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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


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

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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.


Sergey Kryhevich was supported by Russian Foundation for Basic Researches, grant 12-01-00275-a, by Centre for Research and by FEDER funds through COMPETEOperational 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 ("FCTFundação 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.


  1. [1]  Babitsky, V.I. (1998), Theory of Vibro-Impact Systems and Applications, Springer, Berlin.
  2. [2]  Kozlov, V.V. and Treschev, D.V. (1991), Billiards. A genetic introductionto the dynamics of systems with impacts, Translations of Mathematical Monographs, 89, American Mathematical Society, Providence, RI.
  3. [3]  Luo, A.C.J. (2009), Discontinuous Dynamical Systems on Time-varying Domains, Higher Education Press and Springer, Beijing-Heidenberg.
  4. [4]  Peterka, F. (1974), Laws of impact motion of mechanical systems with one degree of freedom, Acta Technika CSAV, 4, 462-473.
  5. [5]  Pavlovskaia, E. and Wiercigroch, M. (2004), Analytical drift reconstruction for visco-elastic impact oscillators operating in periodic and chaotic regimes, Chaos, Solitons and Fractals, 19 (1), 151-161.
  6. [6]  Schatzman, M. (1998), Uniqueness and continuous dependence on data for one-dimensional impact problem, Mathematical and Computer Modelling, 28(4-8), 1-18.
  7. [7]  Akhmet, M.U. (2009), Li-Yorke chaos in systems with impacts, Journal of Mathematical Analysis and Applications, 351 (2), 804-810.
  8. [8]  Banerjee, S., Yorke, J.A., and Grebogi, C. (1998), Robust chaos, Physical Review Letters, 80 (14), 3049-3052.
  9. [9]  Chillingworth, D.R.J. (2010), Dynamics of an impact oscillator near a degenerate graze, Nonlinearity, 23 (11), 2723-2748.
  10. [10]  Chin, W., Ott, E., Nusse, H.E., and Grebogi, C. (1995), Universal behavior of impact oscillators near grazing incidence, Physical Letters A, 201 (2-3), 197-204.
  11. [11]  Fredriksson, M.H. and Nordmark, A.B. (1997), Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators, Proceedings of the Royal Society of London Series A, 453 (1961), 1261-1276.
  12. [12]  Gorbikov, S.P. and Men'shenina, A.V. (2007), Statistical description of the limiting set for chaotic motion of the vibro-impact system, Automation and remote control, 68 (10), 1794-1800.
  13. [13]  Do, Y.H. and Lai, Y.C. (2008), Multistability and arithmetically period-adding bifurcations in piecewise smooth dynamical systems, Chaos, 18 (4), 043107
  14. [14]  Holmes, P.J. (1982), The dynamics of repeated impacts with a sinusoidally vibrating table, Journal of Sound and Vibration, 84 (10), 173-189.
  15. [15]  Ivanov, A.P. (1996), Bifurcations in impact systems, Chaos, Solitons and Fractals, 7 (10), 1615-1634.
  16. [16]  Kryzhevich, S.G. (2008), Grazing bifurcation and chaotic oscillations of single-degree-of-freedom dynamical systems, Journal of Applied Mathematics and Mechanics, 72 (4), 539-556.
  17. [17]  Kryzhevich, S.G. and Pliss, V.A. (2005), Chaotic modes of oscillations of a vibro-impact system, Journal of Applied Mathematics and Mechanics, 69 (1), 15-29.
  18. [18]  Lenci, S. and Rega, G. (2000), Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos, Solitons and Fractals, 11 (15), 2453-2472.
  19. [19]  Molenaar, J., van de Water, W., and de Wegerand, J. (2000), Grazing impact oscillations, Physical Review E, 62 (2), 2030-2041.
  20. [20]  Nordmark, A.B. (1991), Non-periodic motion caused by grazing incidence in an impact oscillator, Journal of Sound and Vibration, 145 (2), 279-297.
  21. [21]  Nordmark, A.B. (2001), Existence of periodic orbits in grazing bifurcations of impacting mechanical oscillators, Nonlinearity, 14 (6), 1517-1542.
  22. [22]  Thomson, J.M.T. and Ghaffari, R. (1983), Chaotic dynamics of an impact oscillator, Physical Review A, 27 (3), 1741-1743.
  23. [23]  Whiston, G.S. (1987), Global dynamics of a vibro-impacting linear oscillator, Journal of Sound and Vibration, 118 (3), 395-429.
  24. [24]  Paoli, L. (2012),Mathematical aspects of vibro-impact problems: existence, (non-)uniqueness and approximation, Nonsmooth Contact Mechanics, Modelling and Simulation, Summer School, Aussois, September, 9-14.
  25. [25]  Devaney, R.L. (1987), An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City.
  26. [26]  Smale, S. (1965), Diffeomorfisms with many periodic points, Differential and Combinatoric Topology, Princeton University Press, Princeton.
  27. [27]  Palis, J. and di Melo, W. (1982), Geometric Theory of Dynamical Systems, Springer, New York.
  28. [28]  Anosov, D.V. (1967), Geodesic Flows on Riemann Manifolds of Negative Curvature, Nauka, Moscow.