Nonlinear Dynamics

, Volume 87, Issue 1, pp 503–510 | Cite as

Control problems of Chen–Lee system by adaptive control method

Original Paper


This paper investigates the control problems of Chen–Lee system. Based on control theory of nonlinear systems, the adaptive controllers were proposed to achieve stabilization, synchronization, coexistence of synchronization and anti-synchronization, and projective synchronization of such system. It should be pointed out that the obtained controllers are not only simpler than the existing ones, but also are physical. It is also noted that the obtained methods can be extended to a general chaotic system. Numerical simulations are used to verify the validity and effectiveness of the obtained results.


Chen–Lee system Stabilization Synchronization Coexistence Anti-synchronization Projective synchronization 



This work was supported by National Natural Science Foundation of China [61304133, 61305130, 61374074, 61473173].


  1. 1.
    Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)CrossRefGoogle Scholar
  2. 2.
    Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)CrossRefGoogle Scholar
  3. 3.
    Grandmont, J.: On endogenous competitive business cycles. Econometrica 53, 995–1045 (1985)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ott, E., Gerbogi, C., Yorke, J.A.: Controlling Chaos. Phys. Rev. Lett. 64(11), 1196–1199 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ren, H.P., Baptista, M.S., Grebogi, C.: Wireless communication with chaos. Phys. Rev. Lett. 110, 184101 (2013)CrossRefGoogle Scholar
  6. 6.
    Bhatnagar, G., Wu, Q.M.J.: A novel chaos based secure transmission of biometric data. Neurocomputing 147, 444–455 (2015)CrossRefGoogle Scholar
  7. 7.
    Pecora, L., Carroll, T.: Synchronization in Chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Carroll, T.L., Pecora, L.M.: Synchronization in chaotic circuits. IEEE Trans. Circ. Syst. 38, 453–456 (1991)CrossRefGoogle Scholar
  9. 9.
    Idowu, B.A., Vincent, U.E., Njah, A.N.: Anti-synchronization of chaos in nonlinear gyros via active control. J. Math. Control Sci. Appl. 1, 191–200 (2007)Google Scholar
  10. 10.
    Chen, M., Wang, F., Wang, C.: Synchronizing strict-feeedback and general strict-feedback chaotic systems via a single controller. Chaos Solitons Fract. 20(2), 235–243 (2004)CrossRefMATHGoogle Scholar
  11. 11.
    Auerbach, D., Grebogi, C., Ott, E., Yorke, J.A.: Controlling chaos in high dimensional systems. Phys. Rev. Lett. 69(24), 3479–3482 (1992)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Sieber, J., Chenko, E.O., Wolfrum, M.: Controlling unstable chaos: stabilizing chimera states by feedback. Phys. Rev. Lett. 112, 054102 (2014)CrossRefGoogle Scholar
  13. 13.
    Chen, M., Han, Z.: Controlling and synchronizing chaotic Genesio system via nonlinear feedback control. Chao. Solitons Fract. 17(4), 709–716 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fujisaka, H., Yamada, T.: Stability theory of synchronized motion in coupled-oscillator system. Prog. Theor. Phys. 69, 32–47 (1983)Google Scholar
  15. 15.
    Chen, M., Wang, F., Wang, C.: Synchronizing strict-feedback and general strict feedback chaotic systems via a single controller. Chaos Solitons Fract. 20, 235–243 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Guo, R.W.: Simultaneous synchronization and anti-synchronization of two identical new 4D chaotic systems. Chin. Phys. Lett. 28, 040205–040209 (2001)CrossRefGoogle Scholar
  17. 17.
    Chen, H., Lee, C.: Anti-control of chaos in rigid body motion. Chaos Solitons Fract. 21, 957–965 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Hammami, S., Benrejeba, M., Fekib, M., Borne, P.: Feedback control design for Rössler and Chen chaotic systems anti-synchronization. Phys. Lett. A 374(28), 2835–2840 (2010)CrossRefMATHGoogle Scholar
  19. 19.
    Ren, L., Guo, R.W.: Synchronization and antisynchronization for a class of chaotic systems by a simple adaptive controller. Math. Probl. Eng. 2015, 7 (2015)Google Scholar
  20. 20.
    Barbashin, E.A.: Introduction to the Theory of Stability. Wolters-Noordhoff Publishing, Groningen (1970)MATHGoogle Scholar
  21. 21.
    Wu, L., Zhu, S.Q.: Coexistence and switching of anticipating synchronization and lag synchronization in an optical system. Phys. Lett. A 315, 101–108 (2003)CrossRefGoogle Scholar
  22. 22.
    Yau, H.T.: synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs. Nonlinear Anal.: Real World Appl. 9, 2253–2261 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhang, Q., Lü, J.H., Chen, S.H.: Coexistence of anti-phase and complete synchronization in the generalized Lorenz system. Commu. Nonlinear Sci. Numer. Simul. 15(10), 3067–3072 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Ren, L., Guo, R.W.: Synchronization and anti-synchronization for a class of chaotic systems by a simple adaptive controller. Math. Probl. Eng. 434651, 1–7 (2015)MathSciNetGoogle Scholar
  25. 25.
    Mainieri, R., Rehacke, J.: Projective synchronization in three-dimensional chaotic oscillators. Phys. Rev. Lett. 82, 3042–3045 (1999)CrossRefGoogle Scholar
  26. 26.
    Xu, D.L.: Control of projective synchronization in chaotic system. Phys. Rev. Lett. 63, 027201–027204 (2001)Google Scholar
  27. 27.
    Chang, C., Chen, H.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee system. Nonlinear Dyn. 62, 851–858 (2010)CrossRefMATHGoogle Scholar
  28. 28.
    Chen, J.H.: controlling chaos and chaotification in the Chen–Lee system by multiple time delays. Chaos Solitons Fract. 36, 843–852 (2008)CrossRefGoogle Scholar
  29. 29.
    Tam, L.M., Si Tou, W.M.: Parametric study of the fractional order Chen–Lee system. Chaos Solitons Fract. 37, 817–826 (2008)CrossRefGoogle Scholar
  30. 30.
    Chen, J., chen, H., Lin, Y.: Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems. Chaos Solitons Fract. 39, 707–716 (2009)CrossRefMATHGoogle Scholar
  31. 31.
    Sheu, L.J., Tam, L.M., Chen, H.K., Lao, S.K.: alternative implementation of the chaotic Chen–Lee system. Chaos Solitons Fract. 41, 1923–1929 (2009)CrossRefGoogle Scholar
  32. 32.
    Guo, R.: A simple adaptive controller for chaos and hyperchaos synchronization. Phys. Lett. A 372(34), 5593–5597 (2008)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of ScienceQilu University of TechnologyJinanChina

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