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杨兆虎, 李险峰, 李登辉, 周碧柳. 拟周期激励下分段不对称振子的全局动力学. 力学学报, 待出版. DOI: 10.6052/0459-1879-24-099
引用本文: 杨兆虎, 李险峰, 李登辉, 周碧柳. 拟周期激励下分段不对称振子的全局动力学. 力学学报, 待出版. DOI: 10.6052/0459-1879-24-099
Yang Zhaohu, Li Xianfeng, Li Denghui, Zhou Biliu. Global dynamics of a piecewise asymmetric oscillator under quasi-periodic excitation. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-24-099
Citation: Yang Zhaohu, Li Xianfeng, Li Denghui, Zhou Biliu. Global dynamics of a piecewise asymmetric oscillator under quasi-periodic excitation. Chinese Journal of Theoretical and Applied Mechanics, in press. DOI: 10.6052/0459-1879-24-099

拟周期激励下分段不对称振子的全局动力学

GLOBAL DYNAMICS OF A PIECEWISE ASYMMETRIC OSCILLATOR UNDER QUASI-PERIODIC EXCITATION

  • 摘要: 分段线性振子是一类强非线性系统, 广泛存在于工程和数学领域. 例如用永磁铁和悬臂梁构造分段线性能量阱, 用分段线性模型逼近非线性振子, 这类系统具有复杂的动力学行为. 由于目前关于分段线性振子的研究主要考虑单频激励情形, 对于拟周期激励系统研究较少. 然而在实际工程中, 外部激励通常为多频激励. 文章通过广义的Melnikov方法, 研究了拟周期激励下一类分段不对称振子的全局动力学, 得到了同宿轨道横截相交的条件, 由此给出了系统发生Smale马蹄型混沌的阈值. 通过时间历程图、相图、Poincaré截面图和最大Lyapunov指数图验证了理论结果, 并分析了阻尼、弹簧刚度及外激励幅值对系统混沌运动的影响. 此外, 发现了系统的拟周期吸引子经环面倍化路径和分形路径演化为奇异非混沌吸引子的特殊现象. 奇异非混沌吸引子是一类具有分形几何结构, 但最大Lyapunov指数非正的吸引子, 这类吸引子可看作一种介于拟周期吸引子和混沌吸引子之间的特殊吸引子. 对奇异非混沌吸引子的产生机理和拓扑结构的研究有助于更好地理解多频激励系统中的分岔和混沌现象.

     

    Abstract: Piecewise linear oscillators are a class of strongly nonlinear systems. They are prevalently found in the fields of practical engineering and other interdisciplinary research fields. For example, piecewise linear energy sinks, which are constructed with permanent magnets and cantilever beams, and piecewise linear models, which are generally used to approximate nonlinear oscillators. Such systems always exhibit complex dynamical behavior. However, the present studies on the piecewise linear oscillators mainly consider the cases of single frequency excitation. There are rare studies on the quasi-periodic excitation systems. In practical engineering and applications, external excitations are often of multi-frequencies. In this paper, the global dynamics of a class of piecewise asymmetrical oscillators with quasi-periodic excitations are studied. The conditions for existence of transverse homoclinic points are obtained by the extended Melnikov method, and then the threshold of Smale horseshoe chaos is given. The theoretical results are verified numerically by the time series, the phase diagrams, the Poincaré sections and the largest Lyapunov exponent. It also demonstrates the effects of damping, spring stiffness and external excitation amplitude on the chaotic motion. But more than that, the special dynamical behavior showing that the quasi-periodic attractors of the system evolve into strange nonchaotic attractors through the routes of torus doubling and fractalization is illustrated. The strange nonchaotic attractors are a class of attractors. The most distinctive mark of them is that they have distinguished fractal structure, whilst the largest Lyapunov exponent of them is not positive. Whereby, the strange nonchaotic attractors can be regarded as a kind of special attractors, which occupies the middle ground between quasi-periodic attractors and chaotic attractors. The study on the generation mechanism and topological structure of strange nonchaotic attractors clarifies the inherent complexities, such as bifurcation scenarios and chaos, in multi-frequency excitation systems.

     

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