Skip to main content
Log in

Combination synchronization of three different order nonlinear systems using active backstepping design

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, two kinds of combination synchronization between two drive systems and one response system are investigated using active backstepping design. Firstly, increased-order combination synchronization between Lorenz system, Rössler system and hyperchaotic Lü system is considered. Secondly, reduced-order combination synchronization between hyperchaotic Lorenz system, hyperchaotic Chen system and Lü system is considered. According to Lyapunov stability theory and active backstepping design method, the corresponding controllers are both designed. Finally, several numerical examples are provided to illustrate the obtained results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)

    Article  MathSciNet  Google Scholar 

  2. Carroll, T.L., Pecora, L.M.: Synchronizing chaotic circuits. IEEE Trans. Circuits Syst. I 38, 453–456 (1991)

    Article  Google Scholar 

  3. Mainieri, R., Rehacek, J.: Projective synchronization in the three-dimensional chaotic system. Phys. Rev. Lett. 82, 3042–3045 (1999)

    Article  Google Scholar 

  4. Xu, D.L., Chee, C.Y.: Controlling the ultimate state of projective synchronization in chaotic systems of arbitrary dimension. Phys. Rev. E 66, 046218 (2002)

    Article  Google Scholar 

  5. Ma, Z.J., Liu, Z.R., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos 16, 023103 (2006)

    Article  MathSciNet  Google Scholar 

  6. Sun, W.G., Yang, Y.Y., Li, C.P., Liu, Z.R.: Synchronization inside complex dynamical networks with double time-delays and nonlinear inner-coupling functions. Int. J. Mod. Phys. B 25, 1531–1541 (2011)

    Article  Google Scholar 

  7. Xu, Y.H., Zhou, W.N., Fang, J.A., Sun, W.: Adaptive synchronization of uncertain chaotic systems with adaptive scaling function. J. Franklin Inst. 348, 2406–2416 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Rao, P.C., Wu, Z.Y., Liu, M.: Adaptive projective synchronization of dynamical networks with distributed time delays. Nonlinear Dyn. 67, 1729–1736 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guan, J.B.: Function projective synchronization of a class of chaotic systems with uncertain parameters. Math. Probl. Eng. 2012, 431752 (2012)

    Article  Google Scholar 

  10. Yu, W.W., Chen, G.R., Cao, J.D.: Adaptive synchronization of certain and uncertain coupled complex networks. Asian J. Control 13, 418–429 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun, W.G., Zhang, J.Y., Li, C.P.: Synchronization analysis of two coupled complex networks with time delays. Discrete Dyn. Nat. Soc. 2011, 209321 (2011)

    MathSciNet  Google Scholar 

  12. Deng, L.P., Wu, Z.Y.: Impulsive cluster synchronization in community network with nonidentical nodes. Commun. Theor. Phys. 58, 525–530 (2012)

    Article  MATH  Google Scholar 

  13. Ricardo, F., Gualberto, S.P.: Synchronization of chaotic systems with different order. Phys. Rev. E 65, 036226 (2002)

    Article  Google Scholar 

  14. Ho, M.C., Hung, Y.C., Liu, Z.Y., Jiang, I.M.: Reduced order synchronization of chaotic systems with parameters unknown. Phys. Lett. A 348, 251–259 (2006)

    Article  Google Scholar 

  15. Mossa Al-sawalha, M., Noorani, M.S.M.: Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 15, 3022–3034 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mossa Al-sawalha, M., Noorani, M.S.M.: Chaos reduced-order anti-synchronization of chaotic systems with fully unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 17, 1908–1920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Vincent, E., Guo, R.W.: A simple adaptive control for full and reduced-order synchronization of uncertain timevarying chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 14, 3925–3932 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hu, M.F., Xu, Z.Y., Zhang, R., Hu, A.H.: Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order. Phys. Lett. A 365, 315–327 (2007)

    Article  MathSciNet  Google Scholar 

  19. Cai, N., Li, W.Q., Jing, Y.W.: Finite-time generalized synchronization of chaotic systems with different order. Nonlinear Dyn. 64, 385–393 (2011)

    Article  MathSciNet  Google Scholar 

  20. Motallebzadeh, F., Motlagh, M.R.J., Cherati, Z.R.: Synchronization of different-order chaotic systems: adaptive active vs optimal control. Commun. Nonlinear Sci. Numer. Simul. 17, 3643–3657 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mossa Al-sawalha, M., Noorani, M.S.M.: Adaptive increasing-order synchronization and anti-synchronization of chaotic systems with uncertain parameters. Chin. Phys. Lett. 28, 110507 (2011)

    Article  Google Scholar 

  22. Bowong, S.: Stability analysis for the synchronization of chaotic systems with different order: application to secure communications. Phys. Lett. A 326, 102–113 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Luo, R.Z., Wang, Y.L., Deng, S.C.: Combination synchronization of three classic chaotic systems using active backstepping design. Chaos 21, 043114 (2011)

    Article  Google Scholar 

  24. Luo, R.Z., Wang, Y.L.: Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication. Chaos 22, 023109 (2012)

    Article  Google Scholar 

  25. Wang, H., Wang, X., Zhu, X.J., Wang, X.H.: Linear feedback controller design method for time-delay chaotic systems. Nonlinear Dyn. 70, 355–362 (2012)

    Article  MATH  Google Scholar 

  26. Kwon, O.M., Park, J.H., Lee, S.M.: Secure communication based on chaotic synchronization via interval time-varying delay feedback control. Nonlinear Dyn. 63, 239–252 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Li, G.H., Zhou, S.P., Yang, K.: Generalized projective synchronization between two different chaotic systems using active backstepping control. Phys. Lett. A 355, 326–330 (2006)

    Article  Google Scholar 

  28. Njah, A.N.: Tracking control and synchronization of the new hyperchaotic Liu system via backstepping techniques. Nonlinear Dyn. 61, 1–9 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yu, Y.G., Li, H.X.: Adaptive hybrid projective synchronization of uncertain chaotic systems based on backstepping design. Nonlinear Anal., Real World Appl. 12, 388–393 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Nana, B., Woafo, P., Domngang, S.: Chaotic synchronization with experimental application to secure communications. Commun. Nonlinear Sci. Numer. Simul. 14, 2266–2276 (2009)

    Article  Google Scholar 

  31. Smaoui, N., Karouma, A., Zribi, M.: Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16, 3279–3293 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Stefanovska, A., Haken, H., McClintock, P.V.E., Hožič, M., Bajrović, F., Ribarič, S.: Reversible transitions between synchronization states of the cardiorespiratory system. Phys. Rev. Lett. 85, 4831–4834 (2000)

    Article  Google Scholar 

  33. Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  Google Scholar 

  34. Rössler, O.E.: An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  Google Scholar 

  35. Chen, M., Lu, J.A., Lü, J.H., Yu, S.M.: Generating hyperchaotic Lü attractor via state feedback control. Physica A 364, 103–110 (2006)

    Article  Google Scholar 

  36. Jia, Q.: Hyperchaos generated from the Lorenz chaotic system and its control. Phys. Lett. A 366, 217–222 (2007)

    Article  MATH  Google Scholar 

  37. Li, Y.X., Tang, W.K.S., Chen, G.R.: Generating hyperchaos via state feedback control. Int. J. Bifurc. Chaos 15, 3367–3376 (2005)

    Article  Google Scholar 

  38. Lü, J.H., Chen, G.R.: A new chaotic attractor coined. Int. J. Bifurc. Chaos Appl. Sci. Eng. 12, 659–661 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their very helpful suggestions and comments. This research is jointly supported by the Tianyuan Special Funds of NSFC Grant 11226242, the NSFC grant 11072136, the Natural Science Foundation of Jiangxi Province of China (20122BAB211006), the Shanghai Univ. Leading Academic Discipline Project (A.13-0101-12-004), and a grant of “The First-class Discipline of Universities in Shanghai”.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhaoyan Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, Z., Fu, X. Combination synchronization of three different order nonlinear systems using active backstepping design. Nonlinear Dyn 73, 1863–1872 (2013). https://doi.org/10.1007/s11071-013-0909-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-013-0909-5

Keywords

Navigation