单自由度碰撞振动系统的奇异非混沌动力学和多稳态共存

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摘要基于一类单自由度受拟周期激励的具有悬臂结构碰撞振动系统，通过数值模拟研究了系统在参数变化下的奇异非混沌动力学及多稳态共存现象。采用相敏感函数、奇异连续谱、有理数频率逼近、状态变量的傅立叶变换部分和在复平面上的路径等工具刻画了系统在特定参数范围内出现的奇异非混沌吸引子(Strange Nonchaotic Attractors, 简记为SNAs)的奇异性。通过最大Lyapunov指数验证了SNAs非混沌特性。发现了系统存在分形和阵发两种通向SNAs的路径，揭示了这两种路径的演化过程和规律特征。结合系统状态变量的时间序列，分析了系统暂态及稳态SNAs与拟周期吸引子的共存、稳态SNAs与混沌吸引子的共存情况，进一步揭示了SNAs的多稳态共存现象及转换规律。该研究可为含悬臂结构的碰撞振动系统的优化设计提供理论依据。

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	吴鑫
	李高磊
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关键词 ：碰撞振动系统, 奇异非混沌动力学, 分形路径, 阵发路径, 多稳态共存

Abstract：Based on a single-degree-of-freedom vibro-impact system with cantilever forced quasiperiodic excitation, the strange nonchaotic dynamics and multistable coexistence phenomena of the system with parameter changes were studied by numerical simulation. The singularity of the strange nonchaotic attractors (SNAs) in the specific parameter range of the system was characterized using such tools as phase sensitivity function, singular continuous spectrum, rational frequency approximations, path of the partial Fourier sum of the state variable in the complex plane. The nonchaotic properties of SNAs were verified by the top Lyapunov exponents. The fractal and intermittency routes to SNAs were discussed, and the evolutions and regularities of these routes were revealed. The coexistence of transient SNAs and quasiperiodic attractors, stable SNAs and quasiperiodic attractors, and stable SNAs and chaotic attractors were analyzed by using the time series of system state variables. Furthermore, the multistable coexistences and transition rules of SNAs were revealed. The research above may be provided theoretical basis for the optimal design of the vibro-impact system with cantilever.

Key words： vibro-impact system strange nonchaotic dynamics fractal route intermittency route multistable coexistence

收稿日期: 2020-10-12 出版日期: 2022-01-28

引用本文:

吴鑫，李高磊，乐源. 单自由度碰撞振动系统的奇异非混沌动力学和多稳态共存[J]. 振动与冲击, 2022, 41(2): 45-52.
WU Xin，LI Gaolei，YUE Yuan. Strange nonchaotic dynamics and multistable coexistence phenomena of a single-degree-of-freedom vibro-impact system. JOURNAL OF VIBRATION AND SHOCK, 2022, 41(2): 45-52.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2022/V41/I2/45

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