Nonlinear Dynamics

, Volume 53, Issue 3, pp 261–271 | Cite as

Chaos control and synchronization of the Φ6-Van der Pol system driven by external and parametric excitations

  • Li Ruihong
  • Xu Wei
  • Li Shuang
Original Paper


The dynamical behavior of the Φ6-Van der Pol system subjected to both external and parametric excitation is investigated. The effect of parametric excitation amplitude on the routes to chaos is studied by numerical analysis. It is found that the probability of chaos happening increases along with the parametric excitation amplitude increases while the external excitation amplitude fixed. Based on the invariance principle of differential equations, the system is lead to desirable periodic orbit or chaotic state (synchronization) with different control techniques. Numerical simulations are provided to validate the proposed method.


Φ6-Van der Pol system Parametric excitation Bifurcation Largest Lyapunov exponent Chaos control Synchronization Invariance principle of differential equations 


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  1. 1.
    Kapitaniak, T.: Chaos for Engineers: Theory, Applications and Control. Springer, New York (1998) MATHGoogle Scholar
  2. 2.
    Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillators. Wiley, New York (1979) Google Scholar
  3. 3.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin (1983) MATHGoogle Scholar
  4. 4.
    Hayashi, C.: Nonlinear Oscillators in Physics Systems. Princeton University Press, Princeton (1985) Google Scholar
  5. 5.
    Wiggins, S.: Global Bifurcations and Chaos. Springer, Berlin (1998) Google Scholar
  6. 6.
    González, J.A., Guerrero, L.E., Bellorín, A.: Self-excited soliton motion. Phys. Rev. E 54, 1265–1273 (1996) CrossRefGoogle Scholar
  7. 7.
    Gulyayev, V.I., Tolbatov, E.Y.: Forced and self-excited vibrations of pipes containing mobile boiling fluid clots. J. Sound Vib. 257, 425–437 (2002) CrossRefGoogle Scholar
  8. 8.
    Dai, X.J., Dong, J.P.: Self-excited vibration of a rigid rotor rubbing with the motion-limiting stop. Int. J. Mech. Sci. 47, 1542–1560 (2005) CrossRefGoogle Scholar
  9. 9.
    Koper, M.T.M.: Bifurcations of mixed-mode oscillations in a three-variable autonomous Van der Pol–Duffing model with a cross-shaped phase diagram. Physica D 80, 72–94 (1995) CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Venkatesan, A., Lakshmanan, M.: Bifurcation and chaos in the double-well Duffing–Van der Pol oscillator: Numerical and analytical studies. Phys. Rev. E 56, 6321–6330 (1997) CrossRefMathSciNetGoogle Scholar
  11. 11.
    Rajasekar, S., Murali, K.: Resonance behaviour and jump phenomenon in a two couple Duffing–Van der Pol oscillators. Chaos Solitons Fractals 19, 925–934 (2004) CrossRefMATHGoogle Scholar
  12. 12.
    Moukam Kakmeni, F.M., Bowong, S., Tchawoua, C., Kaptouom, E.: Strange attractors and chaos control in a Duffing–Van der Pol oscillator with two external periodic forces. J. Sound Vib. 277, 783–799 (2004) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Tchoukuegno, R., Nana Nbendjo, B.R., Woafo, P.: Resonant oscillations and fractal basin boundaries of a particle in a Φ6 potential. Physica A 304, 362–378 (2002) CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Tchoukuegno, R., Nana Nbendjo, B.R., Woafo, P.: Linear feedback and parametric controls of vibrations and chaotic escape in a Φ6 potential. Int. J. Non-Linear Mech. 38, 531–541 (2003) CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Siewe Siewe, M., Moukam Kakment, F.M., Tchawoua, C.: Resonant oscillation and homoclinic bifurcation in a Φ6-Van der Pol oscillator. Chaos Solitons Fractals 21, 841–853 (2004) CrossRefMathSciNetMATHGoogle Scholar
  16. 16.
    Moukam Kakmeni, F.M., Bowong, S., Tchawoua, C., Kaptouom, E.: Chaos control and synchronization of a Φ6-Van der Pol oscillator. Phys. Lett. A 322, 305–323 (2004) CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Siewe Siewe, M., Moukam Kakment, F.M., Tchawoua, C., Woafo, P.: Bifurcations and chaos in the triple-well Φ6-Van der Pol oscillator driven by external and parametric excitations. Physica A 357, 383–396 (2005) CrossRefGoogle Scholar
  18. 18.
    Wolf, A., Swift, J., Swinney, H., Vastano, J.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985) CrossRefMathSciNetMATHGoogle Scholar
  19. 19.
    Zhou, Y.F., Iu, H.H.C., Tse, C.K., Chen, J.N.: Control chaos in DCIDC converter using optimal resonant parametric perturbation. In: IEEE International Symposium Circuits Systems, vol. 3, pp. 2481–2484 (2005) Google Scholar
  20. 20.
    Chen, G., Dong, X.: On feedback control of chaotic continuous-time systems. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 40, 591–600 (1993) CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Hassan, K.K.: Nonlinear Systems. Prentice Hall, Englewood Cliffs (2002) MATHGoogle Scholar
  22. 22.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Chen, G., Dong, X.: From Chaos to Order. World Science, Singapore (1998) MATHGoogle Scholar
  24. 24.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002) CrossRefMathSciNetMATHGoogle Scholar
  25. 25.
    Ge, Z.M., Lin, T.N.: Chaos, chaos control and synchronization of a gyrostat system. J. Sound Vib. 251, 519–542 (2002) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Bu, S.L., Wang, S.Q., Ye, H.Q.: An algorithm based on variable feedback to synchronize chaotic and hyper-chaotic systems. Physica D 164, 45–52 (2002) CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Huang, D.B.: Simple adaptive-feedback controller for identical chaos synchronization. Phys. Rev. E 71, 037203 (2003) CrossRefGoogle Scholar
  28. 28.
    Feki, M.: Synchronization of chaotic systems with parametric uncertainties using sliding observers. Int. J. Bifurc. Chaos 14, 2467–2475 (2004) CrossRefMathSciNetMATHGoogle Scholar
  29. 29.
    Kim, J.H., Park, C.W., Kim, E., Park, M.: Fuzzy adaptive synchronization of uncertain chaotic systems. Phys. Lett. A 334, 295–305 (2005) CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina

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