一类近-Hamilton 多项式系统的极限环分支问题

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 604-618.

一类近-Hamilton 多项式系统的极限环分支问题

顾栒(),熊艳琴^*()

南京信息工程大学数学与统计学院南京 210044

收稿日期:2023-12-24 修回日期:2024-12-20 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 熊艳琴 E-mail:372099143@qq.com;yqxiong@nuist.edu.cn
作者简介:顾栒，E-mail:372099143@qq.com
基金资助:
国家自然科学基金(12371171);江苏省自然科学基金面上项目(BK20221339)

Bifurcations of Limit Cycles in a Class of Near-Hamiltonian Polynomial Systems

Gu Xun(),Xiong Yanqin^*()

School of Mathematics and Statistics, Nanjing university of information science & technology, Nanjing 210044

Received:2023-12-24 Revised:2024-12-20 Online:2025-04-26 Published:2025-04-09
Contact: Yanqin Xiong E-mail:372099143@qq.com;yqxiong@nuist.edu.cn
Supported by:
NSFC(12371171);Natural Science Foundation of Jiangsu Province(BK20221339)

摘要/Abstract

摘要：

该文主要借助 Abel 积分来研究一类近-Hamilton 多项式系统的极限环分支问题. 首先, 借助分析的技巧分别推出 Abel 积分在中心奇点处及异宿环附近的近似展开式并给出系数的计算表达式; 这些结果可以用来研究扰动系统的 Hopf 分支及异宿分支问题. 具体而言, 所讨论的近-Hamilton 多项式系统在中心奇点附近可产生 $[\frac{n+1}{4}]+[\frac{n-1}{4}]+1$ 个极限环及在异宿环附近可分支出 $2[\frac{n+1}{4}]+[\frac{n-1}{4}]$ 个极限环.

关键词: 近-Hamilton 系统, 极限环, Abel 积分, 异宿分支

Abstract:

This article primarily focuses on the study of the limit cycle bifurcation problem of a class of near-Hamiltonian polynomial systems using Abel integral. First, by utilizing analytical techniques, approximate expansions of Abel integral are derived around the central singularity and in the vicinity of the heteroclinic loop, along with the calculated expressions for the coefficients. These results can be utilized to analyze the Hopf bifurcation or heteroclinic bifurcation of the perturbed system. Specifically, it is shown that the discussed near-Hamiltonian polynomial system can produce $[\frac{n+1}{4}]+[\frac{n-1}{4}]+1$ limit cycles near the central singularity and branch out $2[\frac{n+1}{4}]+[\frac{n-1}{4}]$ limit cycles near the heteroclinic loop.

Key words: Near-Hamiltonian system, Limit cycle, Abel integral, Heteroclinic loop bicurcation

中图分类号:

O175.1

顾栒,熊艳琴. 一类近-Hamilton 多项式系统的极限环分支问题[J]. 数学物理学报, 2025, 45(2): 604-618.

Gu Xun,Xiong Yanqin. Bifurcations of Limit Cycles in a Class of Near-Hamiltonian Polynomial Systems[J]. Acta mathematica scientia,Series A, 2025, 45(2): 604-618.

图/表 1

参考文献 11

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