平面系统极限环的局部分支
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摘要
本文主要研究几类平面系统的焦点或中心在多项式扰动下极限环分支问题。利用幂级数及定性分析的方法，确定两类高次对称Liénard系统在奇点附近的小振幅极限环的最大个数，并讨论低次系统在全平面上的极限环个数；研究三次系统存在幂零中心的充要条件，以及一般的具有幂零中心的平面哈密顿系统在小扰动下的极限环分支，考察中心附近的一阶Melnikov函数的光滑性，及其展开式的前几项系数的具体表达式；利用开折及同宿轨改变稳定性方法讨论一类五次对称近哈密顿系统的极限环分支。
全文的主要内容可概括如下：
第一章概述了与本文相关的一些背景和预备知识。在§1．1中，介绍了Hilbert第16问题(后半部分)及弱Hilbert第16问题的研究进展；在§1．2中，介绍了平面系统的分支理论及研究方法；在§1．3中，介绍了我们的工作。
第二章完整地解决了两类高次对称Liénard系统的在指标为+1的奇点的Hopf环性数。我们通过系统的等价变换，构造特殊函数，再借助于幂级数方法来确定Liénard系统在原点的Hopf环性数。我们避开了以改变焦点稳定性来获取极限环的传统方法，灵活地利用已知的定理，通过构造及论证定理的条件来达到我们的目的。另一方面，我们充分利用文献中已有的成果，讨论低次系统在全平面中的极限环最大个数。
第三章讨论了具有幂零中心的平面哈密顿系统的极限环分支问题。首先给出三次系统存在退化的幂零中心的充要条件，其次对于一般的具有幂零中心的哈密顿系统，我们利用巧妙的变换和详细的分析研究了中心附近的一阶Melnikov函数的光滑性，并给出其展开式的前几项系数，最后给出了这类系统新的极限环分支定理及其应用。
第四章讨论了五次扰动近哈密顿系统的极限环个数问题，是第三章幂零中心的其中一种情况。我们利用开折的方法将具有幂零中心的哈密顿系统转化为具有初等中心的哈密顿系统，再通过定性分析和分支理论的技巧，利用改变奇点及奇闭轨线的稳定性产生极限环的方法给出可以出现的极限环个数。通过改变扰动项的系数可以获得更多的极限环，所得结果大大改进了现有的结果。
In this paper, we investigate the bifurcation of limit cycles of several planar systems near a center or a focus. Using the method of power series and qualitative analysis, we obtain the Hopf cyclicity of two symmetric Liénard systems, and a global result is also presented. Next we consider the bifurcation of limit cycles of planar near-Hamiltonian systems near nilpotent centers, study the smooth properties of the first Melnikov function and its first coefficients, obtain a new bifurcation theorem, especially for a class of cubic systems we give the center conditions. Finally we deal with a quintic near-Hamiltonian system, and obtain larger number of limit cycles by using the method of stability-changing than that by a general method.
In Chapter 1, we introduce some results of the second part of Hilbert's 16th problem and its weakened problem, the bifurcation theory and methods of dynamical systems, and then list our main work.
In Chapter 2, we study the maximal number of limit cycles of two types of symmetric polynomial Liénard systems near two singular points of index +1. We make some equivalent transformations to simplify our systems, and using the method of power series, we obtain the Hopf cyclicity of these two systems. For some lower-degree systems, global results are given.
In Chapter 3, we consider Hamiltonian systems with nilpotent centers. We first give nilpotent center conditions for cubic systems. Then for general perturbed Hamiltonian systems near nilpotent centers, we investigate the smooth properties of the first Melnikov function in detail and the first coefficients of its expansion, and obtain a new bifurcation theorem. Finally, we give an example as an application.
In Chapter 4, we investigate limit cycles for a class of quintic near-Hamiltonian systems near nilpotent centers, which is one case of Chapter 3. By using the blowing-up technique and bifurcation theory, we obtain the number of limit cycles near the center, which is more than that by a known result.
全文的主要内容可概括如下：
第一章概述了与本文相关的一些背景和预备知识。在§1．1中，介绍了Hilbert第16问题(后半部分)及弱Hilbert第16问题的研究进展；在§1．2中，介绍了平面系统的分支理论及研究方法；在§1．3中，介绍了我们的工作。
第二章完整地解决了两类高次对称Liénard系统的在指标为+1的奇点的Hopf环性数。我们通过系统的等价变换，构造特殊函数，再借助于幂级数方法来确定Liénard系统在原点的Hopf环性数。我们避开了以改变焦点稳定性来获取极限环的传统方法，灵活地利用已知的定理，通过构造及论证定理的条件来达到我们的目的。另一方面，我们充分利用文献中已有的成果，讨论低次系统在全平面中的极限环最大个数。
第三章讨论了具有幂零中心的平面哈密顿系统的极限环分支问题。首先给出三次系统存在退化的幂零中心的充要条件，其次对于一般的具有幂零中心的哈密顿系统，我们利用巧妙的变换和详细的分析研究了中心附近的一阶Melnikov函数的光滑性，并给出其展开式的前几项系数，最后给出了这类系统新的极限环分支定理及其应用。
第四章讨论了五次扰动近哈密顿系统的极限环个数问题，是第三章幂零中心的其中一种情况。我们利用开折的方法将具有幂零中心的哈密顿系统转化为具有初等中心的哈密顿系统，再通过定性分析和分支理论的技巧，利用改变奇点及奇闭轨线的稳定性产生极限环的方法给出可以出现的极限环个数。通过改变扰动项的系数可以获得更多的极限环，所得结果大大改进了现有的结果。
In this paper, we investigate the bifurcation of limit cycles of several planar systems near a center or a focus. Using the method of power series and qualitative analysis, we obtain the Hopf cyclicity of two symmetric Liénard systems, and a global result is also presented. Next we consider the bifurcation of limit cycles of planar near-Hamiltonian systems near nilpotent centers, study the smooth properties of the first Melnikov function and its first coefficients, obtain a new bifurcation theorem, especially for a class of cubic systems we give the center conditions. Finally we deal with a quintic near-Hamiltonian system, and obtain larger number of limit cycles by using the method of stability-changing than that by a general method.
In Chapter 1, we introduce some results of the second part of Hilbert's 16th problem and its weakened problem, the bifurcation theory and methods of dynamical systems, and then list our main work.
In Chapter 2, we study the maximal number of limit cycles of two types of symmetric polynomial Liénard systems near two singular points of index +1. We make some equivalent transformations to simplify our systems, and using the method of power series, we obtain the Hopf cyclicity of these two systems. For some lower-degree systems, global results are given.
In Chapter 3, we consider Hamiltonian systems with nilpotent centers. We first give nilpotent center conditions for cubic systems. Then for general perturbed Hamiltonian systems near nilpotent centers, we investigate the smooth properties of the first Melnikov function in detail and the first coefficients of its expansion, and obtain a new bifurcation theorem. Finally, we give an example as an application.
In Chapter 4, we investigate limit cycles for a class of quintic near-Hamiltonian systems near nilpotent centers, which is one case of Chapter 3. By using the blowing-up technique and bifurcation theory, we obtain the number of limit cycles near the center, which is more than that by a known result.
引文
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