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Complicated periodic cascades arising from double grazing bifurcations in an impact oscillator with two rigid constraints

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Abstract

In this paper, we study bifurcations of double grazing periodic orbits, namely periodic orbits with two grazing points, in a one-degree-of-freedom impact oscillator constrained by two rigid blocks. By applying the method of Poincaré discontinuity mapping, we approximate the normal Poincaré map near the double grazing orbit to the leading order, which is a two-dimensional piecewise smooth map defined in four regions that has square root singularities. Based on the Poincaré map, the stability and co-dimension-two bifurcation conditions of the double grazing orbit are presented. Furthermore, the results of computing Monte Carlo bifurcation diagrams of the map are compared with those from direct simulations of the original system, and we find that they are in excellent agreement. Through numerical simulations, we find that the system undergoes very complicated bifurcations near the double grazing periodic orbit, such as the period-adding and period-incrementing cascades. Hysteresis effects are also discussed.

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Data Availability

All data included in this study are available upon request by contact with the corresponding author.

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Acknowledgements

The authors are very grateful to the associate editor and the anonymous referees for their careful reading and valuable suggestions, which have notably improved the quality of this paper.

Funding

This work was supported by the National Natural Science Foundation of China (No.11971019, 12071317). F. Luo was also supported by the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2022QZJ02).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, 643000, Sichuan, China

    Fei Luo

  2. Department of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, China

    Zhengdong Du

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  1. Fei Luo
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FL involved in investigation, methodology, validation, data curation, software, and writing—original draft. ZD involved in conceptualization, investigation, methodology, validation, data curation, writing—original draft, and writing—reviewing and editing.

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Correspondence to Zhengdong Du.

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Luo, F., Du, Z. Complicated periodic cascades arising from double grazing bifurcations in an impact oscillator with two rigid constraints. Nonlinear Dyn 111, 13829–13852 (2023). https://doi.org/10.1007/s11071-023-08600-w

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