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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2026, Vol. 42 ›› Issue (6): 1063-1072.doi: 10.3969/j.issn.1005-3085.2025.06.006cstr: 32411.14.cjem.CN61-1269/O1.2025.06.006

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一个复杂四维混沌系统的分析及其应用

张  勇1,  张付臣2,  肖  敏3   

  1. 1. 河南工业职业技术学院基础教学部，南阳 473000
    2. 重庆工商大学数学与统计学院统计智能计算与监测重庆市重点实验室，重庆 400067
    3. 南京邮电大学自动化学院/人工智能学院，南京  210023
  • 收稿日期:2025-03-25 接受日期:2025-07-09 出版日期:2025-12-15 发布日期:2026-02-15
  • 通讯作者: 张付臣 E-mail: zhangfuchen1983@163.com
  • 基金资助:
    国家自然科学基金 (62073172)；江苏省自然科学基金 (BK20221329)；2025年度河南省科技攻关立项研究课题项目(252102110356)；2024年河南省科学技术厅第六批省科技研发计划联合基金(产业类)(245101610063)；重庆市教委科技项目(KJCX2020037)；重庆工商大学科研项目(1960288; 2019ZKYYA122)；重庆工商大学研究生教育教学改革项目(24YJG307)；2025年重庆市高等教育教学改革研究项目(253156).

Analysis of a Complex Four-dimensional Chaotic System and\\ Its Application

ZHANG Yong1,  ZHANG Fuchen2,  XIAO Min3   

  1. 1. Basic Teaching Department of Henan Polytechnic Institute, Nanyang 473000
    2. School of Mathematics and Statistics, Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, Chongqing Technology and Business University, Chongqing 400067
    3. College of Artificial Intelligence and College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210023
  • Received:2025-03-25 Accepted:2025-07-09 Online:2025-12-15 Published:2026-02-15
  • Contact: F. Zhang. E-mail address: zhangfuchen1983@163.com
  • Supported by:
    The National Natural Science Foundation of China (62073172); the Natural Science Foundation of Jiangsu Province (BK20221329); the Research Project of Henan Province Science and Technology Key Project in 2025 (252102110356); the Sixth Batch of Provincial Science and Technology Research and Development Plan Joint Fund Project (Industry Category) of Henan Provincial Department of Science and Technology in 2024 (245101610063); the Chongqing Education Commission Science and Technology Research Project (KJCX2020037); the Program of Chongqing Technology and Business University (1960288; 2019ZKYYA122); the Teaching Reform Project for Postgraduate Students of Chongqing Technology and Business University (24YJG307); the 2025 Chongqing Higher Education Teaching Reform Research Project (253156).

摘要：

研究了一个四维混沌系统的复杂动力学行为包括有界性、全局吸引域和哈密顿能量函数等，计算机仿真结果与理论结果相吻合。最后，把混沌系统的全局指数吸引集的结果用于系统的平衡点$O({0,0,0,0})$稳定性的判定，依据系统的全局指数吸引集的结果可以得到系统的平衡点是全局指数稳定性的。对混沌系统有界性与全局吸引域的研究大部分限于低维混沌系统，由于高维混沌系统的结构复杂性，而对于高维混沌系统的有界性与全局吸引域的研究较少。创新点在于不仅得到了一个高维混沌系统的有界性，还得到了轨线进入吸引域的速率估计式。进一步得到了混沌系统的一簇全局吸引集的数学解析表达式，研究结果为基于混沌同步的保密通信提供了理论依据和支撑。

关键词: 混沌, 稳定性理论, 全局指数稳定, Hamiltonian能量

Abstract:

In this paper, the complex dynamical behaviors of a four-dimensional chaotic system, including boundedness, global attractive domain and Hamiltonian energy function, are studied. We can judge whether the equilibrium point of chaotic system is stable or not according to the result of the globally exponential attractive set. According to the result of the globally exponential attractive set of this system, the equilibrium point of this system is globally exponential stable. As far as the author knows, the study of boundedness and global attraction domain of chaotic systems is mostly limited to low-dimensional chaotic systems, but due to the structural complexity of high-dimensional chaotic systems, there are few studies on the boundedness and the global attraction domain of high-dimensional chaotic systems. The innovation of this paper is that not only the boundedness of a high-dimensional chaotic system is obtained, but also the rate estimation of the attractive domain of the orbit is obtained. The mathematical expression of a cluster of global attractive sets for the chaotic system is obtained. The research results of this paper provide theoretical basis and support for secure communication based on chaos synchronization.

Key words: chaos, stability theory, globally exponential stability, Hamiltonian energy

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