Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations

  • Ruilan Tian
  • Qiliang Wu
  • Xinwei YangEmail author
  • Chundi Si
Regular Article


The smooth-and-discontinuous oscillator under constant excitations (CSD) is an unsymmetrical system with an irrational restoring force, which leads to a barrier to detect the chaotic threshold using conventional nonlinear techniques. The goal of the present work is to overcome the trouble raised by constant excitations and irrational nonlinearity to investigate the necessary and insufficient condition for chaos. A smooth approximate system, whose behaviors are topologically equivalent to the ones of the original system, is proposed to investigate the chaotic boundary. The Hamiltonian system of the approximate system is analyzed and the homoclinic orbits in analytical form are obtained. The Melnikov method is employed to determine the distance between the stable and unstable manifolds under the perturbation of damping and external forcing. Bifurcation diagrams and numerical simulations are used to reveal the motion of chaos and the period for approximate system which is in good agreement with the original system. It is worth recalling that the approach for constructing the approximate function of smoothness is presented, which puts forward a step toward the solution of the irrational nonlinearity system.


Bifurcation Diagram Original System Unstable Manifold Chaotic Attractor Homoclinic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    L. Cveticanin, J. Sound Vib. 261, 169 (2003).MathSciNetADSCrossRefzbMATHGoogle Scholar
  2. 2.
    L. Cveticanin, J. Sound Vib. 270, 441 (2004).ADSCrossRefGoogle Scholar
  3. 3.
    T. Kawamoto, S. Abe, Polyhedron 24, 2676 (2005).CrossRefGoogle Scholar
  4. 4.
    Z.Q. Wu, Y.S. Chen, Appl. Math. Mech-Engl. Ed. 27, 161 (2006).CrossRefzbMATHGoogle Scholar
  5. 5.
    C.W. Lim, S.K. Lai, B.S. Wu, W.P. Sun, Y. Yang, C. Wang, Arch. Appl. Mech. 79, 411 (2009).ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Cveticanin, J. Sound Vib. 330, 976 (2011).ADSCrossRefGoogle Scholar
  7. 7.
    E. Gourdona, N.A. Alexandera, C.A. Taylora, C.H. Lamarqueb, S. Pernot, J. Sound Vib. 300, 522 (2007).ADSCrossRefGoogle Scholar
  8. 8.
    L.I. Manevitch, E. Gourdon, C.H. Lamarque, Chaos Soliton Fract. 31, 900 (2007).ADSCrossRefGoogle Scholar
  9. 9.
    G. Gatti, I. Kovacic, M.J. Brennan, J. Sound Vib. 329, 1823 (2010).ADSCrossRefGoogle Scholar
  10. 10.
    S.K. Lai, Y. Xiang, Comput. Math. Appl. 60, 2078 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    W.P. Sun, B.S. Wu, C.W. Lim, J. Comput. Nonlinear Dyn. 2, 141 (2007).CrossRefGoogle Scholar
  12. 12.
    N. Jamshidi, D.D. Ganji, Curr. Appl. Phys. 10, 484 (2010).ADSCrossRefGoogle Scholar
  13. 13.
    J.H. He, G.C. Wu, F. Austin, Nonlinear Sci. Lett. A 1, 1 (2010).Google Scholar
  14. 14.
    T. Özis, A. Yildirim, J. Sound Vib. 306, 372 (2007).ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    W.P. Sun, B.S. Wu, C.W. Lim, J. Sound Vib. 300, 1042 (2007).ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    H.L. Zhang, Comput. Math. Appl. 58, 2480 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Beléz, A. Hernádez, T. Beléz, M.L. >Ávarez, S. Gallego, M. Ortuñ, C. Neipp, J. Sound Vib. 302, 1018 (2007).ADSCrossRefGoogle Scholar
  18. 18.
    A. Belédez, T. Belédez, C. Neipp, M.L. Ávarez, A. Herández, Chaos Soliton Fract. 39, 746 (2009).ADSCrossRefGoogle Scholar
  19. 19.
    J. Saberi-Nadjafi, A. Ghorbani, Comput. Math. Appl. 58, 2379 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Q.J. Cao, M. Wiercigroch, E.E. Pavlovskaia, J.M.T. Thompson, C. Grebogi, Phys. Rev. E 74, 046218 (2006).MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Q.J. Cao, M. Wiercigroch, E.E. Pavlovskaia, J.M.T. Thompson, C. Grebogi, Phil. Trans. R. Soc. A 366, 635 (2008).MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    Q.J. Cao, M. Wiercigroch, E.E. Pavlovskaia, C. Grebogi, J.M.T. Thompson, Int. J. Non-Linear Mech. 43, 462 (2008).ADSCrossRefGoogle Scholar
  23. 23.
    R.L. Tian, Q.J. Cao, S.P. Yang, Nonlinear Dyn. 59, 19 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R.L. Tian, Q.J. Cao, Z.X. Li, Chin. Phys. Lett. 27, 0747011 (2010).Google Scholar
  25. 25.
    R.L. Tian, X.W. Yang, Q.J. Cao, Q.L. Wu, Chin. Phys. B 21, 020503 (2012).ADSCrossRefGoogle Scholar
  26. 26.
    R.L. Tian, Q.L. Wu, Z.J. Liu, X.W. Yang, Chin. Phys. Lett. 29, 084706 (2012).ADSCrossRefGoogle Scholar
  27. 27.
    H. Hassanabadi, E. Maghsoodi, S. Zarrinkamar, H. Rahimov, Eur. Phys. J. Plus 127, 143 (2012).CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ruilan Tian
    • 1
  • Qiliang Wu
    • 1
  • Xinwei Yang
    • 2
    Email author
  • Chundi Si
    • 1
  1. 1.Department of Mathematics and PhysicsShijiazhuang Tiedao UniversityShijiazhuangChina
  2. 2.School of TrafficShijiazhuang Institute of Railway TechnologyShiJiaZhuangChina

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