ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Through the Looking-Glass of the Grazing Bifurcation: Part I - Theoretical Framework

Discontinuity, Nonlinearity, and Complexity 2(3) (2013) 203--223 | DOI:10.5890/DNC.2013.08.001

James Ing$^{1}$; Sergey Kryzhevich$^{2}$; Marian Wiercigroch$^{1}$

$^{1}$ Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Kings College Aberdeen AB24 3UE, Scotland, UK

$^{2}$ Chebyshev Laboratory and Faculty of Mathematics and Mechanics, Saint-Petersburg State University, 28, Universitetskiy pr., Peterhof, Saint-Petersburg, 198503, Russia, University of Aveiro, Department of Mathematics, 3810−193, Aveiro, Portugal

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. Models of impact Particularly, we show that the same effect can be observed in “soft” models of impacts.

Acknowledgments

Sergey Kryhevich was supported by Russian Foundation for Basic Researches, grant 12-01-00275-a, by Centre for Research and by FEDER funds through COMPETEOperational Programme Factors of Competitiveness ("Programa Operacional Factores de Competitividade") and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology ("FCTFundação para a Ciência e a Tecnologia"), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690 and within project PTDC/MAT/113470/2009. All coauthors are grateful to UK Royal Society for support within joint research project of University of Aberdeen and Saint-Petersburg State University.

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