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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 ›› 2026, Vol. 43 ›› Issue (3): 426-440.doi: 10.3969/j.issn.1005-3085.2026.03.003cstr: 32411.14.cjem.CN61-1269/O1.2026.03.003

• • 上一篇    下一篇

分数阶van der Pol-Rayleigh系统Hopf分岔控制

谢志宽1,2,  解加全1,2,  司嘉琳1,2,  田佳敏1,2,  师  玮3,  焦国辉1,2, 暴一忱4   

  1. 1. 太原师范学院数学与统计学院,晋中 030619
    2. 太原师范学院智能优化计算与区块链技术山西省重点实验室,晋中 030619
    3. 太原理工大学机械与运载工程学院,太原  030024
    4. 华南理工大学计算机科学与工程学院,广州  510006
  • 收稿日期:2023-09-07 接受日期:2024-06-23 出版日期:2026-04-15 发布日期:2026-08-15
  • 通讯作者: 解加全 E-mail: xjq371195982@163.com
  • 基金资助:
    国家自然科学基金(52005360);中央引导地方科技发展资金(YDZJSX2022A053);山西省基础研究计划面上项目(202403021221189);山西省高等教育科技创新项目(2021L403);智能优化计算与区块链技术山西省重点实验室开放基金(IOCBT2025ZZY02);大同市应用基础研究计划(2022060);大同市科技计划(2025009). 

Hopf Bifurcation Control of the Fractional-order van der Pol-Rayleigh System

XIE Zhikuan1,2,  XIE Jiaquan1,2,  SI Jialin1,2,  TIAN Jiamin1,2,  SHI Wei3,  JIAO Guohui1,2,  BAO Yichen4   

  1. 1. College of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619
    2. Intelligent Optimization Computing and Blockchain Technology Shanxi Provincial Key Laboratory, Shanxi Provincial Department of Education, Taiyuan Normal University, Jinzhong 030619
    3. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024
    4. School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006
  • Received:2023-09-07 Accepted:2024-06-23 Online:2026-04-15 Published:2026-08-15
  • Contact: J. Xie. E-mail address: xjq371195982@163.com
  • Supported by:
    The National Natural Science Foundation of China (52005360); the Central Government Funds for Guiding Local Scientific and Technological Development (YDZJSX2022A053); the Basic Research Program of Shanxi Province (202403021221189); the Scientific and Technological Innovation Program of Higher Education Institutions of Shanxi Province (2021L403); the Open Fund of Shanxi Key Laboratory of Intelligent Optimization Computing and Blockchain Technology (IOCBT2025ZZY02); the Applied Basic Research Program of Datong City (2022060); the Science and Technology Program of Datong City (2025009).

摘要:

对分数阶van der Pol-Rayleigh系统在主共振条件下参数变化引起的Hopf分岔进行了深入研究。首先,针对该分数阶非线性振动系统,利用多尺度法和比较系数方法计算了系统的近似解析解,并通过分离其实部和虚部的数学处理手段,得到了系统在主共振条件下的幅频关系解析表达式,为后续稳定性分析奠定了理论基础。其次,基于第一李雅普诺夫指数和雅可比矩阵特征值分析方法,系统地探究了系统的稳定性条件,并根据其特征方程的求解结果精确计算了Hopf分岔点的位置,确定了系统发生周期性振荡的临界参数值。然后,通过静态分岔的理论分析深入讨论了Hopf分岔的类型及其动力学特性,揭示了系统在参数变化过程中的拓扑结构演化规律。最后,通过详细的数值模拟对幅频响应曲线进行了全面分析,有效验证了解析理论推导的正确性和适用性,并借助分岔图、相图、时序图等多种可视化手段,深入分析了不同参数变化对Hopf分岔产生的影响,为分数阶非线性系统的动力学行为预测和参数优化设计提供了重要的理论参考和数值依据。

关键词: van der Pol-Rayleigh系统, 多尺度法, 第一李雅普诺夫指数, 静态分岔, Hopf分岔

Abstract:

This article conducts an in-depth study on the Hopf bifurcation caused by parameter changes in fractional order van der Pol Rayleigh systems under the condition of primary resonance. Firstly, for the fractional order nonlinear vibration system, the approximate analytical solution of the system was calculated using the multiscale method and the comparison coefficient method. By separating its real and imaginary parts through mathematical processing, the analytical expression of the amplitude frequency relationship of the system under the main resonance condition was obtained, laying a theoretical foundation for subsequent stability analysis. Secondly, based on the first Lyapunov exponent and Jacobian matrix eigenvalue analysis method, the stability conditions of the system were systematically explored, and the position of the Hopf bifurcation point was accurately calculated according to the solution results of its characteristic equation, determining the critical parameter values for periodic oscillation of the system. Then, through the theoretical analysis of static bifurcation, the types and dynamic characteristics of Hopf bifurcation were discussed in depth, revealing the topological evolution law of the system during parameter changes. Finally, a comprehensive analysis of the amplitude frequency response curve was conducted through detailed numerical simulations, effectively verifying the correctness and applicability of the analytical theory derivation. With the help of various visualization methods such as bifurcation diagrams, phase diagrams, and time series diagrams, the impact of different parameter changes on Hopf bifurcation was analyzed in depth, providing important theoretical references and numerical basis for the dynamic behavior prediction and parameter optimization design of fractional order nonlinear systems.

Key words: van der Pol-Rayleigh system, multi scale method, first Lyapunov index, static bifurcation, Hopf bifurcation

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