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Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system

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Abstract

We develop a method to compute the Lyapunov spectrum and Lyapunov dimension, which is effective for both symmetric and unsymmetric vibro-impact systems. The Poincaré section is chosen at the moment after impacting, and the six-dimensional Poincaré map is established. The time between two consecutive impacts is determined by the initial conditions and the impact condition, hence the Poincaré map is an implicit map. The Poincaré map is used to calculate all the Lyapunov exponents and the Lyapunov dimension. By numerical simulations, the attractors are represented in the projected Poincaré section, and the Lyapunov spectrum is obtained. The multi-degree-of-freedom vibro-impact system may exhibit complex quasi-periodic attractors, which can be characterized by the Lyapunov dimension.

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Authors and Affiliations

  1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Yuan Yue & Jianhua Xie

  2. College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, 610059, China

    Xuejun Gao

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  1. Yuan Yue
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  2. Jianhua Xie
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  3. Xuejun Gao
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Correspondence to Yuan Yue.

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Yue, Y., Xie, J. & Gao, X. Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system. Nonlinear Dyn 69, 743–753 (2012). https://doi.org/10.1007/s11071-011-0301-2

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  • DOI: https://doi.org/10.1007/s11071-011-0301-2

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