一类四次扰动Liénard系统的极限环分支

数学物理学报 ›› 2021, Vol. 41 ›› Issue (4): 936-953.

一类四次扰动Liénard系统的极限环分支

朱红英¹(),韦敏志¹(),杨素敏^2,*(),蒋曹清¹()

¹ 广西财经学院应用数学系南宁 530003
² 广西机电职业技术学院公共教学部南宁 530003

收稿日期:2020-05-14 出版日期:2021-08-26 发布日期:2021-08-09
通讯作者: 杨素敏 E-mail:zhy71118@163.com;454742516@qq.com;smyang125@126.com;86072787@qq.com
作者简介:朱红英, E-mail: zhy71118@163.com|韦敏志, E-mail: 454742516@qq.com|蒋曹清, E-mail: 86072787@qq.com
基金资助:
国家自然科学基金(11861009);国家自然科学基金(11761011);广西自然科学基金(2020JJB110007);广西高校科研项目(2020KY16020)

Bifurcation of Limit Cycles from a Liénard System of Degree 4

Hongying Zhu¹(),Minzhi Wei¹(),Sumin Yang^2,*(),Caoqing Jiang¹()

¹ Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning 530003
² Department of Public Teaching, Guangxi Technological College of Machinery and Electricity, Nanning 530003

Received:2020-05-14 Online:2021-08-26 Published:2021-08-09
Contact: Sumin Yang E-mail:zhy71118@163.com;454742516@qq.com;smyang125@126.com;86072787@qq.com
Supported by:
the NSFC(11861009);the NSFC(11761011);the NSF of Guangxi(2020JJB110007);the Middle-Aged and Young Teachers' Basic Ability Promotion Project in Guangxi and Scientific Research Project(2020KY16020)

摘要/Abstract

摘要：

该文研究一类四次扰动Liénard系统的极限环分支.根据Chebyshev系统理论，结合多项式代数中的正则链理论，证明了系统的Abel积分的生成元是构成精度为3的Chebyshev系统，得出该系统至多可以分支出6个极限环.根据Abel积分在周期环域中的渐近展开式及分支理论，证明了该系统至少可以分支出3个极限环.

关键词: Liénard系统, Chebyshev系统, Melnikov functions, 弱化的Hilbert第十六问题

Abstract:

In this paper, we study the number of limit cycles by Poincaré bifurcation for some Liénard system of degree 4. We prove that the system can bifurcate at most 6 limit cycles from the periodic annulus, by the tools of regular chain theory in polynomial algebra and Chebyshev criteria, at least 3 limit cycles by asymptotic expansions of the related Abelian integral (first order Melnikov functions).

Key words: Liénard system, Chebyshev system, Melnikov functions, Weak Hilbert's 16th problem

中图分类号:

O175.12

朱红英,韦敏志,杨素敏,蒋曹清. 一类四次扰动Liénard系统的极限环分支[J]. 数学物理学报, 2021, 41(4): 936-953.

Hongying Zhu,Minzhi Wei,Sumin Yang,Caoqing Jiang. Bifurcation of Limit Cycles from a Liénard System of Degree 4[J]. Acta mathematica scientia,Series A, 2021, 41(4): 936-953.

图/表 2

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