含有尖点和零鞍点的复合环附近的极限环分支.pdf

论文题目: 含有尖点和幂零鞍点的复合环附近的极限环分支
学科专业: 应用数学
学位串请人:田焕欢
指导老师: 韩茂安教授
摘要
希尔伯特第十六问题是平面非线性微分方程中最著名和最富挑战性的一个问题,
旨在研宄一般n 次多项式系统的极限环个数和相对位置. 自问题提出以来,国内外许
多数学家为之呕心沥血,
密顿扰动系统的极限环分支问题. 对于这类问题,有一个很重要的研宄工具,被称为
M elnikov函数或阿贝尔积分. 通过研宄此函数在中心或不变环处的展开式来探讨Hopf
分支,同宿异宿分支,
该函数研宄含有尖点及幂零鞍点的复合环附近的极限环分支.
第一章主要介绍所研宄课题的来源、研宄现状、以及本文的研宄方法和主要结论.
第二章主要研宄一类近哈密顿系统的极限环分支,它的未扰系统有一个复合环,此
环包含一个尖点,一个幂零鞍点,
成果和分析的技巧,得到系统在复合环附近的三个M elnikov函数展开式及展开式中各
项系数公式. 其次利用这些系数研宄了系统在复合环附近的极限环个数和分布情况.
第三章研宄一类含参数的L
二章中证明的定理计算得出系统的M elnikov函数展开式系数,并研宄了系统在复合环
附近的极限环个数. 最后证明了这个L i z a r d 系统取不同次数时在复合环附近分别至
少有 11, 13, 15, 18, 1 9 个极限环.
关键 C : 复合环;分支;哈密顿系统;M elnikov函数;极限环.
万方数据
Title: Limit cycle bifurcations by perturbing pound loop with a cusp and a nilpotent saddle
Major: Applied mathematics
Degree Applicant: Tian Huanhuan
Tutor: Professor Han Maoan
A b stra c t
It is well known that, The second part of 16-th problem asked by Hilbert is a most
famous and greatest challenging problem of planar nonlinear differential equations. And
the main target of a polynomial differential system of degree n is the number and location of
limit cycles. These problems has attracted many mathematicians in China and abroad, and
lots of excellent results have been obtained. Where many studies have concentrated on limit
cycle bifurcations of Hamiltonian system under perturbations. For this kind of this systems,
an important tool used to study Hopf bifurcation, homoclinic and heteroclinic bifurcations,
Poincarebifurcation and nilpotent critical point bifurcation under perturbtions, ., is the
so-called Melnikov functions or Abelian integral. In this thesis, using Melnikov function,
we will study the lim it cycle bifurcations of pound loop under perturbations.
In Chapter one, we introduce the background and main topics of our research. Also,
we will desc