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

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Abstract

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.

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References

  1. E. Fermi, Phys. Rev. 75, 1169 (1949)

    Article  ADS  Google Scholar 

  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)

  3. A.J. Lichtenberg, M.A. Lieberman, R.H. Cohen, Physica D 1, 291 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  4. G.A. Luna-Acosta, Phys. Rev. A 42, 7155 (1990)

    Article  ADS  Google Scholar 

  5. V. Zharnitsky, Nonlinearity 13, 1123 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  6. J. de Simoi, Discrete Contin. Dyn. Syst. 25, 719 (2008)

    Article  MathSciNet  Google Scholar 

  7. E.D. Leonel, M.R. Silva, J. Phys. A: Math. Theor. 41, 140 (2009)

    Google Scholar 

  8. A.C.J. Luo, Y. Guo, Math. Prob. Eng. 2009, 298906 (2009)

    Article  Google Scholar 

  9. D.G. Ladeira, E.D. Leonel, Commun. Nonlinear Sci. Numer. Simul. 20, 546 (2015)

    Article  ADS  Google Scholar 

  10. J. de Simoi, D. Dolgopyat, Chaos 22, 593 (2012)

    Article  Google Scholar 

  11. D.R.D. Costa, C.P. Dettmann, E.D. Leonel, Commun. Nonlinear Sci. Numer. Simul. 20, 871 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  12. A. Okniski, B. Radziszewski, Nonlinear Dyn. 67, 1115 (2012)

    Article  Google Scholar 

  13. T. Botari, E.D. Leonel, Phys. Rev. E 87, 574 (2013)

    Article  Google Scholar 

  14. B. van der Pol, Radio Rev. 1, 701 (1920)

    Google Scholar 

  15. J.J. Stoker, Nonlinear vibrations (Interscience, New York, 1950)

  16. X.L. Fu, S.S. Zheng, Commun. Nonlinear Sci. Numer. Simul. 19, 3023 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  17. S.S. Zheng, X.L. Fu, Int. J. Bifurc. Chaos 25, 1550119 (2015)

    Article  Google Scholar 

  18. P.J. Holmes, J. Sound Vib. 84, 173 (1982)

    Article  ADS  Google Scholar 

  19. A.C. Luo, Discontinuous dynamical systems (Springer, Berlin, Heidelberg, 2012)

  20. A.C. Luo, Singularity and dynamics on discontinuous vector fields (Elsevier, Amsterdam, 2006)

  21. A.C. Luo, Discontinuous dynamical systems on time-varying domains (Higher Education Press, Beijing, 2009)

Download references

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Authors and Affiliations

  1. School of Mathematics and Statistics, Shandong Normal University, Ji’nan, 250014, P.R. China

    Xilin Fu & Shasha Zheng

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  1. Xilin Fu
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  2. Shasha Zheng
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Correspondence to Xilin Fu or Shasha Zheng.

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Fu, X., Zheng, S. Acceleration behaviors of Fermi accelerator model excited by Van der Pol oscillator. Eur. Phys. J. Spec. Top. 228, 1421–1439 (2019). https://doi.org/10.1140/epjst/e2019-800236-0

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  • DOI: https://doi.org/10.1140/epjst/e2019-800236-0

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