Periodic orbits of a conservative 2-DOF vibro-impact system by piecewise continuation: bifurcations and fractals

  • Hongcheng TaoEmail author
  • James Gibert
Original Paper


The exact periodic orbits of a conservative 2-degree-of-freedom vibro-impact system with stereo-mechanical impact model is studied using a piecewise continuation method. Feasible initial guesses are extracted from grazing solution points so that each branch of solution can be initiated smoothly from previously solved ones. Frequency-energy plots (FEPs) are produced where an emphasis is placed on enumerating potential bifurcations. Extra critical points are discovered on multiple-period duplicates of existing stable solution branches and lead to cascades of period-multiplying bifurcations. The results indicate that the system’s complete FEP can be viewed through a process of infinite fractals toward zero frequency where pseudo-periodic or chaotic responses are approached. Finally, it is shown both mathematically and through the comparison of FEPs that the impacting system can be represented explicitly as the extreme case of nonlinear systems with an odd-order polynomial internal force. It is thus proposed that as the counterpart to the superposition of linear normal modes, the free responses of a general conservative nonlinear system can be tracked via bifurcations from its nonlinear normal modes.


Nonlinear normal mode Vibro-impact system Continuation Fractals Nonsmooth dynamical system 



This material is based upon work supported by the Purdue Research Foundation and the National Science Foundation under Grant No. CMMI 1662925.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflicts of interest.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Purdue UniversityWest LafayetteUSA

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