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弱化希尔伯特第16问题及其研究现状
  • ISSN号：1000-0917
  • 期刊名称：《数学进展》
  • 时间：0
  • 分类：O175.1[理学—数学;理学—基础数学]
  • 作者机构：[1]北京大学数学科学学院,北京100871
  • 相关基金：国家自然科学基金（No.10671005 No.10831003 和No.10525104）
作者： 李承治[1], 李伟固[1]
关键词： 希尔伯特第16问题, 极限环, 弱化希尔伯特第16问题, 阿贝尔积分, Hilbert＇s 16th Problem, limit cycles, weak Hilbert＇s 16th problem, Abelian integral
中文摘要：

V.I.Arnold多次提出如下问题：对于给定的自然数n≥2,所有n次多项式1-形式,沿一切可能的m≥3次闭代数曲线族的阿贝尔积分的孤立零点的最大个数Z（m,n）=？由Poincare-Pontryagin定理可知,当阿贝尔积分不恒为零时,A（n）=Z（n＋1,n）给出n次Hamilton系统在n次多项式扰动下从原有周期环域分支出极限环的最大个数,因此Arnold把这个问题称为弱化的希尔伯特第16问题.30多年来,对此问题的研究取得了一定进展,也遇到了很大困难.本文拟对这个问题和相关研究工作做一个粗浅的介绍.

英文摘要：

V.I.Arnold posed the following problem several times：for given integers n≥2 and m≥3,what is the maximum number Z（m,n） of isolated zeros of the Abelian integrals of all polynomial 1-forms of degree n along all possible families of closed algebraic curves of degree m？ We know from the Poincare-Pontryagin Theorem that,if the Abelian integral is not identically equal to zero,then A（n） = Z（n ＋1,n） gives the maximal number of limit cycles bifurcated from the periodic annuli of all Hamiltonian systems of degree n under any polynomial perturbations of degree n.Hence,Arnold named this problem the weakened 16th problem of Hilbert.During the past 30 years there was some progress on the research of this problem,but it is still quite far to solve it completely.In this paper we give a brief introduction of the problem and some relative research works.

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期刊信息
  • 《数学进展》
  • 北大核心期刊（2011版）
  • 主管单位:中国科学协术学会
  • 主办单位:中国数学会
  • 主编：丁伟岳
  • 地址：北京大学数学系数学进展编辑部
  • 邮编：100871
  • 邮箱：
  • 电话：
  • 国际标准刊号：ISSN：1000-0917
  • 国内统一刊号：ISSN：11-2312/O1
  • 邮发代号:2-503
  • 获奖情况:
  • 国内外数据库收录:
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  • 被引量:3411