Search Advanced

Message Board

Dear readers, authors and reviewers,you can add a message on this page. We will reply to you as soon as possible!

2020 Volume 45 Issue 9
Article Contents
Previous Article Next Article

Yuan-yuan LIU, Wen-tao HUANG. Limit Cycles and Local Critical Periods for a Class of Generalized Riccati System[J]. Journal of Southwest China Normal University(Natural Science Edition), 2020, 45(9): 6-12. doi: 10.13718/j.cnki.xsxb.2020.09.002
Citation: Yuan-yuan LIU, Wen-tao HUANG. Limit Cycles and Local Critical Periods for a Class of Generalized Riccati System[J]. Journal of Southwest China Normal University(Natural Science Edition), 2020, 45(9): 6-12. doi: 10.13718/j.cnki.xsxb.2020.09.002
shu

Limit Cycles and Local Critical Periods for a Class of Generalized Riccati System

  • 1.

    School of Mathematics and Computational Science, Guilin University of Electronic Technology, Guilin Guangxi 541004, China

  • 2.

    Guilin University of Aerospace Technology, GuilinGuangxi 541004, China

More Information
  • Corresponding author: Wen-tao HUANG
  • Received Date: 15/05/2019
    Available Online: 20/09/2020

  • MSC: O175.12

  • The limit cycle and local critical period bifurcation of a class of generalized Riccati systems at the origin have been investigated. By computing the singular point values of the origin of the system, the necessary conditions for the origin to be the center have been deduced. The sufficient conditions have been proved by the symmetry principle. Moreover, the conditions of the origin to be the six order fine focus have been given. By computing the period constants of the origin of the system, the conditions of the origin to be the three order fine center have been obtained. It is proved respectively that there are six small amplitude limit cycles bifurcated at the origin and three local critical periods bifurcated at the origin. As far as we known, there are the best results of the number of limit cycles and local critical periods for the cubic generalized Riccati systems.
  • 加载中
  • [1] LLOYD N G, PEARSON J M.Conditions for a Centre and the Bifurcation of Limit Cycles in a Class of Cubic Systems [M]// Lecture Notes in Mathematics.Berlin:Springer, 1990:230-242.

    Google Scholar

    [2] LLOYD N G, PEARSON J M.A Cubic Kolmogorov System with Six Limit Cycles[J].Computers and Mathematics with Applications, 2002, 44(3-4):445-455. doi: 10.1016/S0898-1221(02)00161-X

    CrossRef Google Scholar

    [3] LLIBRE J, VALLS C.Algebraic Invariant Curves and First Integrals for Riccati Polynomial Differential Systems[J].Proceedings of the American Mathematical Society, 2014, 142(10):3533-3543. doi: 10.1090/S0002-9939-2014-12085-5

    CrossRef Google Scholar

    [4] LLIBRE J, VALLS C.Liouvillian First Integrals for Generalized Riccati Polynomial Differential Systems[J].Advanced Nonlinear Studies, 2015, 15(4):951-961.

    Google Scholar

    [5] ZHOU Z X, ROMANOVSKI V G, YU J.Centers and Limit Cycles of a Generalized Cubic Riccati System[J].International Journal of Bifurcation and Chaos, 2020, 30(2):2050021. doi: 10.1142/S0218127420500212

    CrossRef Google Scholar

    [6] ROMANOVSKI V G, FERNANDES W, TANG Y L, et al.Linearizability and Critical Period Bifurcations of a Generalized Riccati System[J].Nonlinear Dynamics, 2017, 90(1):257-269. doi: 10.1007/s11071-017-3659-y

    CrossRef Google Scholar

    [7] CHICONE C, JACOBS M.Bifurcation of Critical Periods for Plane Vector Fields[J].Transactions of the American Mathematical Society, 1989, 312(2):433-486. doi: 10.1090/S0002-9947-1989-0930075-2

    CrossRef Google Scholar

    [8] 林怡平, 李继彬.平面自治系统的规范型与闭轨族周期临界点[J].数学学报, 1991, 34(4):490-501.

    Google Scholar

    [9] ROUSSEAU C, TONI B.Local Bifurcations of Critical Periods in the Reduced Kukles System[J].Canadian Journal of Mathematics, 1997, 49(2):338-358. doi: 10.4153/CJM-1997-017-4

    CrossRef Google Scholar

    [10] ZHANG W N, HOU X R, ZENG Z B.Weak Centers and Bifurcation of Critical Periods in Reversible Cubic Systems[J].Computers & Mathematics With Applications, 2000, 40(6-7):771-782.

    Google Scholar

    [11] 刘一戎, 陈海波.奇点量公式的机器推导与一类三次系统的前10个鞍点量[J].应用数学学报, 2002, 25(2):295-302.

    Google Scholar

    [12] HUANG W, LIU Y.Bifurcations of Limit Cycles from Infinity for a Class of Quintic Polynomial System[J].Bulletin Des Sciences Mathématiques, 2004, 128(4):291-302. doi: 10.1016/j.bulsci.2004.02.002

    CrossRef Google Scholar

    [13] LIU Y R, HUANG W T.A New Method to Determine Isochronous Center Conditions for Polynomial Differential Systems[J].Bulletin Des Sciences Mathématiques, 2003, 127(2):133-148. doi: 10.1016/S0007-4497(02)00006-4

    CrossRef Google Scholar

    [14] CHEN T, HUANG W T, REN D C.Weak Centers and Local Critical Periods for a Z2-Equivariant Cubic System[J].Nonlinear Dynamics, 2014, 78(4):2319-2329. doi: 10.1007/s11071-014-1560-5

    CrossRef Google Scholar

  • 加载中
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Export Citation
PDF
XML

Article Metrics

Article views(832) PDF downloads(101) Cited by(0)

Access History

Other Articles By Authors

Limit Cycles and Local Critical Periods for a Class of Generalized Riccati System

    Corresponding author: Wen-tao HUANG
  • Received Date: 15/05/2019
  • Available Online: 20/09/2020

Abstract: The limit cycle and local critical period bifurcation of a class of generalized Riccati systems at the origin have been investigated. By computing the singular point values of the origin of the system, the necessary conditions for the origin to be the center have been deduced. The sufficient conditions have been proved by the symmetry principle. Moreover, the conditions of the origin to be the six order fine focus have been given. By computing the period constants of the origin of the system, the conditions of the origin to be the three order fine center have been obtained. It is proved respectively that there are six small amplitude limit cycles bifurcated at the origin and three local critical periods bifurcated at the origin. As far as we known, there are the best results of the number of limit cycles and local critical periods for the cubic generalized Riccati systems.

    HTML

  • 由于希尔伯特第十六问题的重要性，平面多项式微分系统极限环数的研究一直吸引众多学者的关注，一些特殊类型三次系统的极限环问题常被研究.文献[1-2]分别证明了三次Kules系统和三次Kolmogorov系统在单一细焦点处可分支出6个极限环.最近文献[3-4]将如下形式的系统称为广义Riccati系统

    当f(y)=1时，即为经典的Riccati系统.

    对于三次广义Riccati系统

    其中x，y是实变量且aij，bij是实参数.文献[5]中考虑a03≠0，g2(x)=b02+b12x，g1(x)=b11x+b21x2，g0(x)=x+b20x2+b30x3，得到了系统可线性化的4组充要条件.在文献[6]中，当a03=0，g2(x)=b02+b12x，g1(x)=b11x+b21x2，g0(x)=x+b20x2+b30x3时得到了该系统原点成为中心的7组充要条件，并且证明了在原点处至少可分支出3个极限环和2个局部临界周期分支.

    临界周期分支最先由文献[7]提出.近年来对于三次多项式微分系统局部临界周期分支的研究有许多结论[8-10].

    记a02=a2，a03=a3，b20=b1，b11=b2，b02=b3，b30=b4，b21=b5，b12=b6，再取a3=-1，g2(x)=-b1+b6x，g1(x)=b2x+b5x2，g0(x)=x-b1x2+b4x3，得到如下的三次广义Riccati系统

    证明了该系统在原点处可分支出6个极限环和3个局部临界周期分支.

1.   预备知识

  • 考虑实系统

    其中：Xk(x，y)，Yk(x，y)是关于x，y的k次齐次多项式；λ∈Λ为系统的系数集.通过变换z=x+iy，w=x-iy，T=it，i=$\sqrt{-1}$，可化为如下复系统

    其中z，w，T为复变量，并且

    称系统(2)与(3)互为伴随系统.

    引理1[11]  对系统(3)可逐项确定形式级数M(z，w)=$\sum\limits_{k=2}^{\infty} C_{a \beta} z^{\alpha} w^{\beta}$，且

    其中C00=1，μm是系统(3)在原点的第m个奇点量.对任意的α，β，当α≠β时，

    当α < 0，β < 0或α=β>0，置Cαβ=0.对任意的正整数m，有

    定义1  μm称为系统(3)在原点处的第m个奇点量，若μ1=μ2=…=μm-1=0且μm≠0，称原点为m阶细奇点.

    由文献[12]可知，系统(2)的首个非零的焦点量v2m+1(2π)与其伴随复系统的首个非奇点量μm满足v2m+1(2π)=iπμm，可将系统(2)的焦点量的计算转化为系统(3)的奇点量的计算.

    系统(2)通过变换x=r·cosθ，y=r·sinθ，可得到

    其中k=2，3，4，…，

    记r(θ，h)为满足初始条件r|θ=0=h的系统(4)的解，对于足够小的实常数h，有如下的周期函数

    文献[7]已证明P2k+1=0(k=0，1，2，3…).因此，在原点充分小的邻域内，系统(2)过点(h，0)的闭轨的最小正周期P(h，λ)的展开式如下

    其中P2k称为系统(2)原点的第k个周期常数.如果对某个λ*∈Λ，P2=P4=P6=…=P2k=0，P2k+2≠0，称系统(2)的原点为k阶细中心，当k=0时称之为粗中心.若对于任意的正整数m，有P2m=0，称原点为系统(2)的等时中心.

    引理2[13]  对于系统(3)可找到如下的形式级数

    满足ck+1，k=dk+1，k=0，k=1，2，3….使得

    其中ck，j，dk，j，pj，qj满足如下方程

    记τk=pk+qk为系统(3)原点的复周期常数.系统(2)原点的第一个非零周期常数P2k与它的共轭复系统原点的第一个非零的复周期常数τk有如下的关系P2k=-πτk.

    引理3[14]  假定复系统的周期常数τi依赖于k个独立的参数a1，a2，a3，…，ak，即τi=τi(λ)=τi(a1，a2，a3，…，ak).如果λ*=(a1c，a2c，…，akc)满足

    则在λ=λ*处给予适当的扰动，系统(3)在原点处有k个局部临界周期分支.

2.   伴随复系统奇点量的计算

  • 通过变换z=x+iy，w=x-iy，T=it，i=$\sqrt{-1}$，系统(1)可化为如下的伴随复系统

    由引理1，利用数学软件Mathematica进行计算化简，可得系统(7)前6个奇点量μm的表达式.

    定理1  系统(7)原点的前6个奇点量为

    情形1  若b1=0，则μ3=μ4=μ5=μ6=0.

    情形2  若b1≠0，b2=0，则μ3=μ4=μ5=μ6=0.

    情形3  若b1b2≠0，b6=$\frac{1}{3}$(-15+6b22+7b4)，则

    其中：

    在计算μk时，已置μ1=μ2=…=μk-1=0，k=2，3，4，5，6.

    定理2  系统(7)原点处的前6个奇点量为零当且仅当下面的条件之一成立

    (ⅰ) b5=0，b1=0；

    (ⅱ) b5=0，b2=0.

    证  充分性显然成立，下证必要性.由定理1知，当μ1=0时，b5=-2b1b2.在此条件下有μ2=-$\frac{\mathrm{i} b_{1} b_{2}}{24}$(-15+6b22+7b4-3b6).若定理1中的情形1成立，则条件(ⅰ)成立.同理，由情形2可得条件(ⅱ).针对情形3，通过数学软件Mathematica计算〈F0，F1，F2，F3〉关于独立参数b1，b2，b4的Gröbner基，可得GroebnerBasis[{F0，F1，F2，F3}，{b1，b2，b4}]={1}.即F0=0，F1=0，F2=0，F3=0没有公共根，所以这种情形下前6个奇点量不可能全为零.

    定理3  对于系统(7)，系统原点所有奇点量为零的充要条件为系统前6个奇点量同时为零，即定理2的两个条件为系统(7)原点的中心条件，相应的也是系统(1)原点的中心条件.

    证  当b5=0，b1=0时，系统(1)可化简为

    可知系统关于x轴对称，即证原点是中心.

    当b5=0，b2=0时，同理可知系统关于y轴对称，即证原点是中心.

3.   系统的极限环分支

  • 由定理1情形3和定理3可知，系统(7)在原点处的细奇点的阶数最高是6.原点成为6阶细奇点(即μ1=μ2=μ3=μ4=μ5=0，μ6≠0)当且仅当以下条件成立

    由于当F0=0，F1=0，F2=0时，其精确符号解过于复杂，用数学软件解此方程，可得到16组实数解(精确到小数点后50位)，其中一组解为

    故可分别求出b5，b6

    通过计算

    因此有：

    定理4  对于系统(1)，当系数满足式(8)时，原点为6阶细焦点，通过对系统系数进行扰动，系统在原点处可分支出6个小振幅极限环.

4.   系统的细中心与局部临界周期分支

  • 据引理2中的递推算法，分别计算式(ⅰ)、式(ⅱ)条件下伴随复系统(7)的周期常数，系统(1)原点成为3阶细中心(即τ1=τ2=τ3=0，τ4≠0)当且仅当以下条件之一成立

    当b5=0，b1=0时，可得系统(7)的前4个周期常数如下

    其中：

    在计算τk过程中，令τ1=τ2=…=τk-1=0(k=2，3，4).计算〈M0，M1，M2〉关于变量b2，b6的Gröbner基，可得

    即M0=0，M1=0，M2=0没有公共根，系统(1)原点是3阶细中心.通过数值计算可得到M0=0，M1=0的4组实数解(精确到小数点后50位)，其中一组解为

    代入式(11)可求出b4，

    将b2，b4，b6的解代入M2的表达式可得

    并且

    同理，当b5=0，b2=0时，类似也可证明系统(7)有3个局部临界周期分支.因此有：

    定理5  对于系统(1)，当系数满足式(11)或式(12)时，原点成为3阶细中心，在原点处可分支出3个局部临界周期分支.

5.   结束语

  • 本文讨论了一类三次广义Riccati系统在原点处的极限环分支与局部临界周期分支问题.通过符号计算与数值计算，得到了该系统原点成为6阶细焦点和3阶细中心的充分必要条件，并运用行列式方法证明了该系统在6阶细焦点条件下从原点处可分支出6个极限环以及该系统在3阶细中心条件下从原点处可分支出3个局部临界周期分支.据我们所知，这是三次广义Riccati系统关于极限环数和局部临界周期分支数的最好结果.

Reference (14)

Catalog

    Copyright Journal of Southwest China Normal University(Natural Science Edition)

    Address： Tel：023-68252540，68254576，68254478 E-mail：xbbjb@swu.edu.cn

    Supported by:Beijing Renhe Information Technology Co. Ltd

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return