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Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system

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  • Guo, Xiuying
  • Tian, Ruilan
  • Xue, Qiang
  • Zhang, Xiaolong

Abstract

In this paper, a double pendulum model with multi-point collision is established to study the sub-harmonic bifurcation of high-dimensional coupled non-smooth systems. Considering the coupling and non-smoothness of the system, a two-step decoupling method is proposed to detect the sub-harmonic bifurcation of a two-degree-of-freedom non-smooth coupled system. The core view is to introduce energy-time scale transformation to overcome the obstacle of the system coupled term. In the first step, a reversible transformation is introduced to decouple the system. This transformation enables the coupled form of the impact term, which presents novel obstacles to the high-dimensional non-smooth system. By introducing energy-time scale transformation in the second step, the system is expressed as a smooth decoupling form of the energy coordinate, and the trouble of impact term coupled is solved. Furthermore, the sub-harmonic Melnikov function which depends on frequency, amplitude of excitation and impact recovery coefficient is derived by using the two-step decoupling method. Hence, the sub-harmonic Melnikov function is extended to the high-dimensional non-smooth system, which reveals the influence of the impact recovery coefficient on the existence of sub-harmonic periodic orbits. The innovation of this method is that it solves the coupled problem of non-smooth terms, quantifies the impact of impact recovery coefficient on the dynamic behavior of the system, and provides a theoretical basis for the actual parameter design and control in engineering. The obtained theoretical results are verified through the numerical simulations.

Suggested Citation

  • Guo, Xiuying & Tian, Ruilan & Xue, Qiang & Zhang, Xiaolong, 2022. "Sub-harmonic Melnikov function for a high-dimensional non-smooth coupled system," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    • Handle: RePEc:eee:chsofr:v:164:y:2022:i:c:s0960077922008104
    DOI: 10.1016/j.chaos.2022.112629
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    References listed on IDEAS

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    1. Luo, Albert C.J. & Chen, Lidi, 2005. "Periodic motions and grazing in a harmonically forced, piecewise, linear oscillator with impacts," Chaos, Solitons & Fractals, Elsevier, vol. 24(2), pages 567-578.
    2. Castro, Jose & Alvarez, Joaquin & Verduzco, Fernando & Palomares-Ruiz, Juan E., 2017. "Chaotic behavior of driven, second-order, piecewise linear systems," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 8-13.
    3. Peng, Yuanyuan & Fan, Jinjun & Gao, Min & Li, Jianping, 2021. "Discontinuous dynamics of an asymmetric 2-DOF friction oscillator with elastic and rigid impacts," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    4. Zhou, Liangqiang & Ji, Peng & Chen, Fangqi, 2021. "Chaos and subharmonic bifurcation of a composite laminated buckled beam with a lumped mass," Chaos, Solitons & Fractals, Elsevier, vol. 147(C).
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    Cited by:

    1. Zhang, Yifeng & Xu, Huidong & Zhang, Jianwen, 2023. "Global dynamics for impacting cantilever beam supported by oblique springs," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).

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