Abstract
Co-dimension-one grazing bifurcations of 1 / n impact periodic motions are often accompanied by saddle-node bifurcations or period-doubling bifurcations. The presence of certain degenerate grazing bifurcation points plays an important role in transitions between these two different co-dimension-one grazing bifurcation scenarios. When the saddle-node bifurcation line and period-doubling bifurcation line meet at a grazing bifurcation point, a degenerate grazing bifurcation occurs. By considering the existence and stability of the 1 / n impact periodic motions near grazing points in single-degree-of-freedom impact oscillators, an analytical method is developed with the aid of the discontinuity mapping technique to determine the certain degenerate grazing bifurcation points of 1 / n motions. It is found that the linear term in the local discontinuity mapping does not affect the distribution of certain degenerate grazing points. The unfolding in the neighborhood of degenerate grazing bifurcation points is verified by numerical simulations.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China (No. 11672104), the “Chair Professor of Lotus Scholars Program” in Hunan province (No. XJT2015408), and the National Science Fund for Distinguished Young Scholars in China (No. 11225212). The authors also would like to thank the support from the Collaborative Innovation Center of Intelligent New Energy Vehicle, and Hunan Province Cooperative Innovation Center for The Construction & Development of Dongting Lake Ecological Economic Zone. Meanwhile, the authors are very grateful to the reviewers for their valuable comments and suggestions.
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Yin, S., Shen, Y., Wen, G. et al. Analytical determination for degenerate grazing bifurcation points in the single-degree-of-freedom impact oscillator. Nonlinear Dyn 90, 443–456 (2017). https://doi.org/10.1007/s11071-017-3674-z
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DOI: https://doi.org/10.1007/s11071-017-3674-z