周期扰动下分离指标对异宿轨道分支的影响

数学物理学报 ›› 2021, Vol. 41 ›› Issue (5): 1516-1528.

周期扰动下分离指标对异宿轨道分支的影响

龙斌*(),徐珊珊,曹慧,李建全

^{陕西科技大学文理学院数学系西安 710021}

收稿日期:2020-04-09 出版日期:2021-10-26 发布日期:2021-10-08
通讯作者: 龙斌 E-mail:longbin210@126.com
基金资助:
国家自然科学基金(11801343);国家自然科学基金(12071268);国家自然科学基金(11971281);国家自然科学基金(11801342);陕西省自然科学基础研究计划(2018JQ1031);陕西科技大学博士科研启动基金(2017BJ-45)

The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation

Bin Long*(),Shanshan Xu,Hui Cao,Jianquan Li

^{Department of Mathematics, Shaanxi University of Science and Technology, Xi'an 710021}

Received:2020-04-09 Online:2021-10-26 Published:2021-10-08
Contact: Bin Long E-mail:longbin210@126.com
Supported by:
the NSFC(11801343);the NSFC(12071268);the NSFC(11971281);the NSFC(11801342);the Natural Science Basic Research Plan in Shaanxi Province(2018JQ1031);the Scientific Research Initiation Foundation of Shaanxi University of Science and Technology(2017BJ-45)

摘要/Abstract

摘要：

应用Lyapunov-Schmidt约化方法与指数二分性, 该文研究了退化异宿轨道在具有$m$维参数周期扰动下的分支问题. 假设沿着未扰动异宿轨道的变分方程具有$d$个线性无关的有界解. 给出了未扰动异宿轨道的分离指标$s$. 分支函数是从$\mathbb{R} ^{d+m}$到$\mathbb{R} ^{d-s}$的一个映射. 分支函数零点的存在性就对应着扰动系统异宿轨道的存在性. 如果分离指标$s<0$, 则至少需要$1-s$维的周期扰动才能扰开未扰动的异宿轨. 如果分离指标$s\geq0$, 则存在一个一维的周期小扰动即可扰开未扰动的异宿轨.

关键词: 退化的异宿轨分支, Lyapunov-Schmidt约化, 指数二分性

Abstract:

By using the method of Lyapunov-Schmidt reduction and exponential dichotomies, we consider the degenerate heteroclinic orbit bifurcation with $m$ dimensional periodic perturbations. The variational equation along the heteroclinic orbit has $d (d\ge1)$ bounded solutions. The splitting index of the unperturbed heteroclinic orbit is $s$. The bifurcation equation has $d+m$ variables and $d-s$ equations. The zeros of bifurcation function correspond to the existence of heteroclinic orbits for perturbed equation. If the splitting index $s<0$, it needs at least $1-s$ dimensional periodic perturbation can break the unperturbed heteroclinic orbit. If the splitting index $s\geq0$, there is a small perturbation can break the unperturbed heteroclinic orbit.

Key words: Degenerate heteroclinic bifurcation, Lyapunov-Schmidt reduction, Exponential dichotomy

中图分类号:

O175.1

龙斌,徐珊珊,曹慧,李建全. 周期扰动下分离指标对异宿轨道分支的影响[J]. 数学物理学报, 2021, 41(5): 1516-1528.

Bin Long,Shanshan Xu,Hui Cao,Jianquan Li. The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation[J]. Acta mathematica scientia,Series A, 2021, 41(5): 1516-1528.

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