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 journalArticle

1 Citation (Scopus)

Abstract

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
Volume2
Issue number3
DOIs
Publication statusPublished - 1 Jan 2013

Fingerprint

Grazing Bifurcation
grazing
Glass
glass
Periodic Motion
Chaos theory
Symbolic Dynamics
Behavior of Solutions
Chaotic Behavior
Forcing
low speed
chaos
Periodic Solution
Chaos
Bifurcation
Unstable
Vary
Robustness
Distinct
Framework

Keywords

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

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Computational Mechanics
  • Discrete Mathematics and Combinatorics
  • Control and Optimization

Cite this

Through the looking-glass of the grazing bifurcation : Part I - theoretical framework. / Ing, James; Kryzhevich, Sergey; Wiercigroch, Marian.

In: Discontinuity, Nonlinearity, and Complexity, Vol. 2, No. 3, 01.01.2013, p. 203-223.

Research output: Contribution to journalArticle

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