三维系统的两个几何不变式

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Aimin Liu, Yongjian Liu, Xiaoting Lu
{"title":"三维系统的两个几何不变式","authors":"Aimin Liu, Yongjian Liu, Xiaoting Lu","doi":"10.1002/mma.10491","DOIUrl":null,"url":null,"abstract":"The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Two geometrical invariants for three‐dimensional systems\",\"authors\":\"Aimin Liu, Yongjian Liu, Xiaoting Lu\",\"doi\":\"10.1002/mma.10491\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-09-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/mma.10491\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/mma.10491","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0

摘要

KCC理论的研究对象是二阶常微分方程,由于KCC理论的分析要求,有时很难将高维系统转换为等价的二阶系统。本文借助流在黎曼流形上的欧拉-拉格朗日扩展,非常方便地给出了一些三维系统的五个几何不变量,并重点分析了其中两个不变量。结果表明,七种标准形式的三维线性系统对应的双曲平衡点是雅可比不稳定的。这与我们之前在二维系统中得到的雅可比稳定和雅可比不稳定分别对应焦点和节点的结果完全不同。经典吕氏混沌系统和杨-陈混沌系统的所有平衡点都是雅可比不稳定的。同时,在三维线性情况下,轨迹上任何一点的扭转张量都同等于零,但两个非线性系统的扭转张量分量都不为零。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Two geometrical invariants for three‐dimensional systems
The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信