平面多项式微分系统的中心问题与极限环分支
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摘要
本文主要研究平面多项式微分系统退化奇点与无穷远点的可积条件以及中心焦点判定与极限环分支，全文由七章组成。
第一章对平面多项式微分系统极限环分支问题与中心问题的历史背景与研究现状进行了全面综述，并将本文所做的工作作了简单的介绍。
第二章研究了一类三次多项式系统无穷远点的中心条件与赤道极限环分支。通过将实系统转化为复系统研究，给出了计算无穷远点奇点量的递推公式，并在计算机上用Mathematica推导出该系统无穷远点前七个无穷远点奇点量，进一步导出了无穷远点成为中心的条件和七阶细焦点的条件，得到了三次系统无穷远点分支出七个极限环的一个实例(本章内容发表在《Applied Mathematics and Computation》2006，V.177(1)上)。
第三章研究了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枝问题。给出了计算五次多项式系统无穷远点奇点量的线性递推公式，运用这个公式及计算机代数系统Mathematica，计算了一类五次系统无穷远点的前十一个奇点量。同时得到了无穷远点的中心条件。首次构造了一个在无穷远点产生十一个极限环的五次多项式系统(在《Computers and Mathesatics with Applications》上已接收)。
第四章研究了一类七次系统无穷远点的中心条件与赤道极限环分支问题。通过将实系统转化为复系统研究，给出了计算无穷远点奇点量的递推公式与系统无穷远点前十四个奇点量，进一步导出了无穷远点成为中心的条件和十四阶细焦点的条件，在此基础上首次得到了七次系统无穷远点分支出十三个极限环的一个实例。
第五章研究了一类有两个小参数和八个普通参数的五次系统的退化奇点与无穷远点(赤道)的中心条件与极限环分支。通过两个同胚变换将退化奇点与无穷远点转变成初等奇点，进而计算了原点(退化奇点)与无穷远点的Lyapunov常数(奇点量)，并由此得到了退化奇点与无穷远点的中心条件。在原点和无穷远点的同步扰动下，得到了极限环的{(7)，2}和{(2)，6}分布(本章内容发表在《Applied Mathematics and Computation》2006，V.181(1)上)。
第六章研究了一类更广泛的复自治微分系统(其中z，w，T为相互独立的复变量，a_(αβ)，b_(αβ)是复常数，p，q为互质的整数，n为自然数)的原点(它是系统(1)的退化奇点)。定义了这类奇点的广义奇点量并研究了奇点量的结构，给出了计算奇点量的代数递推公式并得出了奇点为广义复中心的充要条件。本章结论是文[41，46]结论的推广。
第七章研究了一类更广泛的多项式微分系统(其中z，w，T为相互独立的复变量，a_(αβ)，b_(αβ)是复常数，p，q为互质的整数，n为自然数)的无穷远点，定义了这类无穷远点的广义奇点量并研究了奇点量的结构，给出了计算无穷远点奇点量的代数递推公式并得出了无穷远点为广义复中心的充要条件。本章结论是文[46]结论的推广。
This thesis is devoted to the problems of integral conditions, center-focus determination and bifurcation of limit cycles at degeneratesingular point and the infinity of planar polynomial differential system. Itis composed of seven chapters.
In chapter 1, the historical background and the present progress ofproblems about center-focus determination and bifurcation of limit cyclesof planar polynomial differential system were introduced and summarized.At the same time, the main work of this paper was concluded.
In chapter 2, center conditions and bifurcation of limit cycles fromthe equator for a class of cubic polynomial system with no singular pointat the infinity were studied. By converting real planar system intocomplex system, the recursion formula for the computation of singularpoint quantities were given, and, with computer algebra systemMathematica, the first 7 singular point quantities were deduced. At thesame time, the conditions for the infinity to be a center and 7 degree finefocus were derived respectively. A cubic system that bifurcates 7 limitcycles from the infinity was obtained. This result was published on《Applied Mathematics and Computation》2006, V. 177(1).
In chapter 3, center conditions and bifurcation of limit cycles fromthe equator for a class of quintic polynomial system with no singularpoint at the infinity were studied. The recursion formula to compute thesingular point quantities of quintic polynomial system at the infinity wasgiven. With this formula, the first eleven singular point quantities of aclass of quintic polynomial differential system at the infinity werecomputed with computer algebra system Mathematica. The conditions forthe infinity to be a center were derived as well. At last, a system that allows the appearance of eleven limit cycles in a small enoughneighborhood of the infinity was constructed at the first time. This resulthas been accepted on《Computers and Mathematics with Applications》.
In chapter 4, center conditions and bifurcation of limit cycles fromthe equator in a class of polynomial system of degree seven were studied.The method was based on converting real planar system into complexsystem, the reeursion formula for the computation of singular pointquantities of the infinity were given, which allows us to compute thegeneralized Lyapunov constants (the singular point quantities) for theinfinity. The first 14 singular point quantities of the infinity were deduced.At the same time, the conditions for the infinity to be a center and 14degree fine focus were derived respectively. A system of degree 7 thatbifurcates 13 limit cycles from infinity was constructed at the first time.
In chapter 5, Center conditions and bifurcation of limit cycles at thedegenerate singular point and infinity (the equator) in a class of quinticpolynomial differential system with two small parameters and eightnormal parameters was studied. The method was based on twohomeomorphic transformations of the infinity and degenerate singularpoint into linear singular point, which allows us to compute thegeneralized Lyapunov constants (the singular point quantities) for theorigin and infinity. The center conditions for the degenerate singular pointand infinity were derived respectively. The limit cycle configurations of{(7), 2} and {(2), 6} were obtained under simultaneous perturbation atthe origin and infinity. This result was published on《AppliedMathematics and Computation》2006, V. 181 (1).
In chapter 6, the origin of a class of general complex autonomouspolynomial differential system (with z, w, T independent complex variables,α_(αβ)，b_(αβ)complex constants, (p, q)=1, n natural)was studied. It is a degenerate singular point. Theextended singular point quantity of the singular point was defined, at thesame time, the construction of the extended singular point quantity werestudied. The linear recursion formula to compute the extended singularpoint quantity was given and necessary and sufficient condition of thesingular point to be a extended complex center was obtained. Theconclusion of this chapter is the expansion of that of [41, 46].
In chapter 7, the infinity of a class of general complex autonomouspolynomial differential system(with z, w, T independent complex variables, a_(αβ), b_(αβ) complex constants, (p, q)=1, n natural)was studied. The extended singular point quantity ofthe infinity was defined, at the same time, the construction of theextended singular point quantity of the infinity were studied. The linearrecursion formula to compute the extended singular point quantity of theinfinity was given and necessary and sufficient condition of the infinity tobe a extended complex center was obtained. The conclusion of thischapter is the expansion of that of [46].
第一章对平面多项式微分系统极限环分支问题与中心问题的历史背景与研究现状进行了全面综述，并将本文所做的工作作了简单的介绍。
第二章研究了一类三次多项式系统无穷远点的中心条件与赤道极限环分支。通过将实系统转化为复系统研究，给出了计算无穷远点奇点量的递推公式，并在计算机上用Mathematica推导出该系统无穷远点前七个无穷远点奇点量，进一步导出了无穷远点成为中心的条件和七阶细焦点的条件，得到了三次系统无穷远点分支出七个极限环的一个实例(本章内容发表在《Applied Mathematics and Computation》2006，V.177(1)上)。
第三章研究了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枝问题。给出了计算五次多项式系统无穷远点奇点量的线性递推公式，运用这个公式及计算机代数系统Mathematica，计算了一类五次系统无穷远点的前十一个奇点量。同时得到了无穷远点的中心条件。首次构造了一个在无穷远点产生十一个极限环的五次多项式系统(在《Computers and Mathesatics with Applications》上已接收)。
第四章研究了一类七次系统无穷远点的中心条件与赤道极限环分支问题。通过将实系统转化为复系统研究，给出了计算无穷远点奇点量的递推公式与系统无穷远点前十四个奇点量，进一步导出了无穷远点成为中心的条件和十四阶细焦点的条件，在此基础上首次得到了七次系统无穷远点分支出十三个极限环的一个实例。
第五章研究了一类有两个小参数和八个普通参数的五次系统的退化奇点与无穷远点(赤道)的中心条件与极限环分支。通过两个同胚变换将退化奇点与无穷远点转变成初等奇点，进而计算了原点(退化奇点)与无穷远点的Lyapunov常数(奇点量)，并由此得到了退化奇点与无穷远点的中心条件。在原点和无穷远点的同步扰动下，得到了极限环的{(7)，2}和{(2)，6}分布(本章内容发表在《Applied Mathematics and Computation》2006，V.181(1)上)。
第六章研究了一类更广泛的复自治微分系统(其中z，w，T为相互独立的复变量，a_(αβ)，b_(αβ)是复常数，p，q为互质的整数，n为自然数)的原点(它是系统(1)的退化奇点)。定义了这类奇点的广义奇点量并研究了奇点量的结构，给出了计算奇点量的代数递推公式并得出了奇点为广义复中心的充要条件。本章结论是文[41，46]结论的推广。
第七章研究了一类更广泛的多项式微分系统(其中z，w，T为相互独立的复变量，a_(αβ)，b_(αβ)是复常数，p，q为互质的整数，n为自然数)的无穷远点，定义了这类无穷远点的广义奇点量并研究了奇点量的结构，给出了计算无穷远点奇点量的代数递推公式并得出了无穷远点为广义复中心的充要条件。本章结论是文[46]结论的推广。
This thesis is devoted to the problems of integral conditions, center-focus determination and bifurcation of limit cycles at degeneratesingular point and the infinity of planar polynomial differential system. Itis composed of seven chapters.
In chapter 1, the historical background and the present progress ofproblems about center-focus determination and bifurcation of limit cyclesof planar polynomial differential system were introduced and summarized.At the same time, the main work of this paper was concluded.
In chapter 2, center conditions and bifurcation of limit cycles fromthe equator for a class of cubic polynomial system with no singular pointat the infinity were studied. By converting real planar system intocomplex system, the recursion formula for the computation of singularpoint quantities were given, and, with computer algebra systemMathematica, the first 7 singular point quantities were deduced. At thesame time, the conditions for the infinity to be a center and 7 degree finefocus were derived respectively. A cubic system that bifurcates 7 limitcycles from the infinity was obtained. This result was published on《Applied Mathematics and Computation》2006, V. 177(1).
In chapter 3, center conditions and bifurcation of limit cycles fromthe equator for a class of quintic polynomial system with no singularpoint at the infinity were studied. The recursion formula to compute thesingular point quantities of quintic polynomial system at the infinity wasgiven. With this formula, the first eleven singular point quantities of aclass of quintic polynomial differential system at the infinity werecomputed with computer algebra system Mathematica. The conditions forthe infinity to be a center were derived as well. At last, a system that allows the appearance of eleven limit cycles in a small enoughneighborhood of the infinity was constructed at the first time. This resulthas been accepted on《Computers and Mathematics with Applications》.
In chapter 4, center conditions and bifurcation of limit cycles fromthe equator in a class of polynomial system of degree seven were studied.The method was based on converting real planar system into complexsystem, the reeursion formula for the computation of singular pointquantities of the infinity were given, which allows us to compute thegeneralized Lyapunov constants (the singular point quantities) for theinfinity. The first 14 singular point quantities of the infinity were deduced.At the same time, the conditions for the infinity to be a center and 14degree fine focus were derived respectively. A system of degree 7 thatbifurcates 13 limit cycles from infinity was constructed at the first time.
In chapter 5, Center conditions and bifurcation of limit cycles at thedegenerate singular point and infinity (the equator) in a class of quinticpolynomial differential system with two small parameters and eightnormal parameters was studied. The method was based on twohomeomorphic transformations of the infinity and degenerate singularpoint into linear singular point, which allows us to compute thegeneralized Lyapunov constants (the singular point quantities) for theorigin and infinity. The center conditions for the degenerate singular pointand infinity were derived respectively. The limit cycle configurations of{(7), 2} and {(2), 6} were obtained under simultaneous perturbation atthe origin and infinity. This result was published on《AppliedMathematics and Computation》2006, V. 181 (1).
In chapter 6, the origin of a class of general complex autonomouspolynomial differential system (with z, w, T independent complex variables,α_(αβ)，b_(αβ)complex constants, (p, q)=1, n natural)was studied. It is a degenerate singular point. Theextended singular point quantity of the singular point was defined, at thesame time, the construction of the extended singular point quantity werestudied. The linear recursion formula to compute the extended singularpoint quantity was given and necessary and sufficient condition of thesingular point to be a extended complex center was obtained. Theconclusion of this chapter is the expansion of that of [41, 46].
In chapter 7, the infinity of a class of general complex autonomouspolynomial differential system(with z, w, T independent complex variables, a_(αβ), b_(αβ) complex constants, (p, q)=1, n natural)was studied. The extended singular point quantity ofthe infinity was defined, at the same time, the construction of theextended singular point quantity of the infinity were studied. The linearrecursion formula to compute the extended singular point quantity of theinfinity was given and necessary and sufficient condition of the infinity tobe a extended complex center was obtained. The conclusion of thischapter is the expansion of that of [46].
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[110] Van Horssen, W.T. Kooij, R.E.,Bifurcation of limit cycles in a particular class of quadratic systems with two centers. J. Diff. Eqs.,114(1994),538-569.
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