The European Physical Journal Special Topics

, Volume 228, Issue 6, pp 1421–1439 | Cite as

Acceleration behaviors of Fermi accelerator model excited by Van der Pol oscillator

  • Xilin FuEmail author
  • Shasha ZhengEmail author
Regular Article Topical issue
Part of the following topical collections:
  1. Discontinuous Dynamical Systems and Synchronization


In this paper, we investigate on dynamics of a generalized Fermi accelerator model with a moving wall described by a nonlinear Van der Pol oscillator. Different acceleration behaviors appear due to the discontinuity brought by impact and the stationary periodic excitation of the moving wall. Utilizing the theory of discontinuous dynamical systems, acceleration mechanism of the particle is studied and the analytical conditions for stick motion and grazing motion are obtained. By the use of generic mappings between different boundaries including stick and non-stick motions, periodic motions of the Fermi accelerator can be constructed, and corresponding local stability and bifurcations can be discussed according to mapping dynamics. Finally, different acceleration behaviors including periodic, chatter and stick motions are presented and illustrated.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Fermi, Phys. Rev. 75, 1169 (1949)ADSCrossRefGoogle Scholar
  2. 2.
    S.M. Ulam, On some statistical properties of dynamical systems, in Proceedings of the fourth Berkeley Symposium on Mathematical Statistics and Probability (University of California Press, Berkeley, CA, 1961)Google Scholar
  3. 3.
    A.J. Lichtenberg, M.A. Lieberman, R.H. Cohen, Physica D 1, 291 (1980)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    G.A. Luna-Acosta, Phys. Rev. A 42, 7155 (1990)ADSCrossRefGoogle Scholar
  5. 5.
    V. Zharnitsky, Nonlinearity 13, 1123 (2000)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    J. de Simoi, Discrete Contin. Dyn. Syst. 25, 719 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    E.D. Leonel, M.R. Silva, J. Phys. A: Math. Theor. 41, 140 (2009)Google Scholar
  8. 8.
    A.C.J. Luo, Y. Guo, Math. Prob. Eng. 2009, 298906 (2009)CrossRefGoogle Scholar
  9. 9.
    D.G. Ladeira, E.D. Leonel, Commun. Nonlinear Sci. Numer. Simul. 20, 546 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    J. de Simoi, D. Dolgopyat, Chaos 22, 593 (2012)CrossRefGoogle Scholar
  11. 11.
    D.R.D. Costa, C.P. Dettmann, E.D. Leonel, Commun. Nonlinear Sci. Numer. Simul. 20, 871 (2015)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    A. Okniski, B. Radziszewski, Nonlinear Dyn. 67, 1115 (2012)CrossRefGoogle Scholar
  13. 13.
    T. Botari, E.D. Leonel, Phys. Rev. E 87, 574 (2013)CrossRefGoogle Scholar
  14. 14.
    B. van der Pol, Radio Rev. 1, 701 (1920)Google Scholar
  15. 15.
    J.J. Stoker, Nonlinear vibrations (Interscience, New York, 1950)Google Scholar
  16. 16.
    X.L. Fu, S.S. Zheng, Commun. Nonlinear Sci. Numer. Simul. 19, 3023 (2014)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    S.S. Zheng, X.L. Fu, Int. J. Bifurc. Chaos 25, 1550119 (2015)CrossRefGoogle Scholar
  18. 18.
    P.J. Holmes, J. Sound Vib. 84, 173 (1982)ADSCrossRefGoogle Scholar
  19. 19.
    A.C. Luo, Discontinuous dynamical systems (Springer, Berlin, Heidelberg, 2012)Google Scholar
  20. 20.
    A.C. Luo, Singularity and dynamics on discontinuous vector fields (Elsevier, Amsterdam, 2006)Google Scholar
  21. 21.
    A.C. Luo, Discontinuous dynamical systems on time-varying domains (Higher Education Press, Beijing, 2009)Google Scholar

Copyright information

© EDP Sciences, Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Shandong Normal UniversityJi’nanP.R. China

Personalised recommendations