Nonlinear Dynamics

, Volume 62, Issue 1–2, pp 453–459 | Cite as

Projective synchronization between two different time-delayed chaotic systems using active control approach

Original Paper

Abstract

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.

Keywords

Projective synchronization Time-delayed chaotic system Active control approach 

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References

  1. 1.
    Pecora, L.M., Carroll, T.C.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990) CrossRefMathSciNetGoogle Scholar
  2. 2.
    Rulkov, N.F., Sushchik, M.M., Tsimring, L.S., Abarbanel, H.D.I.: Generalized synchronization of chaos in directionally coupled chaotic systems. Phys. Rev. E 51, 980–994 (1995) CrossRefGoogle Scholar
  3. 3.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: Phase synchronization of chaotic oscillators. Phys. Rev. Lett. 76, 1804–1807 (1996) CrossRefGoogle Scholar
  4. 4.
    Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999) CrossRefGoogle Scholar
  5. 5.
    Cao, L.Y., Lai, Y.C.: Antiphase synchronism in chaotic systems. Phys. Rev. E 58, 382–386 (1998) CrossRefGoogle Scholar
  6. 6.
    Chee, C.Y., Xu, D.: Secure digital communication using controlled projective synchronisation of chaos. Chaos Solitons Fractals 23, 1063–1070 (2005) MATHGoogle Scholar
  7. 7.
    Xu, D.: Control of projective synchronization in chaotic systems. Phys. Rev. E 63, 27201–27204 (2001) CrossRefGoogle Scholar
  8. 8.
    Jia, Q.: Projective synchronization of a new hyperchaotic Lorenz system. Phys. Lett. A 370, 40–45 (2007) CrossRefGoogle Scholar
  9. 9.
    Wen, G., Xu, D.: Nonlinear observer control for full-state projective synchronization in chaotic continuous-time systems. Chaos Solitons Fractals 26, 71–77 (2005) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feng, C.F., Zhang, Y., Wang, Y.-H.: Projective synchronization in time-delayed chaotic systems. Chin. Phys. Lett. 23, 1418–1421 (2006) CrossRefGoogle Scholar
  11. 11.
    Cao, J., Ho, D.W.C., Yang, Y.: Projective synchronization of a class of delayed chaotic systems via impulsive control. Phys. Lett. A 373, 3128–3133 (2009) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Ghosh, D.: Generalized projective synchronization in time-delayed systems: Nonlinear observer approach. Chaos 19, 013102 (2009) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hu, M., Yang, Y., Xu, Z., Zhang, R., Guo, L.: Projective synchronization in drive-response dynamical networks. Physica A 381, 457–466 (2007) CrossRefGoogle Scholar
  14. 14.
    Feng, C.F., Xu, X.-J., Wang, S.-J., Wang, Y.-H.: Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. Chaos 18, 023117-1-6 (2008) CrossRefGoogle Scholar
  15. 15.
    Li, G.: Generalized projective synchronization between Lorenz system and Chen’s system. Chaos Solitons Fractals 32, 1454–1458 (2007) MATHCrossRefGoogle Scholar
  16. 16.
    Li, G., Zhou, S., Yang, K.: Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys. Lett. A 355, 326–330 (2006) CrossRefGoogle Scholar
  17. 17.
    Li, R., Xu, W., Li, S.: Adaptive generalized projective synchronization in different chaotic systems based on parameter identification. Phys. Lett. A 367, 199–206 (2007) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Traub, R.D., Miles, R., Wong, R.K.S.: Model of the origin of rhythmic population oscillations in the hippocampal slice. Science 243, 1319–1325 (1989) CrossRefGoogle Scholar
  19. 19.
    Foss, J., Longtin, A., Mansour, B., Milton, J.: Multistability and delayed recurrent loops. Phys. Rev. Lett. 76, 708–711 (1996) CrossRefGoogle Scholar
  20. 20.
    Pyragas, K.: Synchronization of coupled time-delay systems: Analytical estimations. Phys. Rev. E 58, 3067–3071 (1998) CrossRefGoogle Scholar
  21. 21.
    Pyragas, K.: Transmission of signals via synchronization of chaotic time-delay systems. Int. J. Bifurc. Chaos 8, 1839–1842 (1998) CrossRefGoogle Scholar
  22. 22.
    Masoller, C.: Spatiotemporal dynamics in the coherence collapsed regime of semiconductor lasers with optical feedback. Chaos 7, 455–462 (1997) MATHCrossRefGoogle Scholar
  23. 23.
    Bai, E.W., Lonngsen, K.E.: Sequential synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 11, 1041–1044 (2000) MATHCrossRefGoogle Scholar
  24. 24.
    Bai, E.W., Lonngsen, K.E.: Synchronization of two Lorenz systems using active control. Chaos Solitons Fractals 8, 51–58 (1997) MATHCrossRefGoogle Scholar
  25. 25.
    Ho, M.C., Hung, Y.C., Chou, C.H.: Phase and anti-phase synchronization of two chaotic systems by using active control. Phys. Lett. A 296, 43–48 (2002) MATHCrossRefGoogle Scholar
  26. 26.
    Agiza, H.N., Yassen, M.T.: Synchronization of Rossler and Chen dynamical systems using active control. Phys. Lett. A 278, 191–197 (2001) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    He, R., Vaiya, P.G.: Analysis and synthesis of synchronous periodic and chaotic systems. Phys. Rev. A 46, 7387–7392 (1992) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Ikeda, K., Kondo, K., Akimoto, O.: Successive higher-harmonic bifurcations in systems with delayed feedback. Phys. Rev. Lett. 49, 1467–1470 (1982) CrossRefGoogle Scholar
  29. 29.
    Voss, H.U.: Dynamic long-term anticipation of chaotic states. Phys. Rev. Lett. 87, 014102-1-4 (2001) Google Scholar
  30. 30.
    Masoller, C.: Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback. Phys. Rev. Lett. 86, 2782–2785 (2001) CrossRefGoogle Scholar
  31. 31.
    Masoller, C., Zanette, D.H.: Anticipated synchronization in coupled chaotic maps with delays. Physica A 300, 359–366 (2001) MATHCrossRefGoogle Scholar
  32. 32.
    Shahverdiev, E.M.: Synchronization in systems with multiple time delays. Phys. Rev. E 70, 067202-1-4 (2004) CrossRefGoogle Scholar
  33. 33.
    Namajūnas, A., Pyragas, K., Tamaševičius, A.: An electronic analog of the Mackey–Glass system. Phys. Lett. A 201, 42–46 (1995) CrossRefGoogle Scholar
  34. 34.
    Kittel, A., Parisi, J., Pyragas, K.: Generalized synchronization of chaos in electronic circuit experiments. Physica D 112, 459–471 (1998) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.College of ScienceWuhan University of Science and EngineeringWuhanChina

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