一类拟齐次多项式中心的极限环分支

数学物理学报 ›› 2018, Vol. 38 ›› Issue (1): 1-9.

• 论文 • 下一篇

一类拟齐次多项式中心的极限环分支

梁海华¹, 陈玉明¹, 岑秀丽²

1. 广东技术师范学院数学与系统科学学院广州 510665;
2. 清华大学数学科学系北京 100084

收稿日期:2016-11-24 修回日期:2017-03-29 出版日期:2018-02-26 发布日期:2018-02-26
通讯作者: 岑秀丽 E-mail:cenxiuli2010@163.com
作者简介:梁海华,lianghhgdin@126.com;陈玉明,blkhpz@126.com
基金资助:
国家自然科学基金（11401255，11401111，11571379，11771101）和广东省自然科学基金（2015A030-313669，2015A030310424）

Limit Cycles Bifurcating from a Class of Quasi-Homogeneous Polynomial Center

Liang Haihua¹, Chen Yuming¹, Cen Xiuli²

1. School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665;
2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084

Received:2016-11-24 Revised:2017-03-29 Online:2018-02-26 Published:2018-02-26
Supported by:
Supported by the NSFC (11401255, 11401111, 11571379, 11771101) and the Natural Science Foundation of Guangdong Province (2015A030313669, 2015A030310424)

摘要/Abstract

摘要： 确定平面拟齐次多项式微分系统具有中心的条件是一个难度很大的课题.该文首先将文献[12]给出的五次拟齐次多项式系统推广到n（奇数）次系统，给出它具有全局中心的充要条件.然后利用一阶Melnikov函数得到中心的周期环域在n次多项式扰动下产生的极限环个数的最小上界.最后证明了该上界适用于所有以m为权指数的（m，1）-（或（1，m）-）拟齐次平面多项式哈密顿系统，在2m-1次多项式扰动下分支出来的极限环个数，其中m为任意正整数.

关键词: 拟齐次多项式, 中心, 极限环, Melnikov函数

Abstract: Determining the condition such that a planar quasi-homogeneous polynomial differential system has a center is a difficult topic. In this paper, we first extend the quintic quasi-homogeneous system, provided by[12], to the quasi-homogeneous polynomial system of degree n (odd number), and then give the necessary and sufficient condition to ensure that it possess a global center. Using the first order Melnikov function, we obtain the least upper bound for the number of limit cycles bifurcating from the period annulus of the center of the system, under the perturbation of polynomial of degree n. Finally, we prove that this conclusion is also true for the limit cycles bifurcating from all the (m,1)-(or (1,m)-) planar quasi-homogeneous polynomial differential Hamiltonian system, under polynomial perturbation of degree 2m-1, where m is any positive integer.

Key words: Quasi-homogeneous polynomial, Center, Limit cycle, Melnikov function

中图分类号:

O175.12

梁海华, 陈玉明, 岑秀丽. 一类拟齐次多项式中心的极限环分支[J]. 数学物理学报, 2018, 38(1): 1-9.

Liang Haihua, Chen Yuming, Cen Xiuli. Limit Cycles Bifurcating from a Class of Quasi-Homogeneous Polynomial Center[J]. Acta mathematica scientia,Series A, 2018, 38(1): 1-9.

参考文献

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一类拟齐次多项式中心的极限环分支

Limit Cycles Bifurcating from a Class of Quasi-Homogeneous Polynomial Center

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