考虑$n$次多项式微分系统

$ \begin{eqnarray} \left\{ \begin{aligned} \frac{{dx}}{{dt}}=\delta\, x-y+P_{n}(x, y, {{\mu}}), \\ \frac{{dy}}{{dt}}=x+\delta\, y+Q_{n}(x, y, {{\mu}}), \end{aligned} \right. \end{eqnarray} $

(1.1)

其中$\max{(\deg{(P_{n})}, \deg{(Q_{n})})}=n\geq 2, ~~ {\mu}=(\mu_{1}, \mu_{2}, \cdots, \mu_{m})\in \mathbf{R}^{m}, {0}\leq{\left|\delta\right|}\ll {1}$.当$\delta=0$时, 由于非线性项的影响, 系统(1.1) 以原点为中心或细焦点.如何区分称为中心焦点判定问题?

当$\delta\neq {0}$时, 系统(1.1) 以原点为粗焦点, 其第$0$阶焦点量为$v_{0}=\delta$; 当$\delta={0}$时, 存在形式幂级数$F(x, y)=x^2+y^2+\sum\limits_{k=3}^{\infty}F_{k}(x, y)$, 使得

$ \left.\dfrac{dF}{dt}\right|_{(1.1)_{\delta=0}}= \sum\limits_{k=1}^{\infty} {v_{k}(\mu)\, (x^2+y^2)^{k+1}}, $

其中$v_{k}(\mu)$称为系统在原点的第$k$阶焦点量.

一方面, 当多项式系统在原点处的各阶焦点量都为零时, 系统以该点为中心; 另一方面由Hilbert有限基定理, 所有焦点量生成的有理数域上的多项式理想是有限生成的, 因此中心焦点问题可在有限步内解决.为了获得系统(1.1) 具有中心的充要条件, 首先需要计算系统(1.1) 的前面各阶非零焦点量并对它们进行零点分解, 从而得到中心的必要条件; 然后利用首次积分、形式首次积分、积分因子、时间可逆性等方法证明所得条件都是充分的.

Bautin解决了二次系统的中心焦点判定问题; Sibirskii解决了一类$Z_{{2}}$对称三次系统的中心判定问题; Sadovskii等^[1]利用Cherkas方法解决了一类可约化为Liénard系统的三次系统的中心判定问题; 然而对于一般三次系统以及三次以上系统, 目前还没有彻底的结论.

近二十多年以来出现了很多焦点量算法, 比如借助奇点量算法^{[2, 3]}、基于伪除的形式幂级数法^[4]和基于摄动的标准形算法^[5].但当$P_{n}, Q_{n}$为非齐次多项式时, 系统(1.1) 的焦点量非常复杂且难于约化, 为此作者^[6]基于重新参数化法给出了焦点量的约化方法.

一般来讲, 中心焦点问题的最终解决依赖于焦点量的计算, 但当计算量过大时, 可以通过增加条件的方法加以解决.例如:刘一戎等^[2]定义基本李不变量, 给出了广义对称原理; Lloyd等^[7]、Cozma^[8]以Gröbner基为工具, 寻找双线性变换将一类多项式系统化为时间可逆系统, 从而确定中心条件.

设$(\delta, \mu)=(0, \mu_{c})$时, 系统(1.1) 以原点为$M\geq {1}$阶细焦点, 则当参数$(\delta, \mu)$通过点$(0, \mu_{c})$时, 系统(1.1) 从原点分支出的小振幅极限环的最大个数$H_{0}(n)$至多为$M$, 其中$H_{0}(n)$也称为系统(1.1) 在原点处的环性.

关于小振幅极限环的构造, 一般总是利用焦点量三角化后解出主变元达到目的; 但当无法精确求解主变元时, 需要借助陆征一等^[9]的实根分离算法实现构造.

对三次系统而言, Chen等^[10]利用正则链理论和三角列分解方法证明了$H_{0}(3)\geq 9$; Yu等^[11]证明$H_{0}(3)\geq 12$, 这是目前已知最好的结果.

引理1.1^[12]  设系统$(1.1)$的前$k$阶焦点量依次为

$ v_{0}=\delta, v_{1}=v_{1}{({\mu})}, \cdots, v_{k}=v_{k}({\mu}). $

如果

(ⅰ)  当$\delta=0, \mu=\mu_{c}$时, $v_{0}=v_{1}=\cdots=v_{k-1}=0, v_{k}\neq {0};$

(ⅱ)  $\det\left[\dfrac{\partial\left(v_{0}, v_{1}, v_{2}, \cdots, v_{k-1}\right)}{\partial{\left(\delta, \mu_{1}, \mu_{2}, \cdots, \mu_{k-1}\right)}}\right]_{\delta=0, \mu=\mu_{c}} \neq 0, $

则对系统$(1.1)_{\delta=0, \mu=\mu_{c}}$进行适当的系数微扰, 相应系统在原点可分支出$k$个小振幅极限环.

考虑一类具有齐五次项的$Z_{2}$对称系统

$ \left\{ \begin{array}{l} \frac{{dx}}{{dt}} = \delta {\mkern 1mu} x - y + \left( {{a_0}{x^5} + {a_1}{x^4}y + {a_2}{x^3}{y^2} + {a_3}{x^2}{y^3} + {a_4}x{y^4} + {a_5}{y^5}} \right), \\ \frac{{dy}}{{dt}} = \;\;x + \delta {\mkern 1mu} y + \left( {{b_0}{x^5} + {b_1}{x^4}y + {b_2}{x^3}{y^2} + {b_3}{x^2}{y^3} + {b_4}x{y^4} + {b_5}{y^5}} \right). \end{array} \right. $

Chavarriga等^[13]给出其在原点可积的若干充分条件; 为了简化计算, Fercec等^[14]转而研究相应的复系统, 对四组仅有8个参数的特例给出了可积的充分条件; Chavarriga等^[15]将中心条件的推导列为公开问题.

考虑一类特殊五次系统

$ \left\{ \begin{array}{l} \frac{{dx}}{{dt}} = \;\;\delta {\mkern 1mu} x - y + ({x^2} + {y^2}){\mkern 1mu} \left( {{a_0}{x^3} + {a_1}{x^2}y + {a_2}x{y^2} + {a_3}{y^3}} \right), \\ \frac{{dy}}{{dt}} = \;\;\;\;x + \delta {\mkern 1mu} y + ({x^2} + {y^2}){\mkern 1mu} \left( {{b_0}{x^3} + {b_1}{x^2}y + {b_2}x{y^2} + {b_3}{y^3}} \right). \end{array} \right. $

(1.2)

下面将给出系统(1.2) 以原点为中心的充要条件, 并证明其从原点至多可分支出6个小振幅极限环, 最后给出具有6个极限环的实例.

根据文[6]的计算方法, 系统$(1.2)_{\delta=0}$的前6阶非零约化焦点量(不计非零常数因子)为

$ v_{2}\;\;=\;\;{\dfrac{6}{5}}\, b_{{3}}+{\dfrac{2}{5}}\, b_{{1}}+{\dfrac{2}{5}}\, a_{{2}}+{\dfrac{6}{5}}\, a_{{0}}, \\ v_{4}\;\;=\;\;{\dfrac {16}{189}}\, \left( 3\, a_{{1}}-9\, a_{{3}}-9\, b_{{0}}+3\, b_{{2}} \right) a_{{0}}+{\dfrac {16}{189}}\, \left( 2\, a_{{1}}+2\, b_{{2}} \right) a_{{2}}\\ \;\;\;\;\;\;\;\;+{\dfrac {16}{189}}\, \left( -a_{{1}}-3\, a_{{3}}-3\, b_{ {0}}-b_{{2}} \right) b_{{1}}, \\ v_{6}\;\;=\;\;-{\dfrac {64}{143}}\, {a_{{0}}}^{2}b_{{1}}-{\dfrac {192}{143}}\, {a_{{0}}} ^{2}b_{{3}}-{\dfrac {64}{429}}\, a_{{0}}{b_{{1}}}^{2}-{\dfrac {128}{143}} \, a_{{0}}b_{{1}}b_{{3}}-{\dfrac {192}{143}}\, a_{{0}}{b_{{3}}}^{2}\\ \;\;\;\;\;\;\;\;-{\dfrac {64}{429}}\, {a_{{1}}}^{2}b_{{3}}+{\dfrac {64}{429}}\, a_{{1}}a_{{3 }}b_{{1}}-{\dfrac {64}{143}}\, a_{{1}}b_{{0}}b_{{3}}+{\dfrac {64}{429}}\, a_{{1}}b_{{1}}b_{{2}}+{\dfrac {64}{429}}\, a_{{1}}b_{{2}}b_{{3}}\\ \;\;\;\;\;\;\;\;+{\dfrac{64}{429}}\, a_{{3}}b_{{1}}b_{{2}}-{\dfrac {64}{143}}\, b_{{0}}b_{{2}}b_{ {3}}-{\dfrac {64}{429}}\, {b_{{1}}}^{2}b_{{3}}+{\dfrac {64}{429}}\, b_{{1} }{b_{{2}}}^{2}-{\dfrac {64}{143}}\, b_{{1}}{b_{{3}}}^{2}+{\dfrac {128}{ 429}}\, {b_{{2}}}^{2}b_{{3}}, \\ v_{8}\;\;=\;\;-{\dfrac {512}{12155}}\, b_{{0}}{b_{{1}}}^{2}b_{{3}}-{\dfrac {512}{36465} }\, b_{{0}}b_{{1}}{b_{{2}}}^{2}-{\dfrac {512}{36465}}\, {b_{{1}}}^{2}b_{{ 2}}b_{{3}}-{\dfrac {512}{12155}}\, {b_{{0}}}^{2}b_{{1}}b_{{2}}\\ \;\;\;\;\;\;\;\;-{\dfrac {512}{12155}}\, a_{{0}}{a_{{1}}}^{2}b_{{0}} -{\dfrac {512}{36465}}\, a_{{0} }{a_{{1}}}^{2}b_{{2}}-{\dfrac {1536}{12155}}\, a_{{0}}a_{{1}}{b_{{0}}}^{ 2}+{\dfrac {512}{36465}}\, a_{{0}}a_{{1}}{b_{{2}}}^{2}\\ \;\;\;\;\;\;\;\;-{\dfrac {512}{12155}}\, {a_{{1}}}^{2}b_{{0}}b_{{1}}-{\dfrac {512}{36465}}\, {a_{{1}}}^{ 2}b_{{1}}b_{{2}}-{\dfrac {3072}{12155}}\, {a_{{0}}}^{2}b_{{0}}b_{{1}}-{ \dfrac {4608}{12155}}\, {a_{{0}}}^{2}b_{{0}}b_{{3}}\\ \;\;\;\;\;\;\;\;-{\dfrac {1024}{12155}}\, {a_{{0}}}^{2}b_{{1}}b_{{2}}-{\dfrac {1536}{12155}}\, {a_{{0}}}^{2}b_{ {2}}b_{{3}}-{\dfrac {1536}{12155}}\, a_{{0}}{b_{{0}}}^{2}b_{{2}}-{\dfrac {512}{12155}}\, a_{{0}}b_{{0}}{b_{{1}}}^{2}\\ \;\;\;\;\;\;\;\;+{\dfrac {512}{12155}}\, a_{{0 }}b_{{0}}{b_{{2}}}^{2}-{\dfrac {512}{36465}}\, a_{{0}}{b_{{1}}}^{2}b_{{2 }}-{\dfrac {512}{12155}}\, a_{{1}}{b_{{0}}}^{2}b_{{1}}-{\dfrac {512}{ 36465}}\, a_{{1}}b_{{1}}{b_{{2}}}^{2}\\ \;\;\;\;\;\;\;\;-{\dfrac {4608}{12155}}\, {a_{{0}}}^ {3}b_{{0}}-{\dfrac {1536}{12155}}\, {a_{{0}}}^{3}b_{{2}}+{\dfrac {1024}{ 36465}}\, a_{{0}}{b_{{2}}}^{3}-{\dfrac {2048}{36465}}\, a_{{1}}b_{{0}}b_{ {1}}b_{{2}}\\ \;\;\;\;\;\;\;\;-{\dfrac {3072}{12155}}\, a_{{0}}b_{{0}}b_{{1}}b_{{3}}-{ \dfrac {1024}{12155}}\, a_{{0}}b_{{1}}b_{{2}}b_{{3}}, \\ v_{10}\;\;=\;\;v_{10}(a_{{0}}, a_{{1}}, a_{{3}}, b_{{0}}, b_{{1}}, b_{{2}}, b_{{3}}), \\ v_{14}\;\;=\;\;v_{14}(a_{{0}}, a_{{1}}, a_{{3}}, b_{{0}}, b_{{1}}, b_{{2}}, b_{{3}}), $

其中$v_{10}, v_{14}$分别是五次多项式、七次多项式, 其项数分别为53项、64项.

定理2.1  系统$(1.2)_{\delta=0}$以原点为中心的充要条件是下列$6$组条件之一成立

(ⅰ) $ a_{{2}}=-3\, b_{{3}}-b_{{1}}-3\, a_{{0}}, a_{{3}}={\dfrac {C_{1}}{ \left( b_{{2}}+a_{{1}} \right) \left( 3\, a_{{0}}+b_{{1}} \right) }}, b_{{0}}=-{\dfrac {C_{2}}{ \left( b_{{2}}+a_{{1}} \right) \left( 3\, a_{{0}}+b_{{1}} \right) }};$

(ⅱ) $ a_{{2}}=-3\, b_{{3}}, b_{{1}}=-3\, a_{{0}}, b_{{2}}=-a_{{1}};$

(ⅲ) $ a_{{2}}=-b_{{1}}, a_{{3}}=-b_{{0}}, b_{{2}}=-a_{{1}}, b_{{3}}=-a_{{0}};$

(ⅳ) $ a_{{0}}=0, a_{{2}}=-{\dfrac {a_{{1}}a_{{3}}+a_{{1}}b_{{0}}+a_{{3}}b_{{0}}+{b_{{0}}}^{2}+{b_{{3}}}^{2}}{b_{{3}}}}, \\$

$ \;\;\;\;\;\;b_{{1}}={\dfrac {a_{{1}}a_{{3}}+a_{{1}}b_{{0}}+a_{{3}}b_{{0}}+{b_{{0}}}^{2}-2\, {b_{{3}}}^{2 }}{b_{{3}}}}, $

$\\ \;\;\;\;\;\;b_{{2}}=-{\dfrac {{a_{{1}}}^{2}+a_{{1}}a_{{3}}+2\, a_{{1}}b _{{0}}+a_{{3}}b_{{0}}+{b_{{0}}}^{2}-2\, {b_{{3}}}^{2}}{b_{{0}}+a_{{1}}} };$

(ⅴ) $ a_{{0}}=a_{{2}}=b_{{1}}=b_{{3}}=0;$

(ⅵ) $ a_{{0}}=b_{{3}}=0, a_{{2}}=-b_{{1}}, a_{{3}}=-3\, a_{{1}}, b_{{0}}=-a_{{1}}, b_{{2}}=3\, a_{{1}}, $

其中

$ C_{1}\;\;=\;\;9\, {a_{{0}}}^{3}+6\, {a_{{0}}}^{2}b_{{1}}+9\, {a_{{0}}}^{2}b_{{3}}-3\, a_ {{0}}a_{{1}}b_{{2}}+a_{{0}}{b_{{1}}}^{2}+6\, a_{{0}}b_{{1}}b_{{3}}\\ \;\;\;\;\;\;\;\;-3\, a_{{0}}{b_{{2}}}^{2}-2\, {a_{{1}}}^{2}b_{{3}}-a_{{1}}b_{{1}}b_{{2}}-4\, a _{{1}}b_{{2}}b_{{3}}+{b_{{1}}}^{2}b_{{3}}-b_{{1}}{b_{{2}}}^{2}-2\, {b_{ {2}}}^{2}b_{{3}}, \\ C_{2}\;\;=\;\;9\, {a_{{0}}}^{3}+6\, {a_{{0}}}^{2}b_{{1}}+9\, {a_{{0}}}^{2}b_{{3}}+a_{{0 }}{a_{{1}}}^{2}-a_{{0}}a_{{1}}b_{{2}}+a_{{0}}{b_{{1}}}^{2}+6\, a_{{0}}b _{{1}}b_{{3}}\\ \;\;\;\;\;\;\;\;-2\, a_{{0}}{b_{{2}}}^{2}+{a_{{1}}}^{2}b_{{1}}+a_{{1}}b_{{ 1}}b_{{2}}+{b_{{1}}}^{2}b_{{3}}. $

证  必要性:通过求解多项式集$G=\left\{v_{{2}}, v_{{4}}, v_{{6}}, v_{{8}}, v_{{10}}, v_{{14}}\right\}$, 共得到定理中的6组独立系数条件, 从而必要性得证.

充分性:当条件(ⅰ)成立时, 系统$(1.2)_{\delta=0}$的向量场关于直线

$ \left( 3\, a_{{0}}+b_{{1}} \right) x+ \left( a_{{1}}+b_{{2}}+\sqrt {9 \, {a_{{0}}}^{2}+6\, a_{{0}}b_{{1}}+{a_{{1}}}^{2}+2\, a_{{1}}b_{{2}}+{b_{ {1}}}^{2}+{b_{{2}}}^{2}} \right) y=0 $

对称, 因此它以原点为中心.

当条件(ⅱ)成立时, 系统$(1.2)_{\delta=0}$以

$ \mu(x, y)=\exp\left(\dfrac{1}{2}\, b_{{0}}{x}^{4}-2\, a_{{0}}{x}^{3}y-a_{{1}}{x}^{2}{y}^{2} +2\, b_{{3}}x{y}^{3}-\dfrac{1}{2}\, a_{{3}}{y}^{4}\right) $

为积分因子, 因此它以原点为中心.

当条件(ⅲ)成立时, 系统$(1.2)_{\delta=0}$的向量场关于直线$x+y=0$对称, 因此它以原点为中心.

当条件(ⅳ)成立时, 系统$(1.2)_{\delta=0}$的向量场关于直线

$ \left( b_{{0}}+a_{{1}} \right) x+ \left( -b_{{3}}+\sqrt {{a_{{1}}}^{2 }+2\, a_{{1}}b_{{0}}+{b_{{0}}}^{2}+{b_{{3}}}^{2}} \right) y=0 $

对称, 因此它以原点为中心.

当条件(ⅴ)成立时, 系统$(1.2)_{\delta=0}$的向量场关于$y$轴对称, 因此它以原点为中心.

当条件(ⅵ)成立时, 系统$(1.2)_{\delta=0}$是Hamilton系统, 因此它以原点为中心.定理证毕.

由系统$(1.2)_{\delta=0}$的焦点量结构和定理2.1, 可得

推论2.1  系统(1.2) 在原点邻近至多存在$6$个小振幅极限环.

下面总设$a_{0}=1, a_{3}=b_{0}=-1$.通过计算得到

$ \det \left[\dfrac{\partial(v_{0}, v_{2}, v_{4}, v_{6}, v_{8}, v_{10})}{\partial(\delta\;, a_{{2}}, b_{{1}}, b_{{3}}, a_{{1}}, b_{{2}})}\right]=J(a_{{1}}, a_{{2}}, b_{{1}}, b_{{2}}, b_{{3}}), $

其中$J$是十次多项式, 长达1171项.

定理3.1  设系统(1.2) 的系数满足

$ \delta=0, a_{0}=1, a_{3}=b_{0}=-1, a_{{1}}=b_{{2}}=-a_{{2}}=3, b_{{1}}=-21+12\, \sqrt {2}, b_{{3}}=7-4\, \sqrt {2}, $

则系统以原点为$14$阶细焦点; 对其进行适当的系数扰动, 从原点可分支出$6$个小振幅极限环.

证  在定理的系数条件下, 通过计算可得系统$(1.2)$的前14阶焦点量和$J$依次为

$ v_{0}\;\;=\;\;v_{1}=v_{2}=\cdots=v_{13}=0, \\ v_{14}\;\;=\;\;{\frac {64156073984}{2107575}}-{\frac {45365592064}{2107575}}\, \sqrt {2}\neq {0}, \\ \;\;J\;\;=\;\;-{\dfrac {7097673012735901696}{3218662682235}}+{\frac { 3584865303386914816}{2299044773025}}\, \sqrt {2}\neq {0}, $

从而满足引理1.1的条件, 故定理得证.

定理3.2  假设系统(1.2) 满足

$ a_{0}\;\;=\;\;1, a_{3}=b_{0}=-1, \\ \delta\;\;=\;\;305690112000000\, \left( {\frac {128312147968}{2107575}}-{ \frac {90731184128}{2107575}}\, \sqrt {2} \right) {\epsilon}^{28}, \\ a_{1}\;\;=\;\;3+ \left( {\frac {6197677378207744}{14877}}-{\frac {1458277030166528}{4959}}\, \sqrt {2} \right) {\epsilon}^{12}\\ \;\;\;\;\;\;\;\;+ \left( -{\frac {933090642866077696}{8265}}+{\frac {3732362571464310784}{46835}}\, \sqrt {2} \right) {\epsilon}^{16}, \\ a_{2}\;\;=\;\;-3+ \left( {\frac {15395995607290281984}{46835}}-{\frac { 6531634500062543872}{28101}}\, \sqrt {2} \right) {\epsilon}^{16}\\ \;\;\;\;\;\;\;\;+ \left( -{\frac {14614788922249979101184}{13775}}\, \sqrt {2} +{\frac {20668251789454112456704}{13775}} \right) {\epsilon}^{20}\\ \;\;\;\;\;\;\;\;+ \left( -{\frac {470983147592297050275840}{9367}}+{\frac { 333038292648946449776640}{9367}}\, \sqrt {2} \right) {\epsilon}^{24}, \\ b_{1}\;\;=\;\;-21+12\, \sqrt {2}+ \left( {\frac {144912465152}{145}}\, \sqrt {2} -{\frac {204582303744}{145}} \right) {\epsilon}^{8}\\ \;\;\;\;\;\;\;\;+ \left( -{\frac {15395995607290281984}{46835}}+{\frac {6531634500062543872}{28101}}\, \sqrt {2} \right) {\epsilon}^{16}\\ \;\;\;\;\;\;\;\;+ \left( {\frac {14614788922249979101184}{13775}}\, \sqrt {2}-{\frac { 20668251789454112456704}{13775}} \right) {\epsilon}^{20}, \\ b_{2}\;\;=\;\;3+ \left( -{\frac {6197677378207744}{14877}}+{\frac {1458277030166528} {4959}}\, \sqrt {2} \right) {\epsilon}^{12}, \\ b_{3}\;\;=\;\;7-4\, \sqrt {2}+ \left( {\frac {68194101248}{145}}-{\frac {144912465152 }{435}}\, \sqrt {2} \right) {\epsilon}^{8}, $

则当$0<\vert{\epsilon}\vert\ll {1}$时, 在原点充分小的邻域内, 系统$(1.2)$恰有$6$个小振幅极限环, 其位置分别在圆$x^2+y^2=k^2{\epsilon}^2$附近, $k=1, 2, \cdots, 6$.

证  当$0<\vert{\epsilon}\vert\ll {1}$时, 系统(1.2) 的第0阶至第14阶焦点量依次为

$ \;v_{0}(2\, \pi)\;\;=\;\;611380224000000\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right)\pi\, {\epsilon}^{28}+o(\epsilon^{28}), \\ \;v_{1}(2\, \pi)\;\;=\;\;0, \\ \;v_{2}(2\, \pi)\;\;=\;\;-660708809798400\, \left( {\frac {128312147968}{2107575}}-{\frac { 90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {\epsilon}^{24}+o(\epsilon^{24}), \\ \;v_{3}(2\, \pi)\;\;=\;\;0, \\ \;v_{4}(2\, \pi)\;\;=\;\;50072216862304\, \left( {\frac {128312147968}{2107575}}-{\frac { 90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {\epsilon}^{20}+o(\epsilon^{20}), \\ \;v_{5}(2\, \pi)\;\;=\;\;0, \\ \;v_{6}(2\, \pi)\;\;=\;\;-746739508515\, \left( {\frac {128312147968}{2107575}}-{\frac { 90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {\epsilon}^{16}+o(\epsilon^{16}), \\ \;v_{7}(2\, \pi)\;\;=\;\;0, \\ \;v_{8}(2\, \pi)\;\;=\;\;3112103720\, \left( {\frac {128312147968}{2107575}}-{\frac { 90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {\epsilon}^{12}+o(\epsilon^{12}), \\ \;v_{9}(2\, \pi)\;\;=\;\;0, \\ v_{10}(2\, \pi)\;\;=\;\;-3659110\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128} {2107575}}\, \sqrt {2} \right) \pi \, {\epsilon}^{8}+o(\epsilon^{8}), \\ v_{11}(2\, \pi)\;\;=\;\;v_{12}(2\, \pi)=v_{13}(2\, \pi)=0, \\ v_{14}(2\, \pi)\;\;=\;\; \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}} \, \sqrt {2} \right) \pi+O(\epsilon^{2}), $

所以系统(1.2) 在原点邻域的拟后继函数为

$ L(h)\;\;=\;\;\left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{28}\\ \;\;\;\;\;\;\;\;\;\;\;\;-3659110\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{20}{\epsilon}^{8}\\ \;\;\;\;\;\;\;\;\;\;\;\;+3112103720\, \left( {\frac { 128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{16}{\epsilon}^{12}\\ \;\;\;\;\;\;\;\;\;\;\;\;-746739508515\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{12}{\epsilon}^{16}\\ \;\;\;\;\;\;\;\;\;\;\;\;+50072216862304\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{8}{\epsilon}^{20}\\ \;\;\;\;\;\;\;\;\;\;\;\;-660708809798400\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt {2} \right) \pi \, {h}^{4}{\epsilon}^{24}\\ \;\;\;\;\;\;\;\;\;\;\;\;+611380224000000\, \left( {\frac {128312147968}{2107575}}-{\frac {90731184128}{2107575}}\, \sqrt { 2} \right) \pi {\epsilon}^{28}\\ \;\;\;\;\;\;=\;\;-{\frac {536870912}{2107575}}\, \pi \, \left( 16\, {\epsilon}^{2}+{h}^{2 } \right) \left( 25\, {\epsilon}^{2}+{h}^{2} \right) \left( 4\, { \epsilon}^{2}+{h}^{2} \right) \left( 36\, {\epsilon}^{2}+{h}^{2} \right) \\ \;\;\;\;\;\;\;\;\;\;\;\;\left( 9\, {\epsilon}^{2}+{h}^{2} \right) \left( {\epsilon}^ {2}+{h}^{2} \right) \left( 2275\, {\epsilon}^{4}+{h}^{4} \right) \left( 169\, \sqrt {2}-239 \right) \prod\limits _{k=1}^{6}{(h^2-k^2\epsilon^2)}. $

从而由文[3]知系统(1.2)在原点的充分小邻域内恰有$6$个小振幅极限环, 其位置分别在圆$x^2+y^2=k^2{\epsilon}^2$附近, $k=1, 2, \cdots, 6$.