# Analytic integrability of two lopsided systems111This research is partially
supported by the National Nature Science Foundation of China
(11201211,11371373) and Applied Mathematics Enhancement Program of Linyi
University and the Natural Science and Engineering Research Council of Canada
(No. R2686A02)

Feng Lia,b, Pei Yub,222Corresponding author. Tel: +1 519661-2111; E-mail
address<EMAIL_ADDRESS>Yirong Liuc aSchool of Science, Linyi University, Linyi,
Shandong 276005, P.R. China
bDepartment of Applied Mathematics, Western University, London, Ontario N6A
5B7, Canada
cSchool of Mathematics and Statistic, Central South University, Changsha,
Hunan 410012, China

###### Abstract

In this paper, we present two classes of lopsided systems and discuss their
analytic integrability. The analytic integrable conditions are obtained by
using the method of inverse integrating factor and theory of rotated vector
field. For the first class of systems, we show that there are $n+4$ small-
amplitude limit cycles enclosing the origin of the systems for $n\geq 2$, and
$10$ limit cycles for $n=1$. For the second class of systems, we prove that
there exist $n+4$ small-amplitude limit cycles around the origin of the
systems for $n\geq 2$, and $9$ limit cycles for $n=1$.

###### keywords:

Nilpotent Poincaré systems; analytic integrability; Lyapunov constant; Rotated
vector field.

††journal: IJBC

##  1 Introduction

Integrability is one of the most important and difficult problems in studying
ordinary differential systems. To explain the problem, consider a planar
analytic differential system, described by

$\begin{array}[]{ll}\dot{u}=-v+U(u,v),\\\\[2.15277pt]
\dot{v}=u+V(u,v),\end{array}$ (1.1)

where dot indicates differentiation with respect to time $t$, $U$ and $V$ are
real analytic functions whose series expansions in a neighborhood of the
origin start at least from second-order terms. By the Poincaré-Lyapunov
theorem, system (1.1) has a center at the origin if and only if there exists a
first integral, given in the form of

$\phi(u,v)=u^{2}+v^{2}+\sum\limits_{k+j=3}^{\infty}\phi_{kj}u^{k}v^{j},$ (1.2)

where the series converges in a neighborhood of the origin. Determining
whether the origin of system (1.1) is a center or focus is called center
problem. Another important problem in study of system (1.1) is the existence
of analytical first integral in a small neighborhood of the origin of system
(1.1). If there exists such an analytical first integral, the origin of system
(1.1) is a center, in particular, called an analytic center, see [1].

It is well known that it is difficult to distinguish focus from center when
the singular point is degenerate. Many research works have been done in this
direction. For example, analytic systems having a nilpotent singular point at
the origin were studied by Andreev [2] in order to obtain their local phase
portraits. However, Andreev’s results do not distinguish focus from center.
Takens [3] provided a normal form for nilpotent center of foci. Later, Moussu
[4] found the $C^{\infty}$ normal form for analytic nilpotent centers.
Further, Berthier and Moussu [5] studied the reversibility of nilpotent
centers. Teixeria and Yang [6] analysed the relationship between reversibility
and the center-focus problem, expressed in a convenient normal form, and
studied the reversibility of certain types of polynomial vector fields. Han et
al. considered polynomial Hamiltonian systems with a nilpotent singular point,
and they obtained necessary and sufficient conditions for quadratic and cubic
Hamiltonian systems with a nilpotent singular point which may be a center, a
cusp or a saddle, see [7]. In particular, the local analytic integrability for
nilpotent centers was investigated [8], for the differential systems in the
form of

$\begin{array}[]{l}\dot{x}=y+P_{3}(x,y),\\\\[2.15277pt]
\dot{y}=Q_{3}(x,y),\end{array}$

which has a local analytic first integral, where $P_{3}$ and $Q_{3}$ represent
homogeneous polynomials of degree three. For third-order nilpotent singular
points of a planar dynamical system, the analytic center problem was solved by
using the integrating factor method, see for example [9].

The Kukles system, as a well-known example, has been investigated intensively
on the existence of its limit cycles as well as its integrability. For the
following particular Kukles system,

$\begin{array}[]{l}\dot{x}=y,\\\\[2.15277pt]
\dot{y}=-x+a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+a_{4}x^{3}+a_{5}x^{2}y+a_{6}xy^{2}+a_{7}y^{3},\end{array}$

the conditions under which the origin of the system is a center have been
examined in [10, 11, 12, 13, 14, 15, 16]. More details about the Kukles system
can be found in [17]. The so-called extended Kukles system,

$\begin{array}[]{l}\dot{x}=y(1+kx),\\\\[2.15277pt]
\dot{y}=-x+a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+a_{4}x^{3}+a_{5}x^{2}y+a_{6}xy^{2}+a_{7}y^{3},\end{array}$

has also been considered to obtain the center conditions [18, 19]. Recently,
center problem for some more generalized Kukles type systems have been studied
[20, 21, 22]. A kind of Li$\acute{e}$nard systems of type $(n,4)$ for $3\leq
n\leq 27$ was investigated and they obtained the lower bound of the maximal
number of limit cycles for this kind of system in [23].

Research on Hilbert s sixteenth problem in general usually proceeds by the
investigation on specific classes of polynomial systems, much effort has been
devoted in recent years to the investigation of various systems such as
poincare system, Able equation, lopsided system and so on. The Kukles system
is perhaps the earliest example of lopsided systems which have the following
forms

$\begin{array}[]{l}\dot{x}=-y,\\\\[2.15277pt] \dot{y}=x+P(x,y),\end{array}$

or

$\begin{array}[]{l}\dot{x}=-y+P(x,y),\\\\[2.15277pt] \dot{y}=x.\end{array}$

Since then, lopsided systems have drawn more and more attention to
researchers. Lopsided quartic and quintic polynomial vector fields have been
studied and center conditions were obtained [24, 25]. Furthermore, Gine [26]
proved that there is exactly one isochronous system for lopsided quartic
system, and the origin never can be an isochronous center for lopsided quintic
system. For seventh-degree lopsided system Soriano and Salih [27] showed that
the origin is a center if and only if the system is time-reversible and if it
is not, no more than seven local limit cycles can bifurcate from the origin
under certain conditions. However when the origin is a degenerate singular
point, there are fewer results because it is difficult to compute the Lyapunov
constants. The cubic lopsided system with a nilpotent singular point has been
investigated intensively. For example, Alvarez and Gasull [28] proved that
three limit cycles can bifurcate from a nilpotent singular point of the
following system:

$\begin{array}[]{l}\dot{x}=-y,\\\\[2.15277pt]
\dot{y}=a_{1}x^{2}+a_{2}xy+a_{3}y^{2}+a_{4}x^{3}+a_{5}x^{2}y+a_{6}xy^{2}+a_{7}y^{3},\end{array}$
(1.3)

via an analysis based on normal forms. Then, Liu and Li [29] showed that by
making a small perturbation to the linear terms of (1.3), it can exhibit four
small-amplitude limit cycles. Bifurcation of limit cycles and center
conditions for the following two families of lopsided systems with nilpotent
singularities,

$\begin{array}[]{l}\dot{x}=-y+P_{4}(x,y),\\\\[2.15277pt]
\dot{y}=-2x^{3},\end{array}$

and

$\begin{array}[]{l}\dot{x}=-y+P_{5}(x,y),\\\\[2.15277pt]
\dot{y}=-2x^{3},\end{array}$

have been considered by Li et al. [30], where $P_{4}(x,y)$ and $P_{5}(x,y)$
represent homogeneous polynomials in $x$ and $y$ of degree four and five,
respectively. Their results show that it is more difficult to distinguish
focus from center when the singular point is degenerate. As far as analytic
center of lopsided system is concerned, it is more challenging to distinguish
it from focus. So, in this paper, we shall discuss analytic center conditions
and bifurcation of limit cycles for two classes of lopsided systems with a
cubic-order nilpotent singular point, given by

$\begin{array}[]{l}\dot{x}=y+H_{3}(x,y)+H_{2n+3}(x,y),\\\\[2.15277pt]
\dot{y}=-2x^{3},\end{array}$ (1.4)

and

$\begin{array}[]{l}\dot{x}=y,\\\\[2.15277pt]
\dot{y}=-2x^{3}+H_{3}(x,y)+H_{2n+3}(x,y),\end{array}$ (1.5)

where $H_{k}(x,y)$ represent a $k$th-degree homogeneous polynomial in $x$ and
$y$.

The main goal of this paper is to apply the method of integrating factor and
theory of rotating vector fields to distinguish analytic integrability
conditions and to find the conditions for analytic centers. This work is a
continuation of that for the Kukles system with a degenerate singular point.
In next section, we present some known results which are necessary for proving
the main result. We derive the analytic center conditions for the centers of
systems (1.4) and (1.5) in Sections 3 and 4, respectively. Finally, conclusion
is drawn in Section 5.

##  2 Preliminary results

In this section, we present some relative notions and results taken from [31,
32], which will be used in the following sections. A system whose origin is a
cubic-order monodromic singular point can be written as

$\begin{array}[]{l}\dot{x}=y+\mu
x^{2}+\sum\limits_{i+2j=3}^{\infty}a_{ij}x^{i}y^{j}=X(x,y),\\\\[2.15277pt]
\dot{y}=-2x^{3}+2\mu
xy+\sum\limits_{i+2j=4}^{\infty}b_{ij}x^{i}y^{j}=Y(x,y).\end{array}$ (2.1)

######  Theorem 2.1.

For any positive integer $s$ and a given number sequence $\,\\{c_{0\beta}\\},\
\beta\geq 3$, a formal series can be constructed successively in terms of the
coefficients $c_{\alpha\beta}$ ($\alpha\neq 0$) as

$M(x,y)=y^{2}+\sum\limits_{\alpha+\beta=3}^{\infty}c_{\alpha\beta}x^{\alpha}y^{\beta}=\sum\limits_{k=2}^{\infty}M_{k}(x,y),$
(2.2)

satisfying

$\left(\frac{\partial X}{\partial x}+\frac{\partial Y}{\partial
y}\right)M-(s+1)\left(\frac{\partial M}{\partial x}X+\frac{\partial
M}{\partial y}Y\right)=\sum\limits_{m=3}^{\infty}\omega_{m}(s,\mu)x^{m},$
(2.3)

where $M_{k}(x,y)$ is a $k$th-degree homogeneous polynomial in $x$ and $y$,
satisfying $s\mu=0$ for all $k$.

######  Theorem 2.2.

For $\alpha\geq 1,\alpha+\beta\geq 3$ in (2.2) and (2.3), $c_{\alpha\beta}$
can be uniquely determined by the recursive formula,

$c_{\alpha\beta}=\frac{1}{(s+1)\alpha}(A_{\alpha-1,\beta+1}+B_{\alpha-1,\beta+1}).$
(2.4)

For $m\geq 1$, $\omega_{m}(s,\mu)$ can be uniquely determined by the recursive
formulae:

$\omega_{m}(s,\mu)=A_{m,0}+B_{m,0},$ (2.5)
$\lambda_{m}=\frac{\omega_{2m+4}(s,\mu)}{2m-4s-1}.$ (2.6)

where

$\begin{array}[]{l}A_{\alpha\beta}=\sum\limits_{k+j=2}^{\alpha+\beta-1}[k-(s+1)(\alpha-k+1)]a_{kj}c_{\alpha-k+1,\beta-j},\\\
B_{\alpha\beta}=\sum\limits_{k+j=2}^{\alpha+\beta-1}[j-(s+1)(\beta-j+1)]b_{kj}c_{\alpha-k,\beta-j+1}.\end{array}$
(2.7)

######  Theorem 2.3.

The origin of system (2.1) is an analytic center if and only if the origin of
system (2.1) is a center of $\infty$-class, namely, the origin of system (2.1)
is a center for any natural number $s$.

##  3 Analytic centers of system (1.4)

Now, we discuss the analytic centers of system (1.4) in two cases.

###  3.1 Case 1: $n=1$.

For this case, system (1.4) can be written as

$\begin{array}[]{ll}\dot{x}=\\!\\!\\!&y+a_{30}x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{50}x^{5}+a_{41}x^{4}y\\\\[4.30554pt]
&+a_{32}x^{3}y^{2}+a_{23}x^{2}y^{3}+a_{14}xy^{4}+a_{05}y^{5},\\\\[4.30554pt]
\dot{y}=\\!\\!\\!&-2x^{3}.\end{array}$ (3.1)

According to Theorem 2.1, we can find a formal series
$M(x,y)=x^{4}+y^{2}+o((x^{2}+y^{2})^{2})$ for system (3.1), such that (2.3)
holds. Applying the recursive formulae in Theorem 2.2 to system (3.1), with
the help of Mathematica, we obtain

$\begin{split}\omega_{3}&=\omega_{4}=\omega_{5}=0,\\\
\omega_{6}&=(4s-1)a_{30},\\\ \omega_{7}&=3(a+1)c_{03},\\\
\omega_{8}&=-\frac{1}{5}(4s-3)(2a_{12}+5a_{50}),\\\ \omega_{9}&=0,\\\
\omega_{10}&=-\frac{1}{7}(4s-5)(2a_{32}+3a_{21}a_{50}),\end{split}$
$\begin{split}\omega_{11}&=\frac{15}{4}(s+1)c_{05},\\\
\omega_{12}&=-\frac{1}{45}(4s-7)(12a_{14}+30a_{03}a_{50}+5a_{41}a_{50}),\\\
\omega_{13}&=0,\\\
\omega_{14}&=-\frac{3\,a_{50}}{77}(4s-9)(6a_{23}+a_{21}a_{41}-10a_{50}^{2}),\\\
\omega_{15}&=\frac{35}{8}(s+1)c_{07},\\\
\omega_{16}&=-\frac{a_{50}}{117}(4s-11)(60a_{05}+10a_{03}a_{41}+a_{41}^{2}-3a_{21}a_{50}^{2}),\\\
\omega_{17}&=0,\\\
\omega_{18}&=\frac{a_{50}}{1155}(4s-13)(2a_{21}a_{41}^{2}+300a_{03}a_{50}^{2}+9a_{21}^{2}a_{50}^{2}+100a_{41}a_{50}^{2}),\\\
\omega_{19}&=\frac{315}{64}(s+1)c_{09},\\\
\omega_{20}&=-\frac{a_{50}}{895050}(4s-15)(28a_{21}a_{41}^{4}+252a_{21}^{2}a_{41}^{2}a_{50}^{2}+800a_{41}^{3}a_{50}^{2}+567a_{21}^{3}a_{50}^{4}\\\
&\hskip 86.72377pt+3600a_{21}a_{41}a_{50}^{4}+4500a_{50}^{6}),\\\
\omega_{21}&=0,\\\
\omega_{22}&=-\frac{4\,a_{50}}{235125}(4s-17)(4a_{21}^{2}a_{41}^{4}+36a_{21}^{3}a_{41}^{2}a_{50}^{2}+100a_{21}a_{41}^{3}a_{50}^{2}+81a_{21}^{4}a_{50}^{4}\\\
&\hskip 86.72377pt+450a_{21}^{2}a_{41}a_{50}^{4}-125a_{41}^{2}a_{50}^{4}),\\\
\omega_{23}&=\frac{693}{128}(s+1)c_{011},\\\
\omega_{24}&=\frac{a_{50}}{42089726250000(s+1)}\,f_{1},\end{split}$ (3.2)

where

$\begin{split}f_{1}=&-15174868212a_{21}^{6}a_{41}^{4}-84454927200a_{21}^{4}a_{41}^{5}+22768748000a_{21}^{2}a_{41}^{6}-136573813908a_{21}^{7}a_{41}^{2}a_{50}^{2}\\\
&-1193662008000a_{21}^{5}a_{41}^{3}a_{50}^{2}-2087643726000a_{21}^{3}a_{41}^{4}a_{50}^{2}+651216400000a_{21}a_{41}^{5}a_{50}^{2}\\\
&-307291081293a_{21}^{8}a_{50}^{4}-3661266760200a_{21}^{6}a_{41}a_{50}^{4}-9313504335000a_{21}^{4}a_{41}^{2}a_{50}^{4}\\\
&+5946721200000a_{21}^{2}a_{41}^{3}a_{50}^{4}+23826000000a_{41}^{4}a_{50}^{4}-17785962180a_{21}^{6}a_{41}^{4}s\\\
&-98929404000a_{21}^{4}a_{41}^{5}s+26934900000a_{21}^{2}a_{41}^{6}s-160073659620a_{21}^{7}a_{41}^{2}a_{50}^{2}s\\\
&-1398534984000a_{21}^{5}a_{41}^{3}a_{50}^{2}s-2442981870000a_{21}^{3}a_{41}^{4}a_{50}^{2}s+18810000000a_{41}^{4}a_{50}^{4}s\\\
&+770106000000a_{21}a_{41}^{5}a_{50}^{2}s-360165734145a_{21}^{8}a_{50}^{4}s+6998670000000a_{21}^{2}a_{41}^{3}a_{50}^{4}s\\\
&-4290086997000a_{21}^{6}a_{41}a_{50}^{4}s-10903637205000a_{21}^{4}a_{41}^{2}a_{50}^{4}s+4967473392a_{21}^{6}a_{41}^{4}s^{2}\\\
&+27628675200a_{21}^{4}a_{41}^{5}s^{2}-7529168000a_{21}^{2}a_{41}^{6}s^{2}+44707260528a_{21}^{7}a_{41}^{2}a_{50}^{2}s^{2}\\\
&+390585888000a_{21}^{5}a_{41}^{3}a_{50}^{2}s^{2}+682203816000a_{21}^{3}a_{41}^{4}a_{50}^{2}s^{2}-215262400000a_{21}a_{41}^{5}a_{50}^{2}s^{2}\\\
&+100591336188a_{21}^{8}a_{50}^{4}s^{2}+1198155823200a_{21}^{6}a_{41}a_{50}^{4}s^{2}+3044973060000a_{21}^{4}a_{41}^{2}a_{50}^{4}s^{2}\\\
&-1955419200000a_{21}^{2}a_{41}^{3}a_{50}^{4}s^{2}-5016000000a_{41}^{4}a_{50}^{4}s^{2}.\end{split}$

Based on (2.6) and (3.2), it is easy to find the first ten quasi-Lyapunov
constants of system (3.1).

######  Theorem 3.1.

The first ten quasi-Lyapunov constants at the origin of system (3.1) are given
by

$\begin{split}\lambda_{1}&=a_{30},\\\
\lambda_{2}&=\frac{1}{5}(2a_{12}+5a_{50}),\\\
\lambda_{3}&=\frac{1}{7}(2a_{32}+3a_{21}a_{50}),\\\
\lambda_{4}&=\frac{1}{45}(12a_{14}+30a_{03}a_{50}+5a_{41}a_{50}),\\\
\lambda_{5}&=\frac{3\,a_{50}}{77}(6a_{23}+a_{21}a_{41}-10a_{50}^{2}),\\\
\lambda_{6}&=-\frac{a_{50}}{117}(60a_{05}+10a_{03}a_{41}+a_{41}^{2}-3a_{21}a_{50}^{2}),\\\
\lambda_{7}&=-\frac{a_{50}}{1155}(2a_{21}a_{41}^{2}+300a_{03}a_{50}^{2}+9a_{21}^{2}a_{50}^{2}+100a_{41}a_{50}^{2}),\\\
\lambda_{8}&=-\frac{a_{50}}{895050}(28a_{21}a_{41}^{4}+252a_{21}^{2}a_{41}^{2}a_{50}^{2}+800a_{41}^{3}a_{50}^{2}+567a_{21}^{3}a_{50}^{4}\\\
&~{}~{}~{}+3600a_{21}a_{41}a_{50}^{4}+4500a_{50}^{6}),\\\
\lambda_{9}&=-\frac{4\,a_{50}}{235125}(4a_{21}^{2}a_{41}^{4}+36a_{21}^{3}a_{41}^{2}a_{50}^{2}+100a_{21}a_{41}^{3}a_{50}^{2}+81a_{21}^{4}a_{50}^{4}\\\
&~{}~{}~{}+450a_{21}^{2}a_{41}a_{50}^{4}-125a_{41}^{2}a_{50}^{4}),\\\
\lambda_{10}&=-\frac{a_{50}}{42089726250000(s+1)(4s-19)}\,f_{1},\end{split}$
(3.3)

where $\lambda_{k-1}=0$ for $k=2,\cdots,10$ have been used in the computation
of $\lambda_{k}$.

It follows from Theorem 3.1 that the following assertion holds.

######  Proposition 3.1.

For $n=1$, the origin of system (3.1) is an analytic center if and only if the
following conditions are satisfied:

$a_{30}=a_{12}=a_{32}=a_{14}=a_{50}=0.$ (3.4)

Proof. By setting $\lambda_{1}=\lambda_{2}=\cdots=\lambda_{10}=0$, it is easy
to get the conditions in (3.4). Assume $a_{50}\neq 0$, and denote

$\begin{split}f_{2}=&28a_{21}a_{41}^{4}+252a_{21}^{2}a_{41}^{2}a_{50}^{2}+800a_{41}^{3}a_{50}^{2}+567a_{21}^{3}a_{50}^{4}+3600a_{21}a_{41}a_{50}^{4}+4500a_{50}^{6},\\\
f_{3}=&4a_{21}^{2}a_{41}^{4}+36a_{21}^{3}a_{41}^{2}a_{50}^{2}+100a_{21}a_{41}^{3}a_{50}^{2}+81a_{21}^{4}a_{50}^{4}+450a_{21}^{2}a_{41}a_{50}^{4}-125a_{41}^{2}a_{50}^{4}.\end{split}$
(3.5)

Then, we have

$\begin{array}[]{l}\begin{split}R_{1}=&\ {\rm
Resultant}[f_{2},f_{3},a_{21}]\\\ =&\
252226880859375a_{50}^{28}(37a_{41}^{6}+36000a_{41}^{3}a_{50}^{4}+864000a_{50}^{8}),\\\
R_{2}=&\ {\rm Resultant}[f_{2},f_{1},a_{21}]\\\ =&\
-750785873641864353168750000000000000000a_{50}^{44}(-879390304066912a_{41}^{12}\\\
&+47983547106994035360a_{41}^{9}a_{50}^{4}+49445533255803715842660a_{41}^{6}a_{50}^{8}\\\
&+1456057532744172471928500a_{41}^{3}a_{50}^{12}+6498810664995012399669375a_{50}^{16}\\\
&-2759277767198304a_{41}^{12}s+169297706825726316960a_{41}^{9}a_{50}^{4}s\\\
&+173470743593716632941700a_{41}^{6}a_{50}^{8}s+5108374765584631369552500a_{41}^{3}a_{50}^{12}s\\\
&+22851124455468570581840625a_{50}^{16}s-2080601609429376a_{41}^{12}s^{2}\\\
&+151804543373289707520a_{41}^{9}a_{50}^{4}s^{2}+154422423262500291638820a_{41}^{6}a_{50}^{8}s^{2}\\\
&+4547533910484627929569500a_{41}^{3}a_{50}^{12}s^{2}+20400917084512169654885625a_{50}^{16}s^{2}\\\
&+610343850576768a_{41}^{12}s^{3}-33200851039501326720a_{41}^{9}a_{50}^{4}s^{3}\\\
&-34214947244661526011900a_{41}^{6}a_{50}^{8}s^{3}-1007536381850376947992500a_{41}^{3}a_{50}^{12}s^{3}\\\
&-4496729306788344912103125a_{50}^{16}s^{3}+573878672057856a_{41}^{12}s^{4}\\\
&-49876076993258065920a_{41}^{9}a_{50}^{4}s^{4}-50424620011202182011120a_{41}^{6}a_{50}^{8}s^{4}\\\
&-1484973484158264557562000a_{41}^{3}a_{50}^{12}s^{4}\\!-\\!6678213700042142946607500a_{50}^{16}s^{4}\\\
&-217264690435584a_{41}^{12}s^{5}+18276082133674805760a_{41}^{9}a_{50}^{4}s^{5}\\\
&+18496906029642804955200a_{41}^{6}a_{50}^{8}s^{5}+544720023506226322440000a_{41}^{3}a_{50}^{12}s^{5}\\\
&+2448661869754899507450000a_{50}^{16}s^{5}+19914634381312a_{41}^{12}s^{6}\\\
&-1701986627481384960a_{41}^{9}a_{50}^{4}s^{6}-1721646399680525295360a_{41}^{6}a_{50}^{8}s^{6}\\\
&-50701299015707667936000a_{41}^{3}a_{50}^{12}s^{6}+227963727065799050760000a_{50}^{16}s^{6}).\end{split}\end{array}$

With the aid of Mathematica, we obtain for $\forall s\in Z^{+}$

$\begin{split}G_{1}=&\ {\rm Resultant}[R_{1},R_{2},a_{41}]\\\ =&\
-182848672642886912449902102931881129741668701171875a_{50}^{96}(1+s)^{6}(-19+4s)^{6}\\\
&\ \times(12242160594943288477497249258950767957\\\
&+57187190996418911124243473597501985540s\\\
&+84210057837841105190444817587559944702s^{2}\\\
&+22053341878592957414426973876225026580s^{3}\\\
&-34746447450361087057581921863631440523s^{4}\\\
&-7190180552428847800895138514692327280s^{5}\\\
&+8952012886140489676856041653019558112s^{6}\\\
&-1982180847477328550724618213150339840s^{7}\\\
&+138354459536790840295491820367594752s^{8})^{3}\neq 0.\end{split}$

So there are no solutions for the set of equations, $f_{1}=f_{2}=f_{3}=0$,
implying that there do not exist other analytic center conditions for system
(3.1) if $a_{50}\neq 0$.

Under the conditions in (3.4), system (3.1) becomes

$\begin{array}[]{l}\begin{split}\dot{x}&=y+a_{21}x^{2}y+a_{03}y^{3}+a_{41}x^{4}y+a_{23}x^{2}y^{3}+a_{05}y^{5},\\\
\dot{y}&=-2x^{3}.\end{split}\end{array}$ (3.6)

Obviously, system (3.6) is symmetric with the $y$-axis. According to Theorem
11 in [9], the origin is an analytic center of system (3.1).

Proposition 3.1 implies that

######  Theorem 3.2.

The necessary and sufficient conditions for the origin of system (3.1) being
an analytic center are determined from vanishing of the first ten quasi-
Lyapunov constants, that is, the conditions given in Proposition 3.1 are
satisfied.

When the cubic-order nilpotent singular point, $O(0,0)$ is a $10$th-order weak
focus, it is easy to show that the perturbed system of (3.1), given by

$\begin{array}[]{l}\begin{split}\dot{x}=&\ \delta
x+y+a_{30}x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{50}x^{5}+a_{41}x^{4}y\\\
&+a_{32}x^{3}y^{2}+a_{23}x^{2}y^{3}+a_{14}xy^{4}+a_{05}y^{5},\\\ \dot{y}=&\
\delta y-2x^{3},\end{split}\end{array}$ (3.7)

can generate ten limit cycles enclosing an elementary node at the origin of
system (3.9).

Theorem 2.2 in [32] implies the following result,

######  Theorem 3.3.

If the origin of system (3.7) is a $10$th-order weak focus, then within a
small neighborhood of the origin, for $0<\delta\ll 1$, perturbing the
coefficients of system (3.7) can yield ten small-amplitude limit cycles
bifurcating from the elementary node $O(0,0)$.

Proof. The origin of system (3.7) is a $10$th-order weak focus if and only if

$\begin{split}a_{30}&=0;a_{12}=-\frac{5a_{50}}{2};a_{32}=-\frac{3a_{21}a_{50}}{2};\\\
a_{14}&=-\frac{5}{12}(6a_{03}a_{50}+a_{41}a_{50});\\\
a_{23}&=\frac{1}{6}(-a_{21}a_{41}+10a_{50}^{2});\\\
a_{05}&=\frac{1}{60}(-10a_{03}a_{41}-a_{41}^{2}+3a_{21}a_{50}^{2});\\\
a_{03}&=\frac{1}{(300a_{50}^{2})}(-2a_{21}a_{41}^{2}-9a_{21}^{2}a_{50}^{2}-100a_{41}a_{50}^{2}).\end{split}$

and

$\begin{split}J_{0}&=\frac{\partial(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8},\lambda_{9})}{\partial(a_{30},a_{12},a_{32},a_{14},a_{23},a_{05},a_{03},a_{21},a_{41})}\\\
&=-\frac{2048a_{50}^{3}}{56772622027996875}(224a_{21}^{2}a_{41}^{7}+3024a_{21}^{3}a_{41}^{5}a_{50}^{2}+11000a_{21}a_{41}^{6}a_{50}^{2}\\\
&~{}~{}+13608a_{21}^{4}a_{41}^{3}a_{50}^{4}+108900a_{21}^{2}a_{41}^{4}a_{50}^{4}+123500a_{41}^{5}a_{50}^{4}\\\
&~{}~{}+20412a_{21}^{5}a_{41}a_{50}^{6}+311850a_{21}^{3}a_{41}^{2}a_{50}^{6}+783000a_{21}a_{41}^{3}a_{50}^{6}\\\
&~{}~{}+200475a_{21}^{4}a_{50}^{8}+1022625a_{21}^{2}a_{41}a_{50}^{8}+450000a_{41}^{2}a_{50}^{8}).\end{split}$

Furthermore,

$\begin{array}[]{l}\begin{split}R_{5}=&\ {\rm
Resultant}[f_{2},\frac{J_{0}}{a_{50}^{3}},a_{21}]\\\ =&\
97384a_{41}^{9}-98391600a_{41}^{6}a_{50}^{4}+24582976875a_{41}^{3}a_{50}^{8}+80858250000a_{50}^{12},\\\
&\ {\rm Resultant}[R_{5},R_{1},a_{21}]\\\ =&\
1928337060674939567063811915624524516160347438902935330714052761390000\\\ &\
00000000000000000000000a_{50}^{72}\neq 0.\end{split}\end{array}$

So Theorem 2.2 in [32] yields the conclusion holds.

###  3.2 Case 2: $n\geq 2$.

For this case, system (1.4) can be written as

$\begin{array}[]{l}\begin{split}\dot{x}=&\
y+x(a_{30}x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{2n+3,0}x^{2n+3}\\\
&+a_{2n+2,1}x^{2n+2}y+a_{2n+1,2}x^{2n+1}y^{2}+\cdots+a_{1,2n+2}xy^{2n+2}\\\
&+a_{0,2n+3}y^{2n+3})\equiv\ X_{1}(x,y),\\\ \dot{y}=&\
-2x^{3}.\end{split}\end{array}$ (3.8)

######  Theorem 3.4.

For $n\geq 2$, the origin of system (3.8) is at most a $(n\\!+\\!4)$th-order
weak focus. If the origin of system (3.8) is a $(n\\!+\\!4)$th-order weak
focus, then within a small neighborhood of the origin, perturbing the
coefficients of system (3.8) can yield $n+4$ small-amplitude limit cycles
around the elementary node $O(0,0)$.

Proof. For a nilpotent system, in order to study the dynamical behavior in the
neighborhood of the origin, we could consider $y$ and $x^{2}$ to be
infinitesimal equivalence in the neighborhood of the origin, see [32].
Construct a comparison system,

$\begin{array}[]{l}\begin{split}\dot{x}&=y+x(a_{21}x^{2}y+a_{03}y^{3}+a_{2n+2,1}x^{2n+2}y+\cdots+a_{1,2n+2}xy^{2n+2})\\\
&\equiv X_{2}(x,y),\\\ \dot{y}&=-2x^{3},\end{split}\end{array}$ (3.9)

which shows that the system is symmetric with the $x$-axis, and so the origin
$O(0,0)$ is a center.

Next, we compute the determinant of system (3.7) to obtain

$\begin{array}[]{rl}J_{1}=\\!\\!\\!&\det\left[\begin{array}[]{cccc}X_{1}(x,y)&-2x^{3}\\\
X_{2}(x,y)&-2x^{3}\end{array}\right]\\\\[12.91663pt]
=\\!\\!\\!&-2x^{4}(a_{30}x^{2}+a_{12}y^{2}+a_{2n+3,0}x^{2n+2}\\\
&+a_{2n+1,2}x^{2n}y^{2}+\cdots+a_{3,2n}x^{2}y^{2n}+a_{1,2n+2}y^{2n+2}).\end{array}$

By treating the $y$ and $x^{2}$ as infinitesimal equivalence in the
neighborhood of the origin, we have

$\begin{split}J_{1}&=-2x^{4}(a_{30}x^{2}+a_{12}x^{4}+a_{2n+3,0}x^{2n+2}+a_{2n+1,2}x^{2n+4}\\\
&+\cdots+a_{3,2n}x^{4n+2}+a_{1,2n+2}x^{4n+4}),\end{split}$ (3.10)

which implies that
$a_{30},\,a_{12},\,a_{2n+3,0},\,a_{2n+1,2},\,\cdots,\,a_{3,2n},\,a_{1,2n+2}$
could be taken as the focus values of system (3.7). So for $n\geq 2$, the
origin of system (3.8) is at most an $(n\\!+\\!4)$th-order weak focus.
According to Theorem 4.1.5 in [31], within a small neighborhood of the origin,
perturbing the coefficients of system (3.8) can yield $n+4$ small-amplitude
limit cycles around the elementary node $O(0,0)$.

Furthermore, similar to Proposition 3.1, we have the following result.

######  Theorem 3.5.

For $n\geq 2$, the origin of system (3.8) is an analytic center if and only if

$a_{30}=a_{12}=a_{2n+3,0}=a_{2n+1,2}=\cdots=a_{3,2n}=a_{1,2n+2}=0.$ (3.11)

Proof. When
$a_{30}=a_{12}=a_{2n+3,0}=a_{2n+1,2}=\cdots=a_{3,2n}=a_{1,2n+2}=0,$ system
(3.8) could be rewritten as

$\begin{array}[]{l}\begin{split}\dot{x}&=y+a_{21}x^{2}y+a_{03}y^{3}+a_{41}x^{4}y+a_{23}x^{2}y^{3}+a_{05}y^{5}\\\
&+\cdots+a_{2n+2,1}x^{2n+2}y+\cdots+a_{0,2n+3}y^{2n+3},\\\
\dot{y}&=-2x^{3}.\end{split}\end{array}$ (3.12)

Obviously, system (3.12) is symmetric with the $y$-axis. According to Theorem
11 in [9], the origin is an analytic center of system (3.8).

##  4 Analytic centers of system (1.5)

Now we turn to discuss the analytic center conditions for system (1.5). It
also has two cases.

###  4.1 Case A: $n=1$.

For this case, system (1.5) can be written as

$\begin{array}[]{l}\begin{split}\dot{x}&=y,\\\
\dot{y}&=-2x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{50}x^{5}+a_{41}x^{4}y\\\
&~{}~{}~{}+a_{32}x^{3}y^{2}+a_{23}x^{2}y^{3}+a_{14}xy^{4}+a_{05}y^{5},\\\
\end{split}\end{array}$ (4.1)

for which we can find a formal series
$M(x,y)=x^{4}+y^{2}+o((x^{2}+y^{2})^{2})$ according to Theorem 2.1, provided
that (2.3) holds. Carrying out calculations with help of Mathematica and
applying the recursive formulae in Theorem 2.2 to system (4.1), we obtain

$\begin{split}\omega_{3}&=\omega_{4}=\omega_{5}=0,\\\
\omega_{6}&=-\frac{1}{3}(4s-1)a_{21},\\\ \omega_{7}&=3(s+1)c_{03},\\\
\omega_{8}&=-\frac{1}{5}(4s-3)(6a_{03}+a_{41}),\\\ \omega_{9}&=0,\\\
\omega_{10}&=-\frac{1}{7}(4s-5)(2a_{03}a_{12}-2a_{23}+3a_{03}a_{50}),\\\
\omega_{11}&=\frac{15}{4}(s+1)c_{05},\\\
\omega_{12}&=\frac{1}{30}(4s-5)(40a_{05}-4a_{03}a_{32}-2a_{03}a_{12}a_{50}-5a_{03}a_{50}^{2}),\\\
\omega_{13}&=0,\\\
\omega_{14}&=\frac{a_{03}}{154}(4s-9)(48a_{03}^{2}-40a_{14}+12a_{12}a_{32}+6a_{12}^{2}a_{50}\\\
&\hskip 36.135pt+12a_{32}a_{50}+21a_{12}a_{50}^{2}+18a_{50}^{3}).\end{split}$

Then, for $a_{12}+2a_{50}\neq 0$,

$\begin{split}\omega_{15}&=\frac{35}{8}(s+1)c_{07},\\\
\omega_{16}&=\frac{a_{03}}{520}(4s-11)(64a_{03}^{2}a_{12}+16a_{32}^{2}+128a_{03}^{2}a_{50}+16a_{12}a_{32}a_{50}\\\
&\hskip
36.135pt+4a_{12}^{2}a_{50}^{2}+32a_{32}a_{50}^{2}+20a_{12}a_{50}^{3}+23a_{50}^{4}),\\\
\omega_{17}&=0,\\\
\omega_{18}&=-\frac{a_{03}}{61600(a_{12}+2a_{50})}(4s-13)(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})(112a_{12}^{2}a_{32}-432a_{32}^{2}\\\
&\hskip
36.135pt+56a_{12}^{3}a_{50}-96a_{12}a_{32}a_{50}+200a_{12}^{2}a_{50}^{2}-640a_{32}a_{50}^{2}\\\
&\hskip 36.135pt+120a_{12}a_{50}^{3}-85a_{50}^{4}),\\\
\omega_{19}&=\frac{315}{64}(1+s)c_{09},\\\
\omega_{20}&=-\frac{a_{03}}{40840800(a_{12}+2a_{50})^{2}}(4s-15)(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})(14372996a_{12}^{4}a_{32}\\\
&\hskip
36.135pt-63894256a_{12}^{2}a_{32}^{2}+34076160a_{32}^{3}+7186498a_{12}^{5}a_{50}-10734116a_{12}^{3}a_{32}a_{50}\\\
&\hskip
36.135pt-12772032a_{12}a_{32}^{2}a_{50}+28572751a_{12}^{4}a_{50}^{2}-99036264a_{12}^{2}a_{32}a_{50}^{2}\\\
&\hskip
36.135pt+45544768a_{32}^{2}a_{50}^{2}+26958196a_{12}^{3}a_{50}^{3}-39087216a_{12}a_{32}a_{50}^{3}),\\\
\omega_{21}&=0,\end{split}$ (4.2)
$\begin{split}\omega_{22}&=\frac{a_{03}}{11639628000(a_{12}+2a_{50})^{2}}(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})\,f_{4};\end{split}$

and for $a_{12}+2a_{50}=0$,

$\begin{split}\omega_{16}&=\frac{a_{03}}{520}(4s-11)(-4a_{32}+a_{50}^{2})(4a_{32}+a_{50}^{2}),\\\
\omega_{17}&=0,\end{split}$

and in addition if $a_{32}=\frac{a_{50}^{2}}{4}$,

$\begin{split}\omega_{18}&=\frac{9\,a_{03}}{7700}(4s-13)(4a_{32}+a_{50}^{2})(24a_{03}^{2}+a_{50}^{3}),\\\
\omega_{19}&=\frac{315}{64}(s+1)c_{09},\\\
\omega_{20}&=-\frac{7\,a_{03}}{13260}(4s-15)a_{50}^{6},\\\ \omega_{21}&=0,\\\
\omega_{22}&=-\frac{a_{03}a_{50}^{7}}{13856700(1+s)}(5391-205861s+66718s^{2});\end{split}$

if $a_{32}=-\frac{a_{50}^{2}}{4}$,

$\begin{split}\omega_{18}&=0,\\\ \omega_{19}&=\frac{315}{64}(s+1)c_{09},\\\
\omega_{20}&=\frac{2\,a_{03}}{5525}(4s-15)(16a_{03}^{2}+a_{50}^{3})(27a_{03}^{2}+2a_{50}^{3}),\\\
\omega_{21}&=0,\\\
\omega_{22}&=-\frac{4\,a_{03}a_{50}}{1154725(1+s)}(16a_{03}^{2}+a_{50}^{3})(27a_{03}^{2}+2a_{50}^{3}),\\\
\end{split}$

where

$\begin{split}f_{4}=&-175151884140096a_{12}^{5}a_{32}+900479057104608a_{12}^{3}a_{32}^{2}-870691837997952a_{12}a_{32}^{3}\\\
&-87575942070048a_{12}^{6}a_{50}+191710117401504a_{12}^{4}a_{32}a_{50}+108696440458488a_{12}^{2}a_{32}^{2}a_{50}\\\
&-135343772601984a_{32}^{3}a_{50}-348204560750520a_{12}^{5}a_{50}^{2}+1449441463187484a_{12}^{3}a_{32}a_{50}^{2}\\\
&-1227387674864544a_{12}a_{32}^{2}a_{50}^{2}-328545834076764a_{12}^{4}a_{50}^{3}+705733010157654a_{12}^{2}a_{32}a_{50}^{3}\\\
&-180965555611680a_{32}^{2}a_{50}^{3}-196530256579516a_{12}^{5}a_{32}s+956545497558256a_{12}^{3}a_{32}^{2}s\\\
&-775708121404800a_{12}a_{32}^{3}s-98265128289758a_{12}^{6}a_{50}s+188198718433828a_{12}^{4}a_{32}a_{50}s\\\
&-92031891970176a_{32}^{3}a_{50}s-390699835897045a_{12}^{5}a_{50}^{2}s+1519172249991516a_{12}^{3}a_{32}a_{50}^{2}s\\\
&-1080068393470816a_{12}a_{32}^{2}a_{50}^{2}s-368634374118786a_{12}^{4}a_{50}^{3}s+690504693129726a_{12}^{2}a_{32}a_{50}^{3}s\\\
&-123060065895200a_{32}^{2}a_{50}^{3}s+80500800862640a_{12}^{5}a_{32}s^{2}-396831061224512a_{12}^{3}a_{32}^{2}s^{2}\\\
&+336523546570752a_{12}a_{32}^{3}s^{2}+40250400431320a_{12}^{6}a_{50}s^{2}-79597335692936a_{12}^{4}a_{32}a_{50}s^{2}\\\
&-56511998888512a_{12}^{2}a_{32}^{2}a_{50}s^{2}+43311880631808a_{32}^{3}a_{50}s^{2}+160035098537960a_{12}^{5}a_{50}^{2}s^{2}\\\
&-632261525950008a_{12}^{3}a_{32}a_{50}^{2}s^{2}+470151299998208a_{12}a_{32}^{2}a_{50}^{2}s^{2}+150997488382038a_{12}^{4}a_{50}^{3}s^{2}\end{split}$
$\begin{split}&+142715605813496a_{12}^{2}a_{32}^{2}a_{50}s-292287689531688a_{12}^{2}a_{32}a_{50}^{3}s^{2}+57905489716480a_{32}^{2}a_{50}^{3}s^{2}.\end{split}$

Based on (2.6) and (4.2), it is easy to find the first nine quasi-Lyapunov
constants of system (4.1).

######  Theorem 4.1.

The first nine quasi-Lyapunov constants evaluated at origin of system (4.1)
are given by

$\begin{split}\lambda_{1}&=-\frac{1}{3}a_{21},\\\
\lambda_{2}&=-\frac{1}{5}(6a_{03}+a_{41}),\\\
\lambda_{3}&=-\frac{1}{7}(2a_{03}a_{12}-2a_{23}+3a_{03}a_{50}),\\\
\lambda_{4}&=\frac{1}{30}(40a_{05}-4a_{03}a_{32}-2a_{03}a_{12}a_{50}-5a_{03}a_{50}^{2}),\\\
\lambda_{5}&=\frac{a_{031}}{154}(48a_{03}^{2}-40a_{14}+12a_{12}a_{32}+6a_{12}^{2}a_{50}+12a_{32}a_{50}+21a_{12}a_{50}^{2}+18a_{50}^{3}).\end{split}$

Then, for $a_{12}+2a_{50}\neq 0$,

$\begin{split}\lambda_{6}&=\frac{a_{03}}{520}(64a_{03}^{2}a_{12}+16a_{32}^{2}+128a_{03}^{2}a_{50}+16a_{12}a_{32}a_{50}+4a_{12}^{2}a_{50}^{2}+32a_{32}a_{50}^{2}\\\
&~{}~{}~{}~{}+20a_{12}a_{50}^{3}+23a_{50}^{4}),\\\
\lambda_{7}&=-\frac{a_{03}}{61600(a_{12}+2a_{50})}(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})(112a_{12}^{2}a_{32}-432a_{32}^{2}+56a_{12}^{3}a_{50}\\\
&-96a_{12}a_{32}a_{50}+200a_{12}^{2}a_{50}^{2}-640a_{32}a_{50}^{2}+120a_{12}a_{50}^{3}-85a_{50}^{4}),\\\
\lambda_{8}&=-\frac{a_{03}}{40840800(a_{12}+2a_{50})^{2}}(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})(14372996a_{12}^{4}a_{32}\\\
&\ \ \
-63894256a_{12}^{2}a_{32}^{2}+34076160a_{32}^{3}+7186498a_{12}^{5}a_{50}-10734116a_{12}^{3}a_{32}a_{50}\\\
&\ \ \
-12772032a_{12}a_{32}^{2}a_{50}+28572751a_{12}^{4}a_{50}^{2}-99036264a_{12}^{2}a_{32}a_{50}^{2}\\\
&\ \ \
+45544768a_{32}^{2}a_{50}^{2}+26958196a_{12}^{3}a_{50}^{3}-39087216a_{12}a_{32}a_{50}^{3}),\\\
\lambda_{9}&=\frac{a_{03}}{11639628000(a_{12}+2a_{50})^{2}}(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})\,f_{4};\end{split}$

while for $a_{12}+2a_{50}=0$,

$\begin{split}\lambda_{6}&=\frac{a_{03}}{520}(-4a_{32}+a_{50}^{2})(4a_{32}+a_{50}^{2}),\\\
\end{split}$

and in addition if $a_{32}=\frac{a_{50}^{2}}{4}$,

$\begin{split}\lambda_{7}&=\frac{9\,a_{03}}{7700}(4a_{32}+a_{50}^{2})(24a_{03}^{2}+a_{50}^{3}),\\\
\lambda_{8}&=-\frac{7\,a_{03}}{13260}a_{50}^{6},\\\
\lambda_{9}&=-\frac{a_{03}a_{50}^{7}}{13856700(1+s)}(5391-205861s+66718s^{2});\end{split}$

if $a_{32}=-\frac{a_{50}^{2}}{4}$,

$\begin{split}\lambda_{7}&=0,\\\
\lambda_{8}&=\frac{2\,a_{03}}{5525}(16a_{03}^{2}+a_{50}^{3})(27a_{03}^{2}+2a_{50}^{3}),\\\
\lambda_{9}&=-\frac{4\,a_{03}a_{50}}{1154725(1+s)}(16a_{03}^{2}+a_{50}^{3})(27a_{03}^{2}+2a_{50}^{3}),\\\
\end{split}$

where $\lambda_{k-1}=0$ for $k=2,\cdots,9$ have been used in computing
$\lambda_{k}$.

Furthermore, the following result can be easily obtained.

######  Proposition 4.1.

For $n=1$, origin of system (4.1) is an analytic center if and only if one of
the following conditions holds:

$a_{21}=a_{03}=a_{41}=a_{23}=a_{05}=0;$ (4.3)
$\begin{split}&a_{21}=a_{14}=a_{05}=0,\quad a_{41}=-6a_{03},\quad\\\
&a_{23}=\textstyle\frac{1}{2}(2a_{12}+3a_{50})a_{03},\quad
a_{03}^{2}=-\frac{1}{16}a_{50}^{3};\end{split}$ (4.4)
$\begin{split}&a_{21}\\!=\\!a_{05}\\!=\\!0,\ a_{41}\\!=\\!-6a_{03},\
a_{12}\\!=\\!-2a_{50},\ a_{23}\\!=\\!-\textstyle\frac{1}{2}a_{03}a_{50},\\\ &\
a_{32}\\!=\\!\frac{1}{4}a_{50}^{2},\
a_{03}^{2}\\!=\\!-\frac{2}{27}a_{50}^{3},\
a_{14}\\!=\\!-\frac{1}{72}a_{50}^{3}.\end{split}$ (4.5)

Proof. It is easy to get the conditions (4.3)-(LABEL:eq4.5) by setting
$\lambda_{1}=\lambda_{2}=\cdots=\lambda_{9}=0$. When $a_{50}\neq 0$, let

$\begin{split}f_{5}=&\
112a_{12}^{2}a_{32}-432a_{32}^{2}+56a_{12}^{3}a_{50}-96a_{12}a_{32}a_{50}+200a_{12}^{2}a_{50}^{2}-640a_{32}a_{50}^{2}\\\
&+120a_{12}a_{50}^{3}-85a_{50}^{4},\\\ f_{6}=&\
14372996a_{12}^{4}a_{32}-63894256a_{12}^{2}a_{32}^{2}+34076160a_{32}^{3}+7186498a_{12}^{5}a_{50}\\\
&-10734116a_{12}^{3}a_{32}a_{50}-12772032a_{12}a_{32}^{2}a_{50}+28572751a_{12}^{4}a_{50}^{2}\\\
&-99036264a_{12}^{2}a_{32}a_{50}^{2}+45544768a_{32}^{2}a_{50}^{2}\\\
&+26958196a_{12}^{3}a_{50}^{3}-39087216a_{12}a_{32}a_{50}^{3}.\end{split}$

Then, we obtain

$\begin{split}R_{3}=&\ {\rm Resultant}[f_{4},f_{5},a_{32}]\\\
=&-200206652313600a_{50}^{3}(a_{12}+2a_{50})^{4}(72a_{12}^{5}+652a_{12}^{4}a_{50}+2694a_{12}^{3}a_{50}^{2}\\\
&+6043a_{12}^{2}a_{50}^{3}+7092a_{12}a_{50}^{4}+3463a_{50}^{5}),\\\ R_{4}=&\
{\rm Resultant}[f_{4},f_{6},a_{32}]\\\
=&-289298612593152000a_{50}^{3}(a_{12}+2a_{50})^{4}(278734084992a_{12}^{7}+3588989330016a_{12}^{6}a_{50}\\\
&+21026493958464a_{12}^{5}a_{50}^{2}+71854303647672a_{12}^{4}a_{50}^{3}+152324731255716a_{12}^{3}a_{50}^{4}\\\
&+197669760539760a_{12}^{2}a_{50}^{5}+144211704399495a_{12}a_{50}^{6}+45502270176438a_{50}^{7}\\\
&+1651136278704a_{12}^{7}s+18838630495800a_{12}^{6}a_{50}s+98153819816532a_{12}^{5}a_{50}^{2}s\\\
&+294846821571666a_{12}^{4}a_{50}^{3}s+534120897148782a_{12}^{3}a_{50}^{4}s+565346904516561a_{12}^{2}a_{50}^{5}s\\\
&-310952885702031a_{12}a_{50}^{6}s+62534145621954a_{50}^{7}s+1859553268056a_{12}^{7}s^{2}\end{split}$
$\begin{split}&+17577034105268a_{12}^{6}a_{50}s^{2}+76054378966082a_{12}^{5}a_{50}^{2}s^{2}+181916843290105a_{12}^{4}a_{50}^{3}s^{2}\\\
&+238094684972342a_{12}^{3}a_{50}^{4}s^{2}+147267056136244a_{12}^{2}a_{50}^{5}s^{2}+19507558179230a_{12}a_{50}^{6}s^{2}\\\
&-6987410240074a_{50}^{7}s^{2}-1714957345728a_{12}^{7}s^{3}-17270801713568a_{12}^{6}a_{50}s^{3}\\\
&-80278393721648a_{12}^{5}a_{50}^{2}s^{3}-212296048411048a_{12}^{4}a_{50}^{3}s^{3}-328607615124608a_{12}^{3}a_{50}^{4}s^{3}\\\
&-285216632755972a_{12}^{2}a_{50}^{5}s^{3}-121914622605938a_{12}a_{50}^{6}s^{3}-19568176640258a_{50}^{7}s^{3}\\\
&+301352777088a_{12}^{7}s^{4}+3072037299008a_{12}^{6}a_{50}s^{4}+14465726297408a_{12}^{5}a_{50}^{2}s^{4}\\\
&+38896515312256a_{12}^{4}a_{50}^{3}s^{4}+61684793716376a_{12}^{3}a_{50}^{4}s^{4}+55602984609100a_{12}^{2}a_{50}^{5}s^{4}\\\
&+25319000517368a_{12}a_{50}^{6}s^{4}+4451109045332a_{50}^{7}s^{4}).\end{split}$

Further, with the aid of Mathematica, we obtain for $\forall s\in Z^{+}$

$\begin{split}G_{2}=&\ {\rm Resultant}[R_{3},R_{4},a_{12}]\\\ =&\
30984189289342953910272000a_{50}^{35}(1+s)^{5}(-17+4s)^{5}\\\
&\times(-123287750793562256929839075859953216\\\
&-33902812452688795016920021129342044624s\\\
&-180855325034978657368019444342423080236s^{2}\\\
&-1067066959204615961659004741488392865575s^{3}\\\
&-3328343437962444375340762099992891472110s^{4}\\\
&-4773196954655562390848005555854946921459s^{5}\\\
&+14241540803784759916727236436410335714320s^{6}\\\
&-10732088467496096467063502795815305475120s^{7}\\\
&+3721436248399857364295558363131840668032s^{8}\\\
&-625676462230475741935920059840273863168s^{9}\\\
&+41454785818979861302571809000901003264s^{10})\neq 0.\\\ \end{split}$

The above calculations indicate that the equations $f_{4}=f_{5}=f_{6}=0$ do
not have real solutions, namely, there do not exist other analytic center
conditions for system (4.1) if $a_{50}\neq 0$.

When the conditions in (4.3) hold, system (4.1) becomes

$\begin{array}[]{l}\begin{split}\dot{x}&=y,\\\
\dot{y}&=-2x^{3}+a_{12}xy^{2}+a_{50}x^{5}+a_{32}x^{3}y^{2}+a_{14}xy^{4}.\\\
\end{split}\end{array}$ (4.6)

Obviously, this system is symmetric with the $y$-axis, implying that the
origin of system (4.6) is an analytic center due to Theorem 11 in [9].

When the conditions in (4.4) are satisfied, system (4.1) becomes

$\begin{array}[]{l}\begin{split}\dot{x}=&\ y,\\\ \dot{y}=&\
\frac{1}{4}(-8x^{3}+4a_{50}x^{5}-24a_{03}x^{4}y+4a_{12}xy^{2}-2a_{12}a_{50}x^{3}y^{2}\\\
&\quad\,-5a_{50}^{2}x^{3}y^{2}+4a_{03}y^{3}+4a_{03}a_{12}x^{2}y^{3}+6a_{03}a_{50}x^{2}y^{3}).\end{split}\end{array}$
(4.7)

Introducing the transformation,

$x=x,\quad y=\frac{(-2+a_{50}x^{2})z}{2(-2+a_{50}x^{2}+a_{03}xz)},$

and time scaling,

$T=\frac{2(-2+a_{50}x^{2})^{3}t}{-2+a_{50}x^{2}-2a_{03}xy},$

into system (4.7) results in

$\begin{array}[]{l}\begin{split}\frac{dx}{dT}=&\ z(a_{50}x^{2}-2)^{2},\\\
\frac{dz}{dT}=&-\frac{1}{4}x\big{(}128x^{2}-192a_{50}x^{4}+96a_{50}^{2}x^{6}-16a_{50}^{3}x^{8}-16a_{12}z^{2}\\\
&+24a_{12}a_{50}x^{2}z^{2}+20a_{50}^{2}x^{2}z^{2}-96a_{03}^{2}x^{4}z^{2}-12a_{12}a_{50}^{2}x^{4}z^{2}\\\
&-20a_{50}^{3}x^{4}z^{2}+48a_{03}^{2}a_{50}x^{6}z^{2}+2a_{12}a_{50}^{3}x^{6}z^{2}+5a_{50}^{4}x^{6}z^{2}\\\
&+32a_{03}^{3}x^{5}z^{3}+2a_{03}a_{50}^{3}x^{5}z^{3}),\end{split}\end{array}$
(4.8)

which is symmetric with the $z$-axis because
$a_{03}^{2}=-\frac{a_{50}^{3}}{16}$. Thus, according to Theorem 11 in [9], the
origin of system (4.7) is an analytic center.

Similarly, when the conditions in (LABEL:eq4.5) hold, system (4.1) becomes

$\begin{array}[]{l}\begin{split}\dot{x}=&\ y,\\\ \dot{y}=&\
\frac{1}{72}(-144x^{3}+72a_{50}x^{5}-432a_{03}x^{4}y-144a_{50}xy^{2}-18a_{50}^{2}x^{3}y^{2}\\\
&+72a_{03}y^{3}-36a_{03}a_{50}x^{2}y^{3}-a_{50}^{3}xy^{4}),\end{split}\end{array}$
(4.9)

for which there exists an analytic integrating factor,

$u(x,y)=\frac{e^{-\frac{3}{8}a_{50}^{2}x^{4}}}{(1-\frac{1}{2}a_{50}x^{2}+\frac{3}{4}a_{50}xy)^{4}},$

indicating that the origin of system (4.9) is an analytic center.

Therefore, Proposition 4.1 implies the following result.

######  Theorem 4.2.

The necessary and sufficient conditions for the origin of system (4.1) being
an analytic center are given by the vanishing of the first nine quasi-Lyapunov
constants, that is, one of the conditions in Proposition 4.1 is satisfied.

Similarly, when the cubic-order nilpotent singular point $O(0,0)$ is a $9$th-
order weak focus, it is easy to prove that the perturbed system of (4.1),
given by

$\begin{array}[]{l}\begin{split}\dot{x}=&\ \delta x+y,\\\ \dot{y}=&\ \delta
y-2x^{3}+a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{50}x^{5}+a_{41}x^{4}y\\\
&+a_{32}x^{3}y^{2}+a_{23}x^{2}y^{3}+a_{14}xy^{4}+a_{05}y^{5},\\\
\end{split}\end{array}$ (4.10)

can generate nine limit cycles enclosing an elementary node at the origin. The
proof is similar to that for Theorem 6.

######  Theorem 4.3.

If the origin of system (4.10) is a $9$th-order weak focus, then within a
small neighborhood of the origin, for $0<\delta\ll 1$, system (4.10) can yield
nine small-amplitude limit cycles around the elementary node $O(0,0)$.

Proof. The origin of system (4.10) is a $9$th-order weak focus if and only if

$\begin{split}a_{21}&=0;\\\ a_{41}&=-6a_{03};\\\
a_{23}&=\frac{1}{2}a_{03}(2a_{12}+3a_{50});\\\
a_{05}&=\frac{1}{40}(4a_{03}a_{32}+2a_{03}a_{12}a_{50}+5a_{03}a_{50}^{2});\\\
a_{14}&=\frac{3}{40}(16a_{03}^{2}+4a_{12}a_{32}+2a_{12}^{2}a_{50}+4a_{32}a_{50}+7a_{12}a_{50}^{2}+6a_{50}^{3});\\\
a_{03}^{2}&=\frac{1}{64(a_{12}+2a_{50})}(-16a_{32}^{2}-16a_{12}a_{32}a_{50}-4a_{12}^{2}a_{50}^{2}-32a_{32}a_{50}^{2}-20a_{12}a_{50}^{3}-23a_{50}^{4}).\end{split}$

and

$\begin{split}J_{2}&=\frac{\partial(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8})}{\partial(a_{21},a_{41},a_{23},a_{05},a_{14},a_{03},a_{32},a_{12})}\\\
&=-\frac{(4a_{32}+2a_{12}a_{50}+5a_{50}^{2})}{38846808000(a_{12}+2a_{50})^{3}}(16a_{32}^{2}+16a_{12}a_{32}a_{50}+4a_{12}^{2}a_{50}^{2}+32a_{32}a_{50}^{2}+20a_{12}a_{50}^{3}+23a_{50}^{4})^{2}\\\
&\times(9658653312a_{12}^{6}a_{32}^{2}-94084036480a_{12}^{4}a_{32}^{3}+233154747648a_{12}^{2}a_{32}^{4}\\\
&-58883604480a_{32}^{5}+9658653312a_{12}^{7}a_{32}a_{50}-65797494336a_{12}^{5}a_{32}^{2}a_{50}+14579296576a_{12}^{3}a_{32}^{3}a_{50}\\\
&+288547550208a_{12}a_{32}^{4}a_{50}+2414663328a_{12}^{8}a_{50}^{2}+28912166304a_{12}^{6}a_{32}a_{50}^{2}\\\
&-406316235904a_{12}^{4}a_{32}^{2}a_{50}^{2}+936674204416a_{12}^{2}a_{32}^{3}a_{50}^{2}-127467869184a_{32}^{4}a_{50}^{2}\\\
&+19144952176a_{12}^{7}a_{50}^{3}-109760863872a_{12}^{5}a_{32}a_{50}^{3}-244778217136a_{12}^{3}a_{32}^{2}a_{50}^{3}\\\
&+1036308069632a_{12}a_{32}^{3}a_{50}^{3}+49040513184a_{12}^{6}a_{50}^{4}-522094417624a_{12}^{4}a_{32}a_{50}^{4}\\\
&+985367488400a_{12}^{2}a_{32}^{2}a_{50}^{4}-6731934720a_{32}^{3}a_{50}^{4}+18296999196a_{12}^{5}a_{50}^{5}\\\
&-545453749756a_{12}^{3}a_{32}a_{50}^{5}+1124651917440a_{12}a_{32}^{2}a_{50}^{5}-114440868400a_{12}^{4}a_{50}^{6}\\\
&+131607099120a_{12}^{2}a_{32}a_{50}^{6}+107989619520a_{32}^{2}a_{50}^{6}-173404183515a_{12}^{3}a_{50}^{7}\\\
&+352746916240a_{12}a_{32}a_{50}^{7}-62864473110a_{12}^{2}a_{50}^{8}+39921858880a_{32}a_{50}^{8}+12556768140a_{12}a_{50}^{9}).\end{split}$

Furthermore,

$\begin{array}[]{l}\begin{split}R_{6}=&\ {\rm
Resultant}[f_{5},J_{2},a_{32}]\\\ =&\
(49a_{12}^{3}+196a_{12}^{2}a_{50}+357a_{12}a_{50}^{2}+466a_{50}^{3})^{2}\times(225792a_{12}^{9}+3214400a_{12}^{8}a_{50}\\\
&+21345968a_{12}^{7}a_{50}^{2}+87227792a_{12}^{6}a_{50}^{3}+246151320a_{12}^{5}a_{50}^{4}+509498952a_{12}^{4}a_{50}^{5}\\\
&+785491407a_{12}^{3}a_{50}^{6}+864775852a_{12}^{2}a_{50}^{7}+601885864a_{12}a_{50}^{8}+196001002a_{50}^{9}),\\\
&\ {\rm Resultant}[R_{6},R_{3},a_{12}]\\\ =&\
-110020282897692638814502001588125819626479649518678944188134258621907\\\
&66734966784000000a_{50}^{15}\neq 0.\end{split}\end{array}$

So Theorem 2.2 in [32] yields the conclusion holds.

###  4.2 Case B: $n\geq 2$.

For this case, system (1.5) can be written as

$\begin{array}[]{l}\begin{split}\dot{x}=&\ y,\\\
\dot{y}=&-2x^{3}+(a_{21}x^{2}y+a_{12}xy^{2}+a_{03}y^{3}+a_{2n+3,0}x^{2n+3}\\\
&+a_{2n+2,1}x^{2n+2}y+a_{2n+1,2}x^{2n+1}y^{2}+\cdots+a_{1,2n+2}xy^{2n+2}\\\
&+a_{0,2n+3}y^{2n+3})\\\ \equiv&\ Y_{1}(x,y).\end{split}\end{array}$ (4.11)

######  Theorem 4.4.

For $n\geq 2$, the origin of system (4.11) is at most a $(n\\!+\\!4)$th-order
weak focus. If the origin of system (4.11) is a $(n\\!+\\!4)$th-order weak
focus, then within a small neighborhood of the origin of its perturbed
system,, perturbing the coefficients of system (4.11) can yield $n+4$ small-
amplitude limit cycles enclosing the elementary node $O(0,0)$.

Proof. The proof is similar to that for Theorem 3.4. We construct a comparison
system for system (4.1),

$\begin{array}[]{l}\begin{split}\dot{x}&=y,\\\
\dot{y}&=-2x^{3}+x(a_{12}xy^{2}+a_{2n+3,0}x^{2n+3}+\cdots+a_{2,2n+1}x^{2}y^{2n+1})\\\
&\equiv Y_{2}(x,y).\end{split}\end{array}$ (4.12)

It is easy to see that system (4.12) is symmetric with the $x$-axis, and so
$O(0,0)$ is a center.

Next, we compute the determinant of system (4.12), yielding

$\begin{array}[]{rl}J_{3}=\\!\\!\\!&\det\left[\begin{array}[]{cccc}y&Y_{1}(x,y)\\\
y&Y_{2}(x,y)\end{array}\right]\\\\[12.91663pt]
=\\!\\!\\!&a_{21}x^{2}y^{2}+a_{03}y^{4}+a_{2n+2,1}x^{2n+2}y^{2}+a_{2n,3}x^{2n}y^{4}\\\
&+\cdots+a_{2,2n+1}x^{2}y^{2n+2}+a_{0,2n+3}y^{2n+4}.\end{array}$

Similarly, we take the $y$ and $x^{2}$ as infinitesimal equivalence in the
neighborhood of the origin in order to study the dynamical behavior of (4.11)
around the origin. So, $J_{2}$ becomes

$\begin{split}J_{3}=&x^{4}(a_{21}x^{2}+a_{03}x^{4}+a_{2n+2,1}x^{2n+2}+a_{2n,3}x^{2n+2}\\\
&+\cdots+a_{2,2n+1}x^{4n+2}+a_{0,2n+3}x^{4n+4}),\end{split}$ (4.13)

implying that
$a_{21},\,a_{03},\,a_{2n+2,1},\,a_{2n,3},\,\cdots,\,a_{2,2n+1},\,a_{0,2n+3}$
could be considered as the focal values of the system. Therefore, for $n\geq
2$, the origin of system (4.11) is at most a $(n\\!+\\!4)$th-order weak focus.
According to Theorem 2.2 in [32], within a small neighborhood of the origin,
one can perturb the coefficients of system (4.11) to obtain $n+4$ small-
amplitude limit cycles around the elementary node $O(0,0)$.

Moreover, we have a similar theorem for this case.

######  Theorem 4.5.

For $n\geq 2$, the origin of system (4.11) is an analytic center if and only
if

$a_{21}=a_{03}=a_{2n+2,1}=a_{2n,3}=\cdots=a_{2,2n+1}=a_{0,2n+3}=0.$ (4.14)

Proof. When
$a_{21}=a_{03}=a_{2n+2,1}=a_{2n,3}=\cdots=a_{2,2n+1}=a_{0,2n+3}=0,$ system
(4.11) can be rewritten as

$\begin{array}[]{l}\begin{split}\dot{x}=&\ y,\\\
\dot{y}=&-2x^{3}+a_{12}xy^{2}+a_{2n+3,0}x^{2n+3}+a_{2n+1,2}x^{2n+1}y^{2}+\cdots+a_{1,2n+2}xy^{2n+2},\\\
\end{split}\end{array}$ (4.15)

Obviously, system (4.15) is symmetric with the $y$-axis. According to Theorem
11 in [9], the origin is an analytic center of system (4.11).

##  5 Conclusion

In this paper, two classes of lopsided systems have been studied on their
analytic integrable conditions and bifurcation of limit cycles. We have
obtained some analytic integrability conditions for each class of the systems
for case $n=1$. By using certain transformations or integrating factors, we
have proved that all conditions are sufficient and necessary. For case $n\geq
2$, we have constructed different comparison systems for each class of the
systems and shown that $n+4$ limit cycles may bifurcate from the origin of
each system. In addition, conditions for the origin being an analytic center
are obtained simultaneously.

##  6 Appendix

Detailed recursive MATHEMATICA code to compute the quasi-Lyapunov constants at
the origin of system (13): c [0,0]=0, c [1, 0]=0, c [0, 1]=0, c [2, 0]=0, c
[1, 1]=0, c [0, 2]=1; when k¡0 or j¡0, c [k,j]=0; else

$\begin{split}c[k,j]&=-\frac{1}{k(1+s)}(-10a_{50}c[-5+k,1+j]+a_{50}kc[-5+k,1+j]-5a_{50}sc[-5+k,1+j]\\\
&+a_{50}ksc[-5+k,1+j]-8a_{41}c[-4+k,j]+a_{41}kc[-4+k,j]-4a_{41}sc[-4+k,j]\\\
&+a_{41}ksc[-4+k,j]-4c[-4+k,2+j]-2jc[-4+k,2+j]-4sc[-4+k,2+j]\\\
&-2jsc[-4+k,2+j]-6a_{32}c[-3+k,-1+j]+a_{32}kc[-3+k,-1+j]\\\
&-3a_{32}sc[-3+k,-1+j]+a_{32}ksc[-3+k,-1+j]+a_{30}ksc[-3+k,1+j]\\\
&-6a_{30}c[-3+k,1+j]+a_{30}kc[-3+k,1+j]-3a_{30}sc[-3+k,1+j]\\\
&-4a_{23}c[-2+k,-2+j]+a_{23}kc[-2+k,-2+j]-2a_{23}sc[-2+k,-2+j]\\\
&+a_{23}ksc[-2+k,-2+j]-4a_{21}c[-2+k,j]+a_{21}kc[-2+k,j]-2a_{21}sc[-2+k,j]\\\
&+a_{21}ksc[-2+k,j]-2a_{14}c[-1+k,-3+j]+a_{14}kc[-1+k,-3+j]\\\
&-a_{14}sc[-1+k,-3+j]+a_{14}ksc[-1+k,-3+j]-2a_{12}c[-1+k,-1+j]\\\
&+a_{12}kc[-1+k,-1+j]-a_{12}sc[-1+k,-1+j]+a_{12}ksc[-1+k,-1+j]\\\
&+a_{05}kc[k,-4+j]+a_{05}ksc[k,-4+j]+a_{03}kc[k,-2+j]+a_{03}ksc[k,-2+j]).\\\
\end{split}$
$\begin{split}w[m]&=9a_{50}c[-4+m,0]-a_{50}mc[-4+m,0]+4a_{50}sc[-4+m,0]-a_{50}msc[-4+m,0]\\\
&+7a_{41}c[-3+m,-1]-a_{41}mc[-3+m,-1]+3a_{41}sc[-3+m,-1]\\\
&-a_{41}msc[-3+m,-1]+2c[-3+m,1]+2sc[-3+m,1]+5a_{32}c[-2+m,-2]\\\
&-a_{32}mc[-2+m,-2]+2a_{32}sc[-2+m,-2]-a_{32}msc[-2+m,-2]\\\
&+5a_{30}c[-2+m,0]-a_{30}mc[-2+m,0]+2a_{30}sc[-2+m,0]-a_{30}msc[-2+m,0]\\\
&+3a_{23}c[-1+m,-3]-a_{23}mc[-1+m,-3]+a_{23}sc[-1+m,-3]-a_{23}msc[-1+m,-3]\\\
&+3a_{21}c[-1+m,-1]-a_{21}mc[-1+m,-1]+a_{21}sc[-1+m,-1]-a_{21}msc[-1+m,-1]\\\
&+a_{14}c[m,-4]-a_{14}mc[m,-4]-a_{14}msc[m,-4]+a_{12}c[m,-2]-a_{12}mc[m,-2]\\\
&-a_{12}msc[m,-2]-a_{05}c[1+m,-5]-a_{05}mc[1+m,-5]-a_{05}sc[1+m,-5]\\\
&-a_{05}msc[1+m,-5]-a_{03}c[1+m,-3]-a_{03}mc[1+m,-3]-a_{03}sc[1+m,-3]\\\
&-a_{03}msc[1+m,-3]-c[1+m,-1]-mc[1+m,-1]-sc[1+m,-1]-msc[1+m,-1].\end{split}$

Detailed recursive MATHEMATICA code to compute the quasi-Lyapunov constants at
the origin of system (24): c [0,0]=0, c [1, 0]=0, c [0, 1]=0, c [2, 0]=0, c
[1, 1]=0, c [0, 2]=1; when k¡0 or j¡0, c [k,j]=0; else

$\begin{split}c[k,j]&=-\frac{1}{k(1+s)}(2a_{50}c[-6+k,2+j]+a_{50}jc[-6+k,2+j]+2a_{50}sc[-6+k,2+j]\\\
&+a_{50}jsc[-6+k,2+j]+a_{41}jc[-5+k,1+j]+a_{41}sc[-5+k,1+j]\\\
&+a_{41}jsc[-5+k,1+j]-2a_{32}c[-4+k,j]+a_{32}jc[-4+k,j]+a_{32}jsc[-4+k,j]\\\
&-4c[-4+k,2+j]-2jc[-4+k,2+j]-4sc[-4+k,2+j]-2jsc[-4+k,2+j]\\\
&-4a_{23}c[-3+k,-1+j]+a_{23}jc[-3+k,-1+j]-a_{23}sc[-3+k,-1+j]\\\
&+a_{23}jsc[-3+k,-1+j]+a_{21}jc[-3+k,1+j]+a_{21}sc[-3+k,1+j]\\\
&+a_{21}jsc[-3+k,1+j]-6a_{14}c[-2+k,-2+j]+a_{14}jc[-2+k,-2+j]\\\
&-2a_{14}sc[-2+k,-2+j]+a_{14}jsc[-2+k,-2+j]-2a_{12}c[-2+k,j]+a_{12}jc[-2+k,j]\\\
&+a_{12}jsc[-2+k,j]-8a_{05}c[-1+k,-3+j]+a_{05}jc[-1+k,-3+j]\\\
&-3a_{05}sc[-1+k,-3+j]+a_{05}jsc[-1+k,-3+j]-4a_{03}c[-1+k,-1+j]\\\
&+a_{03}jc[-1+k,-1+j]-a_{03}sc[-1+k,-1+j]+a_{03}jsc[-1+k,-1+j]).\\\
\end{split}$
$\begin{split}w[m]&=-a_{50}c[-5+m,1]-a_{50}sc[-5+m,1]+a_{41}c[-4+m,0]+3a_{32}c[-3+m,-1]\\\
&+a_{32}sc[-3+m,-1]+2c[-3+m,1]+2sc[-3+m,1]+5a_{23}c[-2+m,-2]\\\
&+2a_{23}sc[-2+m,-2]+a_{21}c[-2+m,0]+7a_{14}c[-1+m,-3]+3a_{14}sc[-1+m,-3]\\\
&+3a_{12}c[-1+m,-1]+a_{12}sc[-1+m,-1]+9a_{05}c[m,-4]+4a_{05}sc[m,-4]\\\
&+5a_{03}c[m,-2]+2a_{03}sc[m,-2]-c[1+m,-1]-mc[1+m,-1]\\\
&-sc[1+m,-1]-msc[1+m,-1].\end{split}$

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