Through the looking-glass of the grazing bifurcation: Part I - theoretical framework

James Ing, Sergey Kryzhevich*, Marian Wiercigroch

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


It is well-known for vibro-impact systems that the existence of a periodic solution with a low-velocity impact (so-called grazing) may yield complex behavior of the solutions. In this paper we show that unstable periodic motions which pass near the delimiter without touching it may give birth to chaotic behavior of nearby solutions. We demonstrate that the number of impacts over a period of forcing varies in a small neighborhood of such periodic motions. This allows us to use the technique of symbolic dynamics. It is shown that chaos may be observed in a two-sided neighborhood of grazing and this bifurcation manifests at least two distinct ways to a complex behavior. In the second part of the paper we study the robustness of this phenomenon. Particularly, we show that the same effect can be observed in "soft" models of impacts.

Original languageEnglish
Pages (from-to)203-223
Number of pages21
JournalDiscontinuity, Nonlinearity, and Complexity
Issue number3
Publication statusPublished - 1 Jan 2013


  • Grazing
  • Homoclinic point
  • Models of impact
  • Structural stability


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