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Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters

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Abstract

To obtain the correlation between multiple parameters, multi-initial values and multi-stable behaviors for a non-smooth nonlinear system with time-varying parameters, a new method of calculating the bifurcation of multi-stable behaviors in the parametric plane is first proposed based on Poincaré mapping theory, Lyapunov theory and Floquot theory. The bifurcation and distribution of multi-stable behaviors of a nonlinear gear system with time-varying meshing stiffness in a two-parameter plane are studied by using the proposed method. Various multi-stable behaviors and potential hidden bifurcation curves are fully revealed. Double-bifurcation points formed by the intersection of two different bifurcation curves are further investigated. The probability of occurrence of hidden bifurcation curve is calculated and analyzed based on statistical theory. Results indicate that saddle-node bifurcation curves are sensitive to the initial value and change both the type of multi-stable behaviors and the topology of the attraction basin. However, period-doubling bifurcation curves are not sensitive to the initial value, and only change the type of multi-stable behavior, but do not greatly change the topology of the attraction basin. Four different types of multi-stable behaviors are observed around double-bifurcation points. Multi-stable behaviors and bifurcation curves are easily hidden in the parametric plane due to their small occurring probabilities.

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Acknowledgements

This investigation is financially supported by the Natural Science Foundation of Tianjin, China (Grant No. 18JCYBJC88800), by the National Natural Science Foundation of China (Grant No. 51365025) and by the Program for Innovative Research Team in University of Tianjin, China (Grant No. TD13-5037).

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Authors and Affiliations

  1. School of Mechanical Engineering, Tiangong University, Tianjin, 300387, China

    Jian-fei Shi, Xiang-feng Gou & Ling-yun Zhu

  2. Tianjin Key Laboratory of Modern Mechatronics Equipment Technology, Tiangong University, Tianjin, 300387, China

    Jian-fei Shi, Xiang-feng Gou & Ling-yun Zhu

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  1. Jian-fei Shi
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  2. Xiang-feng Gou
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  3. Ling-yun Zhu
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Correspondence to Xiang-feng Gou.

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Shi, Jf., Gou, Xf. & Zhu, Ly. Bifurcation of multi-stable behaviors in a two-parameter plane for a non-smooth nonlinear system with time-varying parameters. Nonlinear Dyn 100, 3347–3365 (2020). https://doi.org/10.1007/s11071-020-05510-z

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