几类微分自治系统的中心条件与极限环分支
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摘要
本文主要研究了几类平面多项式微分系统中心焦点的判定及极限环分支,以及一类非线性方程组的行波解,全文有四章组成:
第一章对平面多项式微分系统的中心焦点的判定与极限环分支问题的历史背景与研究现状进行了综述,并将本文所做的工作进行了简单的介绍.
第二章研究了一类五次系统的奇点量与中心条件,先研究原点的奇点量,在用一同胚变换将无穷远点转变成原点(初等奇点),用计算机代数系统Mathenatics计算了这个多项式系统原点的前6个奇点量和无穷远点前12个奇点量,并由此得到了原点和无穷远点的中心条件.
第三章研究了一类拟三次系统的奇点量、中心焦点判定与极限环分支问题,首先通过适当的变换将系统的原点(无穷远点)转化为原点,再通过变换把拟系统转化为复系统,并在计算机上用Mathenatics软件得到了系统原点的前21个奇点量,进一步导出原点为中心的条件和最高阶细焦点(细奇点)的条件,并分别给出了原点和无穷远点分支出4个极限环的实例.
第四章,随着社会进步和科学研究的不断深入,在工程实际和自然科学各分支学科甚至社会科学领域涌现出大量非线性数学模型,等待各学科的科学工作者去研究.与线性问题不同的是,非线性问题在一般情况下很难求得精确解.孤立波在内的各类有限行波解的研究是一个非常重要的研究课题.出现了大量研究成果和求解方法和技巧,其中包括反射法、Darboux变换法、Hirota双线性法、tanh法等.这些方法的基本想法就在于通过各种变换技巧将方程化为相对易于求解的形式,特别在某些特定情况下求出方程的孤立子等特殊精确解.迄今国内外数学物理研究者利用这些方法已获得了大量研究成果.然而,这些方法除了确定一些特定情况下的精确解外,对方程的参数变化对其孤子解等特殊有限解存在性的影响却不能给出较完整的回答.国内外近年来的最新研究表明,微分方程定性理论和动力系统分叉理论可以弥补上述求精确解方法在这方面的不足,甚至可以用动力系统的观点对某些已知精确解提供更深刻的认识.本章研究了一类非线性联立薛定谔方程组,运用平面动力系统分支理论,证明该方程组存在光滑孤立波解、扭结波解、无穷光滑周期波解.并在不同参数条件下,给出了光滑孤立波解、扭结波解、无穷光滑周期波解的各类充分条件,并给出求上述所有显示精确行波解的方法.
This thesis is devoted to the problems of integral conditions, center- focus determination and bifurcation of limit cycles at the origin and the infinity of planar polynomial differential system. It is composed of four chapters.
In chapter 1, it is introduced and summarized about the historical background and the present progress of problems about center-focus determination and bifurcation of limit cycles of planar polynomial differential system. At the same time, the main work of this paper is concluded.
In chapter 2, it is studied singular quantities and center conditions for a class of quintic polynomial system. First, singular quantities at the origin of the system are discussed, then by means of the transformation, the study of the infinity can be changed into the origin of the system, and, with Mathematical, the first 6 singular quantities at the origin and the first 12 singular quantities at the infinity are deduced. At the same time, the center conditions of the origin and the infinity are derived.
In chapter 3, the center conditions and bifurcation of limit cycles for a class quasi cubic systems are investigated. First the twenty-first singular point are computed and conditions for origin to be a center are deduced as well, then a system that bifurcations four limit cycles at the origin or the infinity are constructed.
In chapter 4, as social progress and ceaseless scientific research, a large number of nonlinear mathematical models appear in actual engineering and various branches of natural sciences, even, in the domain of social sciences, which waiting for being studied deeply by science workers. The explicit solution expressions of these nonlinear issues are difficult obtained, compare to the case of linear equations. it becomes an important topic to study kinds of finity travelling waves which including solitary waves, a lot of techniques and methods have been developed such as the inverse scattering method, Darboux transformation method, Hirota bilinear method, tanh method and so on. However, except determining the explicit solutions under the given conditions, these methods can't give the integrated explanation about the relationship between pa-rameters and the existence of peculiar finity solutions. Latest researches indicate the theory of bifurcation method and qualitative analysis of dynamical system can make up these deficiencies, even, by using the theoretics of dynamical system, we can get much deep, nonlinear simultaneous Schr(o|¨)dinger equation has been studied in the light of the theory of dynamical systems and the theory of bifurcation. The existence of smooth solitary wave and kink and periodic wave solutions have been proved. Conditions sufficient for the existence of smooth solitary wave solutions and kink wave solutions and periodic wave solutions under different parameters have been given, the method of all exact explicit formula of above solutions is also given.
第一章对平面多项式微分系统的中心焦点的判定与极限环分支问题的历史背景与研究现状进行了综述,并将本文所做的工作进行了简单的介绍.
第二章研究了一类五次系统的奇点量与中心条件,先研究原点的奇点量,在用一同胚变换将无穷远点转变成原点(初等奇点),用计算机代数系统Mathenatics计算了这个多项式系统原点的前6个奇点量和无穷远点前12个奇点量,并由此得到了原点和无穷远点的中心条件.
第三章研究了一类拟三次系统的奇点量、中心焦点判定与极限环分支问题,首先通过适当的变换将系统的原点(无穷远点)转化为原点,再通过变换把拟系统转化为复系统,并在计算机上用Mathenatics软件得到了系统原点的前21个奇点量,进一步导出原点为中心的条件和最高阶细焦点(细奇点)的条件,并分别给出了原点和无穷远点分支出4个极限环的实例.
第四章,随着社会进步和科学研究的不断深入,在工程实际和自然科学各分支学科甚至社会科学领域涌现出大量非线性数学模型,等待各学科的科学工作者去研究.与线性问题不同的是,非线性问题在一般情况下很难求得精确解.孤立波在内的各类有限行波解的研究是一个非常重要的研究课题.出现了大量研究成果和求解方法和技巧,其中包括反射法、Darboux变换法、Hirota双线性法、tanh法等.这些方法的基本想法就在于通过各种变换技巧将方程化为相对易于求解的形式,特别在某些特定情况下求出方程的孤立子等特殊精确解.迄今国内外数学物理研究者利用这些方法已获得了大量研究成果.然而,这些方法除了确定一些特定情况下的精确解外,对方程的参数变化对其孤子解等特殊有限解存在性的影响却不能给出较完整的回答.国内外近年来的最新研究表明,微分方程定性理论和动力系统分叉理论可以弥补上述求精确解方法在这方面的不足,甚至可以用动力系统的观点对某些已知精确解提供更深刻的认识.本章研究了一类非线性联立薛定谔方程组,运用平面动力系统分支理论,证明该方程组存在光滑孤立波解、扭结波解、无穷光滑周期波解.并在不同参数条件下,给出了光滑孤立波解、扭结波解、无穷光滑周期波解的各类充分条件,并给出求上述所有显示精确行波解的方法.
This thesis is devoted to the problems of integral conditions, center- focus determination and bifurcation of limit cycles at the origin and the infinity of planar polynomial differential system. It is composed of four chapters.
In chapter 1, it is introduced and summarized about the historical background and the present progress of problems about center-focus determination and bifurcation of limit cycles of planar polynomial differential system. At the same time, the main work of this paper is concluded.
In chapter 2, it is studied singular quantities and center conditions for a class of quintic polynomial system. First, singular quantities at the origin of the system are discussed, then by means of the transformation, the study of the infinity can be changed into the origin of the system, and, with Mathematical, the first 6 singular quantities at the origin and the first 12 singular quantities at the infinity are deduced. At the same time, the center conditions of the origin and the infinity are derived.
In chapter 3, the center conditions and bifurcation of limit cycles for a class quasi cubic systems are investigated. First the twenty-first singular point are computed and conditions for origin to be a center are deduced as well, then a system that bifurcations four limit cycles at the origin or the infinity are constructed.
In chapter 4, as social progress and ceaseless scientific research, a large number of nonlinear mathematical models appear in actual engineering and various branches of natural sciences, even, in the domain of social sciences, which waiting for being studied deeply by science workers. The explicit solution expressions of these nonlinear issues are difficult obtained, compare to the case of linear equations. it becomes an important topic to study kinds of finity travelling waves which including solitary waves, a lot of techniques and methods have been developed such as the inverse scattering method, Darboux transformation method, Hirota bilinear method, tanh method and so on. However, except determining the explicit solutions under the given conditions, these methods can't give the integrated explanation about the relationship between pa-rameters and the existence of peculiar finity solutions. Latest researches indicate the theory of bifurcation method and qualitative analysis of dynamical system can make up these deficiencies, even, by using the theoretics of dynamical system, we can get much deep, nonlinear simultaneous Schr(o|¨)dinger equation has been studied in the light of the theory of dynamical systems and the theory of bifurcation. The existence of smooth solitary wave and kink and periodic wave solutions have been proved. Conditions sufficient for the existence of smooth solitary wave solutions and kink wave solutions and periodic wave solutions under different parameters have been given, the method of all exact explicit formula of above solutions is also given.
引文
[1]Li Jibin.Hilbert's 16th problem and bifurcations of planar polynomial vector fields.International Journal of Bifurcation and Chaos,2003,13:47-106.
[2]史松龄.二次系统(E_2)出现至少四个极限环的例子.中国科学,1979,(11),1051-56.
[3]J.Li,Q.Huang.Bifurcation set and limit cycles forming compound eyes in the cubic system.China.Ann.of Math,1987,8B,391-403.
[4]Liu Yiong,Huang Wentao,A cubic system with twelve small applitude limit,Bull.Sci.math.129(2005)83-98.
[5]Pei Yu.The report on Bifurcation of limit cycles and Hilbert's 16th problem,Jin Hua China,June 2005.
[6]叶彦谦.多项式微分系统定性理论[M].上海:上海科学技术出版社.1995.
[7]Jamue Llibre,Enrique Pone.Bifurcation of a periodic orbit from infinity in Planar piecewise linear vector fields.Nonlinear Analysis,1999,36:623-653.
[8]Blows T R.Roussean C.Bifurcation at infinity in polynomial vector fields.Journal of Differential Equations.1993,104(2):215-242.
[9]刘一戎.一类高次奇点与无穷远点中心焦点理论[J].中国科学,2001,44A(1):37-48.
[10]Liu Yirong,Chen Haibo.Stability.and bifurcations of limit cycles of the equator in a class of cubic polynomial systems.Comuters and Mathematics with applications,2002,44:997-1005.
[11]Zhang Qi,Yirong Liu.A cubic polynomial system with seven limit cycles at Infinity.Applied Mathematics and Computation,2006,177(1),319-329.
[12]刘一戎,赵梅春.一类五次系统赤道环的稳定性与极限环分支[J].数学年刊,23A:1(2002),75-78.
[13]陈海波.刘一戎.一类五次系统的赤道极限环问题[J].数学年刊,24A:2(2003),219-224.
[14]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.
[15]黄文韬,张理.一类五次多项式系统无穷远点的极限环[J].湘潭大学自然科学学报(自然科学版),2006,28(3):12-16.
[16]张齐,李建平.一类五次多项式系统无穷远点的中心条件与赤道极限环分支[J].湖南农业大学学报(自然科学版),2007,33(1):117-121.
[17]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,2002,25(2):195-302.
[18]赵百利,李险峰,曾志辉.一类拟三次系统的中心条件与极限环分支[J].黑龙江科技学院学报,2007,17(3),220-223.
[19]曾志辉,窦玉娥.一类拟五次解析系统的广义焦点量和可积条件[J].黑龙江科技学院学报,2007,19(3),220-223.
[20]Chow S N,Hate J K.Method of Bifurcation Theory[M].New York:Springer-Verlag,1981.
[21]Chow S N,Hale J K.Method of Bifurcation Theory[M].New York:Springer-Verlag,1981.
[22]LI Jibin,LIU Zhengrong.Traveling waves solutions for a class of nonlinearly dispersive equations[J].Chinese Ann Math SerB,2002(3):397-418.
[23]吴小红,拓展的Riccatic方程映射法在非线性联立薛定谔方程中的应用[J],西北师范大学学报,2007(3):34-36.
[24]刘一戎,李继彬.论复自治系统的焦点量[J].中国数学(A),1989,(3):245-255.
[2]史松龄.二次系统(E_2)出现至少四个极限环的例子.中国科学,1979,(11),1051-56.
[3]J.Li,Q.Huang.Bifurcation set and limit cycles forming compound eyes in the cubic system.China.Ann.of Math,1987,8B,391-403.
[4]Liu Yiong,Huang Wentao,A cubic system with twelve small applitude limit,Bull.Sci.math.129(2005)83-98.
[5]Pei Yu.The report on Bifurcation of limit cycles and Hilbert's 16th problem,Jin Hua China,June 2005.
[6]叶彦谦.多项式微分系统定性理论[M].上海:上海科学技术出版社.1995.
[7]Jamue Llibre,Enrique Pone.Bifurcation of a periodic orbit from infinity in Planar piecewise linear vector fields.Nonlinear Analysis,1999,36:623-653.
[8]Blows T R.Roussean C.Bifurcation at infinity in polynomial vector fields.Journal of Differential Equations.1993,104(2):215-242.
[9]刘一戎.一类高次奇点与无穷远点中心焦点理论[J].中国科学,2001,44A(1):37-48.
[10]Liu Yirong,Chen Haibo.Stability.and bifurcations of limit cycles of the equator in a class of cubic polynomial systems.Comuters and Mathematics with applications,2002,44:997-1005.
[11]Zhang Qi,Yirong Liu.A cubic polynomial system with seven limit cycles at Infinity.Applied Mathematics and Computation,2006,177(1),319-329.
[12]刘一戎,赵梅春.一类五次系统赤道环的稳定性与极限环分支[J].数学年刊,23A:1(2002),75-78.
[13]陈海波.刘一戎.一类五次系统的赤道极限环问题[J].数学年刊,24A:2(2003),219-224.
[14]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.
[15]黄文韬,张理.一类五次多项式系统无穷远点的极限环[J].湘潭大学自然科学学报(自然科学版),2006,28(3):12-16.
[16]张齐,李建平.一类五次多项式系统无穷远点的中心条件与赤道极限环分支[J].湖南农业大学学报(自然科学版),2007,33(1):117-121.
[17]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,2002,25(2):195-302.
[18]赵百利,李险峰,曾志辉.一类拟三次系统的中心条件与极限环分支[J].黑龙江科技学院学报,2007,17(3),220-223.
[19]曾志辉,窦玉娥.一类拟五次解析系统的广义焦点量和可积条件[J].黑龙江科技学院学报,2007,19(3),220-223.
[20]Chow S N,Hate J K.Method of Bifurcation Theory[M].New York:Springer-Verlag,1981.
[21]Chow S N,Hale J K.Method of Bifurcation Theory[M].New York:Springer-Verlag,1981.
[22]LI Jibin,LIU Zhengrong.Traveling waves solutions for a class of nonlinearly dispersive equations[J].Chinese Ann Math SerB,2002(3):397-418.
[23]吴小红,拓展的Riccatic方程映射法在非线性联立薛定谔方程中的应用[J],西北师范大学学报,2007(3):34-36.
[24]刘一戎,李继彬.论复自治系统的焦点量[J].中国数学(A),1989,(3):245-255.