{Liu} C.~Liu, M.~ Han; \emph{ Bifurcation of critical periods from the reversible rigidly isochronous centers, } {Nonlinear Anal.} \textbf{95} (2014), 388--403.


\bibitem{LLH} Y.-R.~Liu, J.~Li, W.~Huang;
 \emph{Singular Point values, Center Problem, and Bifurcations of Limit Cycles of Two Dimensional
 Differential Autonomous Systems}.
 Beijing: Science Press, 2008.


\bibitem{RCH} V.~G.~Romanovski,
X. Chen, Z. Hu; \emph{ Linearizability of linear systems perturbed by fifth degree homogeneous polynomials, }
 {J. Phys. A: Math. Theor.} {\bf 40} (2007,) 5905-5919.

\bibitem{RomSh} V.~G.~Romanovski, D.~S.~Shafer;
 \emph{The Center and Cyclicity Problems: A Computational Algebra Approach.}
 Boston: Birkh\"auser, 2009.


\bibitem{RT} C.~Rousseau, B.~Toni; \emph{
 Local bifurcation of critical periods in vector fields with homogeneous nonlinearities of the third
 degree,
 } {Canad.~Math.~Bull.} {\bf 36} (1993), 473--484.

\bibitem{RT2} C.~Rousseau, B.~Toni; \emph{
 Local bifurcation of critical periods in the reduced Kukles system,}
 {Canad.~Math.~Bull.} {\bf 49} (1997), 338--358.

\bibitem{V} J.~Villadelprat; \emph{
 Bifurcation of local critical periods in the generalized Loud's system,}
 {Appl.~Math.~Comput.} {\bf 218} (2012), 6803--6813.


\bibitem{WCZ} Z.~Wang, X.~Chen, W.~Zhang; \emph{
 Local bifurcation of critical periods in a generalized 2D LV system,}
 {Appl.~Math.~Comput.} {\bf 214} (2009), 17--25.

\bibitem{XW} Q.~Xu, W.~Huang; \emph{
 The center conditions and local bifurcation of critical periods for a Li\'enard system,
} { Appl. Math.~Comput.} {\bf 217} (2011), 6637-6643.

\bibitem{Yu} P.~Yu, M.~ Han, J.~ Zhang; \emph{ Critical periods of third-order planar Hamiltonian systems, } {Internat. J. Bifur. Chaos Appl. Sci. Engrg.}\textbf{ 20} (2010), 2213--2224.

\bibitem{ZCZ} L.~Zou, X.~Chen, W.~Zhang; \emph{
 Local bifurcations of critical periods for cubic Li\'enard equations with cubic damping,}
 {J.~Comput.~Appl.~Math.} {\bf 222} (2008), 404--410.

\end{thebibliography}

\end{document}