Two Dimensional Map for Impact Oscillator with Drift

  • Ekaterina E. Pavlovskaia
  • Marian Wiercigroch
Conference paper
Part of the Solid Mechanics and its Applications book series (SMIA, volume 122)


An impact oscillator with a drift is considered. Using a simple co-ordinates transformation the bounded oscillations are separated from the drift. In general the dynamic state of the system is fully described by four variables: time, τ, relative displacement, p, and velocity, y, of the mass and relative displacement of the slider top, q. However, this number can be reduced by two if the beginning of the progression phase is monitored. In this paper a new two dimensional numerical map is developed and its dynamics is discussed.

Key words

Impact oscillator drift two dimensional map bifurcations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J.M.T. Thompson and R. Ghaffari, Phys. Rev. A 27, 1741–1743, 1983.MathSciNetCrossRefGoogle Scholar
  2. [2]
    S.W. Shaw and P.J. Holmes, “A periodically forced piecewise linear oscillator,” J. Sound Vib. 90, 129–155, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    S. Foale and S.R. Bishop, “Dynamic complexities of forced impacting systems,” Philos. Trans. R. Soc. London A 338, 547–556, 1992.MathSciNetzbMATHGoogle Scholar
  4. [4]
    W. Chin, E. Ott, H.E. Nusse and C. Grebogi, Phys. Rev. E 50, 4427–4444, 1994.MathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Banerjee and C. Grebogi, Phys. Rev. E 59, 4052–4061, 1999.CrossRefGoogle Scholar
  6. [6]
    E. Pavlovskaia, M. Wiercigroch and C. Grebogi, “Modeling of an impact system with a drift,” Phys. Rev. E 64, 056224, 2001.Google Scholar
  7. [7]
    E. Pavlovskaia and M. Wiercigroch, “Analytical drift reconstruction for impact oscillator with drift,” Chaos, Solitons and Fractals 19, 151–161, 2004.CrossRefzbMATHGoogle Scholar
  8. [8]
    E. Pavlovskaia and M. Wiercigroch, “Two dimensional map for impact oscillator with drift,” submitted in 2003.Google Scholar
  9. [9]
    M. Oestreich, N. Hinrichs and K. Popp, “Bifurcation and stability analysis for a non-smooth frictional oscillator,” Arch. Appl Mech. 66, 301–314, 1996.CrossRefzbMATHGoogle Scholar
  10. [10]
    U. Galvanetto, “Numerical computation of Lyapunov exponents in discontinous maps implicitly defined,” Computer Physics Communications 131, 1–9, 2000.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    A.B. Nordmark, “Non-periodic motion caused by grazing incidence in an impact oscillator,” J. Sound Vib. 145, 279–297, 1991.CrossRefGoogle Scholar
  12. [12]
    M. di Bernardo, M.I. Feigin, S.J. Hogan and M.E. Holmer, “Local analysis of C-bifurcations in n-dimensional piecewise smooth dynamical system,” Chaos, Solitons and Fractals 10(11), 1881–1908, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    C. Budd and F. Dux, “Chattering and related behaviour in impact oscillators,” Philos. Trans. R. Soc. London A 347, 365–389, 1994.zbMATHGoogle Scholar
  14. [14]
    M. di Bernardo, P. Kowalczyk and A. Nordmark, “Bifurcations of dynamical systems with sliding: derivation of normal-form mapping,” Physica D 170, 175–205, 2002.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Ekaterina E. Pavlovskaia
    • 1
  • Marian Wiercigroch
    • 1
  1. 1.Centre for Applied Dynamics Research, School of Engineering and Physical SciencesKings College University of AberdeenScotland, UK

Personalised recommendations