一类新混沌系统的几何分析

数学物理学报 ›› 2019, Vol. 39 ›› Issue (2): 339-347.

一类新混沌系统的几何分析

黄燮桢^1,²,刘永建^2,*(),黄秋健³

¹ 闽南师范大学数学与统计学院福建漳州 363000
² 玉林师范学院; 广西高校复杂系统优化与大数据处理重点实验室广西玉林 537000
³ 广西民族大学理学院南宁 530006

收稿日期:2018-01-03 出版日期:2019-04-26 发布日期:2019-05-05
通讯作者: 刘永建 E-mail:liuyongjianmaths@126.com
基金资助:
国家自然科学基金(11561069);广西自然科学基金(2016GXNSFBA380170);广西自然科学基金(2017GXNSFAA198234);福建省数学类研究生教育创新基地项目(1013-313009);广西高等学校高水平创新团队及卓越学者计划项目([2018]35)

Geometric Analysis of a Class of the New Chaotic System

Xiezhen Huang^1,²,Yongjian Liu^2,*(),Qiujian Huang³

¹ School of Mathematics and Statistics, Minnan Normal University, Fujian Zhangzhou 363000
² Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Guangxi Yulin 537000
³ School of Science, Guangxi University for Nationalities, Nanning 530006

Received:2018-01-03 Online:2019-04-26 Published:2019-05-05
Contact: Yongjian Liu E-mail:liuyongjianmaths@126.com
Supported by:
the NSFC(11561069);the Guangxi Natural Science Foundation of China(2016GXNSFBA380170);the Guangxi Natural Science Foundation of China(2017GXNSFAA198234);the Educational Innovation Base of the Graduate Students of Mathematics in Fujian(1013-313009);Guangxi University High Level Innovation Team and Distinguished Scholars Program of China([2018]35)

摘要/Abstract

摘要：

基于Poincaré紧致化技术，分析一类三维混沌系统的全局动力学行为.研究表明系统在无穷远处的奇点高度退化且不稳定.该文也通过设计一个不改变系统奇点结构的线性控制器，构造了一个受控系统，研究发现受控系统在特定参数组条件下，存在一簇退化奇异异宿轨.结合数值仿真结果，论文指出，当参数b和c发生微小扰动时，受控系统异宿环破裂，产生新的混沌吸引子.希冀这些研究对解释系统混沌几何机理能提供有益帮助.

关键词: Poincaré紧致化, 无穷远奇点, 奇异退化异宿轨, 混沌系统

Abstract:

Based on Poincaré compactification technology, the global dynamics behavior of three dimensional chaotic system is studied. The results show that the equilibria at infinity are unstable and highly degraded. The controlled system with a linear controller which does not change the singularity structure has a bunch of degenerate singular orbits. The chaotic attractors for the system in the case of small parameters b and c are found numerically, and thus the nearby singularly degenerate heteroclinic cycles. It is hoped that the investigation of this paper will be quite beneficial for further studies of the geometrical structure for the chaotic attractor.

Key words: Poincaré compactification, Singular point at infinity, Singularly degenerate heteroclinic cycles, Chaotic system

中图分类号:

O175

黄燮桢,刘永建,黄秋健. 一类新混沌系统的几何分析[J]. 数学物理学报, 2019, 39(2): 339-347.

Xiezhen Huang,Yongjian Liu,Qiujian Huang. Geometric Analysis of a Class of the New Chaotic System[J]. Acta mathematica scientia,Series A, 2019, 39(2): 339-347.

图/表 6

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条件		O(0, 0)的类型
a_2m≠0	b_n=0	退化的奇点

条件			O(0, 0)的类型
a_2m+1 < 0	b_n≠0	n=m, λ≥0且n是偶数 n < m且n是偶数	结点