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Complete synchronization of delay hyperchaotic Lü system via a single linear input

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Abstract

This work is devoted to investigating the complete synchronization of two identical delay hyperchaotic Lü systems with different initial conditions, and a simple complete synchronization scheme only with a single linear input is proposed. Based on the Lyapunov stability theory, sufficient conditions of synchronization are obtained for both linear feedback and adaptive control approaches. The problem of adaptive synchronization between two nearly identical delay hyperchaotic Lü systems with unknown parameters is also studied. A single input adaptive synchronization controller is proposed, and the adaptive parameter update laws are developed. Numerical simulation results are presented to demonstrate the effectiveness of the proposed chaos synchronization scheme.

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References

  1. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1996)

    Article  MathSciNet  Google Scholar 

  2. Luo, A.C.J.: A theory for synchronization of dynamical systems. Commun. Nonlinear Sci. Numer. Simul. 14, 1901–1951 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahn, C.K.: Output feedback \(\mathcal{H}\infty\) synchronization for delayed chaotic neural networks. Nonlinear Dyn. 59, 319–327 (2010)

    Article  MATH  Google Scholar 

  4. Ma, J., et al.: Complete synchronization, phase synchronization and parameters estimation in a realistic chaotic system. Commun. Nonlinear Sci. Numer. Simul. 16, 3770–3785 (2011)

    Article  MATH  Google Scholar 

  5. Kitio, C.A., Woafo, P.: Dynamics, chaos and synchronization of self-sustained electromechanical systems with clamped-free flexible arm. Nonlinear Dyn. 53, 201–213 (2008)

    Article  MATH  Google Scholar 

  6. Ge, Z.M., Chen, C.C.: Phase synchronization of coupled chaotic multiple time scales systems. Chaos Solitons Fractals 20, 639–647 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chiang, T.Y., et al.: Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity. Nonlinear Anal. 68(9), 2629–2637 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Al-sawalha, M.M., Noorani, M.S.M., Al-dlalah, M.M.: Adaptive anti-synchronization of chaotic systems with fully unknown parameters. Comput. Math. Appl. 59(10), 3234–3244 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Wang, Z.L., Shi, X.R.: Anti-synchronization of Liu system and Lorenz system with known or unknown parameters. Nonlinear Dyn. 57(3), 425–430 (2009)

    Article  MATH  Google Scholar 

  10. Odibat, Z.M.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems. Nonlinear Dyn. 60, 479–487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Feng, C.F., et al.: Generalized projective synchronization in time-delayed chaotic systems. Chaos Solitons Fractals 38, 743–747 (2008)

    Article  MATH  Google Scholar 

  12. Shahverdiev, E.M., Sivaprakasam, S., Shore, K.A.: Lag synchronization in time-delayed systems. Phys. Lett. A 292, 320–324 (2002)

    Article  MATH  Google Scholar 

  13. Wang, L.P., et al.: Lag synchronization of chaotic systems with parameter mismatches. Commun. Nonlinear Sci. Numer. Simul. 16, 987–992 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, Z.L., Shi, X.R.: Chaotic bursting lag synchronization of Hindmarsh–Rose system via a single controller. Appl. Math. Comput. 215, 1091–1097 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, Z., Xu, D.: A secure communication scheme using projective chaos synchronization. Chaos Solitons Fractals 22(2), 477–481 (2004)

    Article  MATH  Google Scholar 

  16. Hu, M., et al.: Parameters identification and adaptive full state hybrid projective synchronization of chaotic (hyperchaotic) systems. Phys. Lett. A 36, 1231–1237 (2007)

    Google Scholar 

  17. Wang, Z., Shi, X.: Adaptive Q-S synchronization of non-identical chaotic systems with unknown parameters. Nonlinear Dyn. 59, 559–567 (2010)

    Article  MATH  Google Scholar 

  18. Hu, M., Xu, Z.: A general scheme for Q-S synchronization of chaotic systems. Nonlinear Anal. 69, 1091–1099 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Banerjee, S., Chowdhury, A.R.: Functional synchronization and its application to secure communications. Int. J. Mod. Phys. B 23, 2285–2295 (2009)

    Article  MATH  Google Scholar 

  20. Ma, J., et al.: Control chaos in the Hindmarsh–Rose neuron by using intermittent feedback with one variable. Chin. Phys. Lett. 25, 3582–3585 (2008)

    Article  Google Scholar 

  21. Shi, X.: Bursting synchronization of Hind–Rose system based on a single controller. Nonlinear Dyn. 59, 95–99 (2010)

    Article  MATH  Google Scholar 

  22. Wang, Z.L.: Chaos synchronization of the energy resource system based on linear control. Nonlinear Anal., Real World Appl. 11(5), 3336–3343 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, Q., Lü, J.H., Chen, S.H.: Coexistence of anti-phase and complete synchronization in the generalized Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 15, 3067–3072 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Chen, J.H., et al.: Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems. Chaos Solitons Fractals 39, 707–716 (2009)

    Article  MATH  Google Scholar 

  25. Pang, S., Liu, Y.: A new hyperchaotic system from the Lü system and its control. J. Comput. Appl. Math. 235, 2775–2789 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ghosh, D., Chowdhury, R., Saha, P.: Multiple delay Rossler system–bifurcation and chaos control. Chaos Solitons Fractals 35, 472–485 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grants No. 11102180 and No. 51178157) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 10KJB120005).

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Authors and Affiliations

  1. School of Mathematical Sciences, Yancheng Teachers University, Yancheng, 224002, China

    Xuerong Shi & Zuolei Wang

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  1. Xuerong Shi
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  2. Zuolei Wang
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Correspondence to Xuerong Shi.

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Shi, X., Wang, Z. Complete synchronization of delay hyperchaotic Lü system via a single linear input. Nonlinear Dyn 69, 2245–2253 (2012). https://doi.org/10.1007/s11071-012-0423-1

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