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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. 43 ›› Issue (2): 327-344.doi: 10.3969/j.issn.1005-3085.2026.02.009cstr: 32411.14.cjem.CN61-1269/O1.2026.02.009

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具有三个二次项的五维混沌系统的余维分岔分析及加密应用

张玲梅,  王雨婷,  王川龙   

  1. 太原师范学院数学与统计学院，晋中  030619
  • 收稿日期:2024-12-30 接受日期:2025-10-19 出版日期:2026-04-15 发布日期:2026-06-15

The Dimensional Bifurcation Analysis and Encryption Application of a Five-dimensional Chaotic System with Three Quadratic Terms

ZHANG Lingmei,  WANG Yuting,  WANG Chuanlong   

  1. School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619
  • Received:2024-12-30 Accepted:2025-10-19 Online:2026-04-15 Published:2026-06-15

摘要：

混沌系统在通信等领域具有重要应用价值，而高维混沌系统因呈现出更复杂的动力学行为，对其开展分析研究面临严峻挑战。为此，对一个具有三个二次项的五维混沌系统的Hopf 分岔进行研究。首先，通过含参中心流形理论及Hopf分岔理论，分析了该五维混沌系统的平衡点稳定性及Hopf分岔等局部动力学行为，得到了系统的平衡点和Hopf分岔的存在条件，表明该系统中存在Hopf分岔。然后，采用严格的数学分析和数值模拟的方法研究了混沌系统余维一、余维二分岔产生的条件。最后，针对传统图像加密方案中密钥敏感性弱、相邻位置像元相关性强等问题，将该系统与DNA编码结合，应用于图像加密，显著提升了图像加密的效果和安全性。

关键词: 混沌系统, 余维, Hopf分岔, 局部稳定性, 图像加密

Abstract:

Chaotic system play an important role in fields such as communication, yet the analysis of high-dimensional chaotic system faces severe challenges due to their complex dynamic behavior. In order to address this issue, this paper investigates the Hopf bifurcation of a five-dimensional chaotic system with three quadratic terms. First, by employing the parameter-dependent center manifold theory and Hopf bifurcation theory, the stability of the equilibrium points and local dynamical behaviors such as Hopf bifurcation in the five-dimensional chaotic system are analyzed. The conditions for the existence of equilibrium points and Hopf bifurcation are obtained, indicating the presence of Hopf bifurcation in the system. Then, rigorous mathematical analysis and numerical simulation methods are used to study the conditions for the generation of one-dimensional and two-dimensional bifurcations in the chaotic system. Finally, in response to issues such as weak key sensitivity and strong correlation between adjacent pixels in image encryption schemes, the system is combined with DNA coding and applied to image encryption. The experimental results show that the new algorithm significantly improves the performance and safety of image encryption.

Key words: chaotic system, codimension, Hopf bifurcation, local stability, image encryption

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