一类拟三次系统的中心、极限环分支及等时中心
|
摘要
本文主要研究拟解析系统的中心条件、极限环分支及等时中心问题,全文共由三章组成。
第一章对平面多项式微分系统的中心-焦点判定、极限环分支及等时中心问题的历史背景及研究现状进行了概述。
第二章研究了一类拟三次系统的中心条件与极限环分支问题。首先通过适当的变换将拟解析系统的原点(或无穷远点)转化为解析系统的原点,推导出了计算原点奇点量的线性递推公式,然后在个人计算机上用Mathematica软件计算出该系统原点的前18个奇点量,从而导出原点成为中心的条件与15阶、18阶细焦点的条件,在此基础上用两种不同的方法,在不构造poincare环域的情况下,给出了拟三系统在原点可以扰动出5个小振幅的极限环的一个实例(本文内容发表在《黑龙江科技学院学报》2007,17(3)上),并证明其中有3个是稳定的。
第三章研究了第二章中的一类拟三次系统的等时中心。首先用一种新方法求出了计算系统原点周期常数的线性递推公式,然后用Mathematica软件求出该系统原点的前9个周期常数,从而得到了中心成为等时中心的必要条件,并利用一些有效途径证明了这些条件的充分性。
This thesis is devoted to the problems of the center conditions, bifurcation of limit cycles and the isochronous conditions for quasi analytic systems. It is composed of three chapters.
In chapter 1, the historical background and the present progress of problem about center conditions, bifurcation of limit cycles and isochronous conditions of planar polynomial differential system are introduced and summarized.
In chapter 2, center condition and bifurcation of limit cycles of quasi cubic system are investigated. By converting the origin or the singular point at infinity of quasi analytic system into the origin of a equivalent complex analytic system, the recursion formula for computation of singular point quantities are given, and, with computer algebra system Mathematica, the first 18 singular point quantities are deduced. At the same time, the conditions for the origin to be a center and 15-order and 18-order fine focus are derived respectively. A quasi cubic system that bifurcated 5 limit cycles, three of which are stable, at the origin is obtained. This result was published on《Journal of Heilongjiang Institute of Science&Technology》2007,17(3).
In chapter 3, the problem of the isochronous conditions of quasi cubic system in chapter 2 is investigated. Firstly, by using a new method, the recursion formula for calculating periodic constants are given, then the first 9 periodic constants are deduced. At the same time, necessary conditions for center to be an isochronous center are given, and proof of isochronous these systems by using some effective methods are given, too.
第一章对平面多项式微分系统的中心-焦点判定、极限环分支及等时中心问题的历史背景及研究现状进行了概述。
第二章研究了一类拟三次系统的中心条件与极限环分支问题。首先通过适当的变换将拟解析系统的原点(或无穷远点)转化为解析系统的原点,推导出了计算原点奇点量的线性递推公式,然后在个人计算机上用Mathematica软件计算出该系统原点的前18个奇点量,从而导出原点成为中心的条件与15阶、18阶细焦点的条件,在此基础上用两种不同的方法,在不构造poincare环域的情况下,给出了拟三系统在原点可以扰动出5个小振幅的极限环的一个实例(本文内容发表在《黑龙江科技学院学报》2007,17(3)上),并证明其中有3个是稳定的。
第三章研究了第二章中的一类拟三次系统的等时中心。首先用一种新方法求出了计算系统原点周期常数的线性递推公式,然后用Mathematica软件求出该系统原点的前9个周期常数,从而得到了中心成为等时中心的必要条件,并利用一些有效途径证明了这些条件的充分性。
This thesis is devoted to the problems of the center conditions, bifurcation of limit cycles and the isochronous conditions for quasi analytic systems. It is composed of three chapters.
In chapter 1, the historical background and the present progress of problem about center conditions, bifurcation of limit cycles and isochronous conditions of planar polynomial differential system are introduced and summarized.
In chapter 2, center condition and bifurcation of limit cycles of quasi cubic system are investigated. By converting the origin or the singular point at infinity of quasi analytic system into the origin of a equivalent complex analytic system, the recursion formula for computation of singular point quantities are given, and, with computer algebra system Mathematica, the first 18 singular point quantities are deduced. At the same time, the conditions for the origin to be a center and 15-order and 18-order fine focus are derived respectively. A quasi cubic system that bifurcated 5 limit cycles, three of which are stable, at the origin is obtained. This result was published on《Journal of Heilongjiang Institute of Science&Technology》2007,17(3).
In chapter 3, the problem of the isochronous conditions of quasi cubic system in chapter 2 is investigated. Firstly, by using a new method, the recursion formula for calculating periodic constants are given, then the first 9 periodic constants are deduced. At the same time, necessary conditions for center to be an isochronous center are given, and proof of isochronous these systems by using some effective methods are given, too.
引文
[1]Dulac,H..Sur les cycles limites[J].Bull.Soc.Math.De France,51(1923):45-188.
[2]Diliberto,S.P..On systems of ordinary differential equations[J],contribution to the theory of nonlinear oscillations,Ⅰ(1950):1-38.
[3]Li Jibin.Hilbert's 16th problem and bifurcations of planar polynomial vector fields[J].International Journal of Bifurcation and Chaos,2003,13:47-106.
[4]史松龄.平面二次系统存在四个极限环的具体例子.中国科学.1979,11:1051-1056.
[5]陈兰荪,王明淑.二次系统极限环的相对位置与个数[J].数学学报,1979,(22):751-758.
[6]叶彦谦.多项式微分系统定性理论[M].上海:上海科技出版社,1995:85-108.
[7]史松龄.关于平面二次系统的极限环[J].中国科学,1980,(8):734-739.
[8]Li Jibin.,Huang Qiming.Bifurcation set and limit cycles forming compound eyes in the cubic system[J].Chin.Ann.of Math,1987,8B:391-403.
[9]Yu Pei,Han Maoan.Twelve limit cycles in a cubic case of the 16~(th)Hilbert problem[J].International Journal of Bifurcation and Chaos,2005,15(7):2191-2205.
[10]Liu Yirong,Huang Wentao.A cubic system with twelve small amplitude limit[J].Bull.Sci.math.129(2005):83-98.
[11]杜超雄,王勤龙,米黑龙.一类四次系统的极限环分枝[J].邵阳学院学报(自然科学版),2006,2(2):2-4
[12]Yu Pei.The report on Bifurcation of limit cycles and Hilbert's 16~(th)problem.Jin Hua China,June 2005.
[13]Smale S..Dynamics retrospective:great problem,attempts that failed.Physica D.1954,51:261-273.
[14]刘尊全,秦朝斌.微分方程公式的机器推导(Ⅰ)[J].中国科学,1980,(8):812-820.
[15]刘尊全,秦朝斌.微分方程公式的机器推导(Ⅱ)[J].科学通讯,1981,(5):257-258.
[16]秦元勋,刘尊全.微分方程公式的机器推导(Ⅲ)[J].科学通讯,1981,(7):288-397.
[17]杜乃林,曾宪武.计算焦点量的一类递推公式[J].科学通报,1994,39(19):1742-1744.
[18]Chavarriga J..Integrable systems in the plane with center type linear part[J].Applicationes Mathematicae,1994,22:285-309.
[19]刘一戎,陈海波.奇点量公式的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.
[20]刘一戎.拟二次系统的广义焦点量与极限环分支[J].数学学报,2002,45(4):671-682.
[21]Loud,W.S.Behavior of the period of solutions of certain plane autonomous systems near centers[J].Contributions to Differential Equation,1964,3:21-36.
[22]Pleshkan,I.A new method of investigating the isochronicity of a system of two differential equations[J].Differential equation,1969,5:796-802.
[23]Christopher,C.J.,Devlin,J.Isochronous centers in planar polynomial systems[J].SIAM J.Math.Anal,1997,28:162-177.
[24]Lloyd,N.G.,Christopher,J.,Devlin,J.,PearsonJ.M.and Uasmim,N..Quadratic like cubic systems[J],Differential Equations Dynamical Systems,1997,5(3-4):329-345.
[25]Chavarriga J,Gine J,Garcia I.Isochronous centers of a linear center perturbed by fourth degree homogrneous polynomial[J].Bull.Sci.Math,1999,123:77-96.
[26]Chavarriga J,Gine J,Garcia I.Isochronous centers of a linear center perturbed by fifth degree homogrneous polynomial[J].J.Comput.Appl.Math,2000,126(12):351-368.
[27]Cairo,L.,Chavarriga,J.,Gine,J.,Llibre,J..A class of reversible cubic systems with an isochronous center[J].Computers and Mathematics with Applications,1999,38:39-53.
[28]Zhang Weinian,Hou X.,Zeng Z..Weak centers and bifurcation of critical periods in reversible cubic systems[J].Computers and Mathematics with Applications,2000,40:771-782.
[29]Romanovski,V.,Robnik,M..The center and isochronicity problems for some cubic systems[J].Journal of Physics A:Mathematical and General,2001,34:10267-10292.
[30]Liu Yirong,Huang Wentao.Center and isochronous center at infinity for differential systems[J].Bull.Sci.math.128(2004):77-89.
[31]黄文韬,刘一戎,朱芳来.一类微分系统无穷远点的中心与拟等时中心[J].中山大学学报,2005,44(4):24-27.
[32]王勤龙,刘一戎.一类五次多项式系统无穷远点等时中心条件与极限环分支[J].高校应用数学学报A辑,2006,21(2):165-173.
[33]Blows T.R.,Rousseau C..Bifurcation at infinity in polynomial vectorfields[J]. Journal of Differential Equations,1993,104(2):215-242.
[34]Liu Yirong,Huang Wentao.A new method to determine isochronous center conditions for polynomial defferential systerms[J].Bull.Sci.math,2003,127(2):133-148.
[35]Liu Yirong,Li Jibin.Periodic constants and Time-angle difference isochronous centers for complex analytic systems[J].International Journal of Bifurcation and Chaos,2006,16(12):3747-3757.
[36]刘一戎.一类高次奇点与无穷远点的中心焦点理论[J].中国科学(A辑),2001,(1):37-48.
[37]刘一戎,李继彬.论复自治微分系统的奇点量[J].中国科学(A辑),1989,(3):245-255.
[38]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2000,3.
[39]黄文韬,刘一戎.一个在无穷远点分支出6个极限环的三次多项式系统[J].中南大学学报(自然科学版),2004,35(4):690-693.
[40]黄文韬,刘一戎,唐清干.一类五次多项式系统的奇点量与极限环分支[J].数学物理学报,2005,25A(5):744-752.
[41]黄文韬,刘一戎.一个在无穷远点分支出八个极限环的多项式微分系统[J].数学杂志,2004,24(5):551-556.
[42]刘一戎,赵梅春.一类五次系统赤道环的稳定性与极限环分枝[J].数学年刊(A 辑),2002,23,(1):75-78.
[43]陈海波,罗交晚.一类平面七次多项式系统赤道环的稳定性与极限环分支[J].应用数学,2002,15(2):22-25.
[44]陈海波,刘一戎,曾宪武.一类平面多项式微分系统赤道环量的代数递推公式[J].数学学报,2005,48(5):963-972.
[45]Zhang Qi,Liu Yirong.A cubic polynomial system with seven limit cycles at infinity[J].Applied Mathematics and Computation,2006,177(1):319-329.
[46]Zhang Qi,Liu Yirong,Chen Haibo.Bifurcation at the equator for a class of quintic polynomial differential system[J],Applied Mathematics and Computation,2006,181(1):747-755.
[47]刘一戎.平面n次微分自治系统的焦点量与几类极限环分枝[J].数学年刊,1992,13A(5):589-597.
[48]王勤龙,刘建平.一类三次系统原点的奇点量与极限环分枝[J].桂林电子工业学院学报,2002,22(3):39-41.
[49]韩茂安,朱德明.微分方程分支理论[M].北京:煤炭工业出版社,1994.
[50]Zhang T.,Zang H.,Han M..Bifurcations of limit cycles in a cubic system[J],Chaos,Solitons and Fractals,20(2004):629-638.
[51]Wang Qinlong,Hunag Wentao,Limit cycles bifurcated from the origin an equator for a class of fifth system s[J].Ann.of Diff.Eqs.20:1(2004):92-98.
[52]Iliev I.D.,Perko L.M..Higher order bifurcations of limit cycles.J.D.E.,1999,154:339-363.
[53]Li J.,Chan HSY,Chung KW.Bifurcations of limit cycles in a Z_6-equivariant planar vector field of degree 5.Sci.in China,series A,2002,45(7):817-826.
[54]Li Jibin,Liu Zhenrong.Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system,Pub.Math.35,(1991):487-506.
[55]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京:科学出版社,2003.
[56]Sibirskii,K.S..The number of limit cycles in the neighborhood of a singular point,Diff.Equs,1965,1:36-47.
[57]秦元勋,史松龄,蔡燧林.关于平面二次系统的极限环.中国科学.1981,(8):929-938.
[58]黄其明,李继彬.平面三次微分系统的极限环复眼分支(Ⅰ).云南大学学报,1985,No.1:7-16.
[59]Zoladek H..Eleven small limit cycles in a cubic vector field.Nonlinearity,1995,8:843-860.
[60]Ye Yanqian.Some problems in the qualitative theory of ordinary differential equations.J.Differential Equations,46(1982):153-164.
[61]Lin Yiping,Li Jibin.Normal and critical points of the period of closed orbits for planar autonomous systems[J].Acta Math.Sinica,34(1991):490-501(in Chinese).
[2]Diliberto,S.P..On systems of ordinary differential equations[J],contribution to the theory of nonlinear oscillations,Ⅰ(1950):1-38.
[3]Li Jibin.Hilbert's 16th problem and bifurcations of planar polynomial vector fields[J].International Journal of Bifurcation and Chaos,2003,13:47-106.
[4]史松龄.平面二次系统存在四个极限环的具体例子.中国科学.1979,11:1051-1056.
[5]陈兰荪,王明淑.二次系统极限环的相对位置与个数[J].数学学报,1979,(22):751-758.
[6]叶彦谦.多项式微分系统定性理论[M].上海:上海科技出版社,1995:85-108.
[7]史松龄.关于平面二次系统的极限环[J].中国科学,1980,(8):734-739.
[8]Li Jibin.,Huang Qiming.Bifurcation set and limit cycles forming compound eyes in the cubic system[J].Chin.Ann.of Math,1987,8B:391-403.
[9]Yu Pei,Han Maoan.Twelve limit cycles in a cubic case of the 16~(th)Hilbert problem[J].International Journal of Bifurcation and Chaos,2005,15(7):2191-2205.
[10]Liu Yirong,Huang Wentao.A cubic system with twelve small amplitude limit[J].Bull.Sci.math.129(2005):83-98.
[11]杜超雄,王勤龙,米黑龙.一类四次系统的极限环分枝[J].邵阳学院学报(自然科学版),2006,2(2):2-4
[12]Yu Pei.The report on Bifurcation of limit cycles and Hilbert's 16~(th)problem.Jin Hua China,June 2005.
[13]Smale S..Dynamics retrospective:great problem,attempts that failed.Physica D.1954,51:261-273.
[14]刘尊全,秦朝斌.微分方程公式的机器推导(Ⅰ)[J].中国科学,1980,(8):812-820.
[15]刘尊全,秦朝斌.微分方程公式的机器推导(Ⅱ)[J].科学通讯,1981,(5):257-258.
[16]秦元勋,刘尊全.微分方程公式的机器推导(Ⅲ)[J].科学通讯,1981,(7):288-397.
[17]杜乃林,曾宪武.计算焦点量的一类递推公式[J].科学通报,1994,39(19):1742-1744.
[18]Chavarriga J..Integrable systems in the plane with center type linear part[J].Applicationes Mathematicae,1994,22:285-309.
[19]刘一戎,陈海波.奇点量公式的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.
[20]刘一戎.拟二次系统的广义焦点量与极限环分支[J].数学学报,2002,45(4):671-682.
[21]Loud,W.S.Behavior of the period of solutions of certain plane autonomous systems near centers[J].Contributions to Differential Equation,1964,3:21-36.
[22]Pleshkan,I.A new method of investigating the isochronicity of a system of two differential equations[J].Differential equation,1969,5:796-802.
[23]Christopher,C.J.,Devlin,J.Isochronous centers in planar polynomial systems[J].SIAM J.Math.Anal,1997,28:162-177.
[24]Lloyd,N.G.,Christopher,J.,Devlin,J.,PearsonJ.M.and Uasmim,N..Quadratic like cubic systems[J],Differential Equations Dynamical Systems,1997,5(3-4):329-345.
[25]Chavarriga J,Gine J,Garcia I.Isochronous centers of a linear center perturbed by fourth degree homogrneous polynomial[J].Bull.Sci.Math,1999,123:77-96.
[26]Chavarriga J,Gine J,Garcia I.Isochronous centers of a linear center perturbed by fifth degree homogrneous polynomial[J].J.Comput.Appl.Math,2000,126(12):351-368.
[27]Cairo,L.,Chavarriga,J.,Gine,J.,Llibre,J..A class of reversible cubic systems with an isochronous center[J].Computers and Mathematics with Applications,1999,38:39-53.
[28]Zhang Weinian,Hou X.,Zeng Z..Weak centers and bifurcation of critical periods in reversible cubic systems[J].Computers and Mathematics with Applications,2000,40:771-782.
[29]Romanovski,V.,Robnik,M..The center and isochronicity problems for some cubic systems[J].Journal of Physics A:Mathematical and General,2001,34:10267-10292.
[30]Liu Yirong,Huang Wentao.Center and isochronous center at infinity for differential systems[J].Bull.Sci.math.128(2004):77-89.
[31]黄文韬,刘一戎,朱芳来.一类微分系统无穷远点的中心与拟等时中心[J].中山大学学报,2005,44(4):24-27.
[32]王勤龙,刘一戎.一类五次多项式系统无穷远点等时中心条件与极限环分支[J].高校应用数学学报A辑,2006,21(2):165-173.
[33]Blows T.R.,Rousseau C..Bifurcation at infinity in polynomial vectorfields[J]. Journal of Differential Equations,1993,104(2):215-242.
[34]Liu Yirong,Huang Wentao.A new method to determine isochronous center conditions for polynomial defferential systerms[J].Bull.Sci.math,2003,127(2):133-148.
[35]Liu Yirong,Li Jibin.Periodic constants and Time-angle difference isochronous centers for complex analytic systems[J].International Journal of Bifurcation and Chaos,2006,16(12):3747-3757.
[36]刘一戎.一类高次奇点与无穷远点的中心焦点理论[J].中国科学(A辑),2001,(1):37-48.
[37]刘一戎,李继彬.论复自治微分系统的奇点量[J].中国科学(A辑),1989,(3):245-255.
[38]张锦炎,冯贝叶.常微分方程几何理论与分支问题[M].北京:北京大学出版社,2000,3.
[39]黄文韬,刘一戎.一个在无穷远点分支出6个极限环的三次多项式系统[J].中南大学学报(自然科学版),2004,35(4):690-693.
[40]黄文韬,刘一戎,唐清干.一类五次多项式系统的奇点量与极限环分支[J].数学物理学报,2005,25A(5):744-752.
[41]黄文韬,刘一戎.一个在无穷远点分支出八个极限环的多项式微分系统[J].数学杂志,2004,24(5):551-556.
[42]刘一戎,赵梅春.一类五次系统赤道环的稳定性与极限环分枝[J].数学年刊(A 辑),2002,23,(1):75-78.
[43]陈海波,罗交晚.一类平面七次多项式系统赤道环的稳定性与极限环分支[J].应用数学,2002,15(2):22-25.
[44]陈海波,刘一戎,曾宪武.一类平面多项式微分系统赤道环量的代数递推公式[J].数学学报,2005,48(5):963-972.
[45]Zhang Qi,Liu Yirong.A cubic polynomial system with seven limit cycles at infinity[J].Applied Mathematics and Computation,2006,177(1):319-329.
[46]Zhang Qi,Liu Yirong,Chen Haibo.Bifurcation at the equator for a class of quintic polynomial differential system[J],Applied Mathematics and Computation,2006,181(1):747-755.
[47]刘一戎.平面n次微分自治系统的焦点量与几类极限环分枝[J].数学年刊,1992,13A(5):589-597.
[48]王勤龙,刘建平.一类三次系统原点的奇点量与极限环分枝[J].桂林电子工业学院学报,2002,22(3):39-41.
[49]韩茂安,朱德明.微分方程分支理论[M].北京:煤炭工业出版社,1994.
[50]Zhang T.,Zang H.,Han M..Bifurcations of limit cycles in a cubic system[J],Chaos,Solitons and Fractals,20(2004):629-638.
[51]Wang Qinlong,Hunag Wentao,Limit cycles bifurcated from the origin an equator for a class of fifth system s[J].Ann.of Diff.Eqs.20:1(2004):92-98.
[52]Iliev I.D.,Perko L.M..Higher order bifurcations of limit cycles.J.D.E.,1999,154:339-363.
[53]Li J.,Chan HSY,Chung KW.Bifurcations of limit cycles in a Z_6-equivariant planar vector field of degree 5.Sci.in China,series A,2002,45(7):817-826.
[54]Li Jibin,Liu Zhenrong.Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system,Pub.Math.35,(1991):487-506.
[55]张芷芬,丁同仁,黄文灶,董镇喜.微分方程定性理论[M].北京:科学出版社,2003.
[56]Sibirskii,K.S..The number of limit cycles in the neighborhood of a singular point,Diff.Equs,1965,1:36-47.
[57]秦元勋,史松龄,蔡燧林.关于平面二次系统的极限环.中国科学.1981,(8):929-938.
[58]黄其明,李继彬.平面三次微分系统的极限环复眼分支(Ⅰ).云南大学学报,1985,No.1:7-16.
[59]Zoladek H..Eleven small limit cycles in a cubic vector field.Nonlinearity,1995,8:843-860.
[60]Ye Yanqian.Some problems in the qualitative theory of ordinary differential equations.J.Differential Equations,46(1982):153-164.
[61]Lin Yiping,Li Jibin.Normal and critical points of the period of closed orbits for planar autonomous systems[J].Acta Math.Sinica,34(1991):490-501(in Chinese).