含平方阻尼项的Mathieu-Duffing系统混沌与分岔研究  

Chaos and bifurcation of Mathieu-Duffing system with square damping term

作　　者：解加全 王海军 师玮 张佳乐 霍逸婷 曹佳琳 高蔷 XIE Jiaquan;WANG Haijun;SHI Wei;ZHANG Jiale;HUO Yiting;CAO Jialin;GAO Qiang(College of Mathematics and Statistics,Taiyuan Normal University,Jinzhong 030619,China;Intelligent Optimization Computing and Blockchain Technology Shanxi Provincial Key Laboratory,Shanxi Provincial Department of Education,Taiyuan Normal University,Jinzhong 030619,China;College of Mechanical and Vehicle Engineering,Taiyuan University of Technology,Taiyuan 030024,China)

机构地区：[1]太原师范学院数学与统计学院,山西晋中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,山西晋中030619 [3]太原理工大学机械与运载工程学院,太原030024

出　　处：《振动与冲击》2024年第7期168-174,195,共8页Journal of Vibration and Shock

基　　金：国家自然科学基金(52005360);中央引导地方科技发展资金项目(YDZJSX2022A053);大同市应用基础研究计划项目(2022060);山西省高等教育科技创新项目(2021L403)。

摘　　要：旨在研究含平方阻尼项Mathieu-Duffing系统的共振与混沌。利用多尺度法探究系统在参数和受迫联合激励作用下主共振的幅频与相频特性。基于Lyapunov第一方法给出定常解的稳定性条件并判定系统存在的周期解支。依据系统异宿轨道参数方程推导系统出现异宿轨道横截相交及系统发生混沌的必要条件。根据分岔图、相轨迹图以及Poincare截面研究激励幅值与激励频率对系统进入混沌运动性态的影响,证实激励频率与激励幅值的变化均可导致系统经倍周期分岔进入混沌状态。Here,resonance and chaos of Mathieu-Duffing system with square damping term were studied.The multi-scale method was used to explore amplitude-frequency and phase-frequency characteristics of main resonance of the system under the combined excitation of parameters and external forces.Based on Lyapunov first method,stability conditions for steady solutions were given,and the system’s periodic solution branches were determined.According to the parameter equation of the system’s heteroclinic orbits,necessary conditions for occurrence of intersection of heteroclinic orbits and chaos in the system were derived.Based on bifurcation diagrams,phase trajectory diagrams and Poincare cross-sections,effects of excitation amplitude and frequency on chaotic motion behavior of the system were studied.It was shown that changes in excitation frequency and excitation amplitude can cause the system to enter a chaotic state from period-doubling bifurcation.

关 键 词：Mathieu-Duffing系统 平方项阻尼 多尺度法 Lyapunov第一方法 倍周期分岔 

分 类 号：O323[理学—一般力学与力学基础] O313.1[理学—力学]