Degenerate grazing bifurcations in a three-degree-of-freedom impact oscillator

  • Shan Yin
  • Jinchen Ji
  • Shuning Deng
  • Guilin WenEmail author
Original Paper


This paper presents the analysis of the degenerate grazing bifurcation in a three-degree-of-freedom impact oscillator by studying the bifurcations of near-grazing period-one impact motion near the degenerate grazing point. Actually, this paper extends the higher-order zero time discontinuity mapping to perform the perturbation analysis of characteristic equation of period-one impact motion and obtains feasible eigenvalue approximation to study the potential bifurcations. The shooting method is applied to verify the validity of the derived approximation and corresponding computation results. In addition to the known bifurcation scenarios of saddle-node and period-doubling, novel Neimark–Sacker bifurcation and related co-dimension two bifurcation points of near-grazing period-one impact motion are also found to arise near the degenerate grazing point in a three-degree-of-freedom impact oscillator. For the in-depth understanding of near-grazing dynamics, the obtained results are compared with the reported results in the single- and two-degree-of-freedom impact oscillators.


Degenerate grazing bifurcation Neimark–Sacker bifurcation Perturbation of eigenvalue Higher-order discontinuity mapping 



The first author is partially supported by the CSC scholarship No. 201706130054. 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 would also like to thank the support from the Collaborative Innovation Center of Intelligent New Energy Vehicle, the Hunan Collaborative Innovation Center for Green Car and Hunan Province Cooperative Innovation Center for the Construction & Development of Dongting Lake Ecological Economic Zone.

Compliance with ethical standards

Conflict of interest

The authors declare that for this article, there is no conflict of interest in authorial ascription to organizations or financial and personal relationships with other people.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • Shan Yin
    • 1
    • 2
  • Jinchen Ji
    • 2
  • Shuning Deng
    • 1
  • Guilin Wen
    • 1
    • 3
    Email author
  1. 1.State Key Laboratory of Advanced Design and Manufacture for Vehicle BodyHunan UniversityChangshaChina
  2. 2.School of Mechanical and Mechatronic Engineering, FEITUniversity of Technology SydneyBroadwayAustralia
  3. 3.School of Mechanical and Electric EngineeringGuangzhou UniversityGuangzhouChina

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