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Adaptive synchronization design for uncertain chaotic systems in the presence of unknown system parameters: a revisit

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Abstract

Recently the synchronization control for chaotic systems with unknown parameters has attracted great attention among the researchers and diverse synchronization schemes have been reported in the literature. In this review article, we carefully revisit several recent articles published from 2010 to the present and find that several reported schemes are problematic. The imperfect synchronization schemes are categorized into five cases according to their defect types. By providing a general theorem for the adaptive synchronization design, we further present modified schemes to correct the defects in these articles. In addition, we have emphasized the significant linear independence condition for ensuring successful identification, as this condition has been neglected in several previous articles. We also summarize three cases when this condition is not valid, and accordingly four approaches are proposed to guarantee the successful parameter estimation for uncertain chaotic systems.

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Notes

  1. We should note that there are some typo errors in the original controller functions designed in [10]. In the controller term u 12(t), the parameter a 2 should be written with a hat \(\bar{a}_{2}\) since a 2 is unknown to users. In the controller term u 13(t), the cross product term x 1 x 3 that was wrongly written in Eq. (18) of [10] should be x 1 x 2. These errors have been corrected in Eq. (8) of this paper.

References

  1. Parlitz, U.: Estimating model parameters from time series by autosynchronization. Phys. Rev. Lett. 76(8), 1232–1235 (1996)

    Article  Google Scholar 

  2. Wang, Y., Guan, Z.-H., Wen, X.: Adaptive synchronization for Chen chaotic system with fully unknown parameters. Chaos Solitons Fractals 19(4), 899–903 (2004)

    Article  MATH  Google Scholar 

  3. Chen, S., Hu, J., Wang, C., Lü, J.: Adaptive synchronization of uncertain Rössler hyperchaotic system based on parameter identification. Phys. Lett. A 321(1), 50–55 (2004)

    Article  MATH  Google Scholar 

  4. Adloo, H., Roopaei, M.: Review article on adaptive synchronization of chaotic systems with unknown parameters. Nonlinear Dyn. 65(1), 141–159 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Zhao, J., Ren, T.: Q–S synchronization between chaotic systems with double scaling functions. Nonlinear Dyn. 62(3), 665–672 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Zheng, S., Dong, G., Bi, Q.: Adaptive modified function projective synchronization of hyperchaotic systems with unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3547–3556 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu, X., Li, S.: Dynamics analysis and hybrid function projective synchronization of a new chaotic system. Nonlinear Dyn. 69(4), 1979–1994 (2012)

    Article  Google Scholar 

  8. Yang, C.C.: Adaptive synchronization of Lü hyperchaotic system with uncertain parameters based on single-input controller. Nonlinear Dyn. 63(3), 447–454 (2011)

    Article  Google Scholar 

  9. Yang, C.-C.: Exponential synchronization of a new Lorenz-like attractor with uncertain parameters via single input. Appl. Math. Comput. 217(14), 6490–6497 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang, X.Y., Sun, P.: Multi-switching synchronization of chaotic system with adaptive controllers and unknown parameters. Nonlinear Dyn. 63(4), 599–609 (2011)

    Article  Google Scholar 

  11. Wu, X.J., Lu, H.T.: Generalized projective lag synchronization between different hyperchaotic systems with uncertain parameters. Nonlinear Dyn. 66(1), 185–200 (2011)

    Article  MathSciNet  Google Scholar 

  12. Wu, X.J., Lu, H.T.: Adaptive generalized function projective lag synchronization of different chaotic systems with fully uncertain parameters. Chaos Solitons Fractals 44(10), 802–810 (2011)

    Article  Google Scholar 

  13. Li, S.Y., Ge, Z.M.: Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control. Nonlinear Dyn. 64(1), 77–87 (2011)

    Article  MathSciNet  Google Scholar 

  14. Ma, H., Xu, B., Lin, W., Feng, J.: Adaptive identification of time delays in nonlinear dynamical models. Phys. Rev. E 82(6), 066210 (2010)

    Article  MathSciNet  Google Scholar 

  15. Sorrentino, F., DeLellis, P.: Estimation of communication-delays through adaptive synchronization of chaos. Chaos Solitons Fractals 45(1), 35–46 (2012)

    Article  Google Scholar 

  16. Bowong, S., Kurths, J.: Parameter estimation based synchronization for an epidemic model with application to tuberculosis in Cameroon. Phys. Lett. A 374(44), 4496–4505 (2010)

    Article  MATH  Google Scholar 

  17. Nguyen, L.H., Hong, K.-S.: Adaptive synchronization of two coupled chaotic Hindmarsh–Rose neurons by controlling the membrane potential of a slave neuron. Appl. Math. Model. 37(4), 2460–2468 (2013)

    Article  MathSciNet  Google Scholar 

  18. Illing, L., Saunders, A.M., Hahs, D.: Multi-parameter identification from scalar time series generated by a Malkus–Lorenz water wheel. Chaos 22(1), 013127 (2012)

    Article  Google Scholar 

  19. Cai, J.: Comment on “Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters”. Commun. Nonlinear Sci. Numer. Simul. 15(2), 469 (2010)

    Article  Google Scholar 

  20. Li, H.-Y., Hu, Y.-A., Mi, Y.-L., Zhu, M.: Comments and modifications on: “Pragmatical adaptive synchronization of different orders chaotic systems with all uncertain parameters via nonlinear control”. Nonlinear Dyn. 69(3), 1489–1491 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sun, Z., Si, G.: Comment on: “Topology identification and adaptive synchronization of uncertain complex networks with adaptive double scaling functions” [Commun. Nonlinear Sci. Numer. Simul. 2011;16:3337–43]. Commun. Nonlinear Sci. Numer. Simul. 17(8), 3461–3463 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, F., Peng, H., Luo, Q., Li, L., Yang, Y.: Parameter identification and projective synchronization between different chaotic systems. Chaos 19(2), 023109 (2009)

    Article  MathSciNet  Google Scholar 

  23. Peng, H., Li, L., Yang, Y., Sun, F.: Conditions of parameter identification from time series. Phys. Rev. E 83(3), 036202 (2011)

    Article  Google Scholar 

  24. Sun, Z., Si, G., Min, F., Zhang, Y.: Adaptive modified function projective synchronization and parameter identification of uncertain hyperchaotic (chaotic) systems with identical or non-identical structures. Nonlinear Dyn. 68(4), 471–486 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, W., Ma, H.F.: Failure of parameter identification based on adaptive synchronization techniques. Phys. Rev. E 75(6), 066212 (2007)

    Article  MathSciNet  Google Scholar 

  26. Sun, F., Peng, H., Xiao, J., Yang, Y.: Identifying topology of synchronous networks by analyzing their transient processes. Nonlinear Dyn. 67(2), 1457–1466 (2012)

    Article  MathSciNet  Google Scholar 

  27. Zhao, J., Zhang, K.: A general scheme for Q–S synchronization of chaotic systems with unknown parameters and scaling functions. Appl. Math. Comput. 216(7), 2050–2057 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, Z.L.: Projective synchronization of hyperchaotic Lü system and Liu system. Nonlinear Dyn. 59(3), 455–462 (2010)

    Article  MATH  Google Scholar 

  29. Xu, Y., Zhou, W., Fang, J.A., Sun, W.: Adaptive synchronization of uncertain chaotic systems with adaptive scaling function. J. Frankl. Inst.-Eng. Appl. Math. 348(9), 2406–2416 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. El-Dessoky, M., Yassen, M., Saleh, E.: Adaptive modified function projective synchronization between two different hyperchaotic dynamical systems. Math. Probl. Eng. (2012). doi:10.1155/2012/810626

    MathSciNet  Google Scholar 

  31. Bai, J., Yu, Y., Wang, S., Song, Y.: Modified projective synchronization of uncertain fractional order hyperchaotic systems. Commun. Nonlinear Sci. Numer. Simul. 17(4), 1921–1928 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, L., Lu, J., Tse, C.K.: Synchronization: an obstacle to identification of network topology. IEEE Trans. Circuits Syst. II, Express Briefs 56(4), 310–314 (2009)

    Article  Google Scholar 

  33. Al-Sawalha, M.M., Noorani, M.: Adaptive reduced-order anti-synchronization of chaotic systems with fully unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 15(10), 3022–3034 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Miao, Q., Tang, Y., Lu, S., Fang, J.: Lag synchronization of a class of chaotic systems with unknown parameters. Nonlinear Dyn. 57(1), 107–112 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, Z.L., Shi, X.R.: Adaptive Q–S synchronization of non-identical chaotic systems with unknown parameters. Nonlinear Dyn. 59(4), 559–567 (2010)

    Article  MATH  Google Scholar 

  37. Li, X.F., Leung, A.C.S., Han, X.P., Liu, X.J., Chu, Y.D.: Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control. Nonlinear Dyn. 63(1), 263–275 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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(4), 1908–1920 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, Z.: Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 14(5), 2366–2372 (2009)

    Article  Google Scholar 

  40. Li, Z., Zhao, X.: The parametric synchronization scheme of chaotic system. Commun. Nonlinear Sci. Numer. Simul. 16(7), 2936–2944 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  41. Li, S.Y., Yang, C.H., Lin, C.T., Ko, L.W., Chiu, T.T.: Adaptive synchronization of chaotic systems with unknown parameters via new backstepping strategy. Nonlinear Dyn. 70(3), 2129–2143 (2012)

    Article  MathSciNet  Google Scholar 

  42. Ioannou, P.A., Sun, J.: Robust Adaptive Control. Prentice-Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

  43. Al-Sawalha, M.M., Noorani, M.: Adaptive anti-synchronization of two identical and different hyperchaotic systems with uncertain parameters. Commun. Nonlinear Sci. Numer. Simul. 15(4), 1036–1047 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Anderson, B.D.O.: Adaptive systems, lack of persistency of excitation and bursting phenomena. Automatica 21(3), 247–258 (1985)

    Article  MATH  Google Scholar 

  45. Wu, X.J.: A new chaotic communication scheme based on adaptive synchronization. Chaos 16(4), 043118 (2006)

    Article  MathSciNet  Google Scholar 

  46. Yu, W., Cao, J., Wong, K.W., Lü, J.: New communication schemes based on adaptive synchronization. Chaos 17(3), 033114 (2007)

    Article  MathSciNet  Google Scholar 

  47. Liu, Y., Tang, W.K.S.: Cryptanalysis of a chaotic communication scheme using adaptive observer. Chaos 18(4), 043110 (2008)

    Article  MathSciNet  Google Scholar 

  48. Liu, Y., Mao, Y., Tang, W.K.S., Kocarev, L.: Cryptanalysis of chaotic communication schemes by dynamical minimization algorithm. Int. J. Bifurc. Chaos 19(7), 2429–2437 (2009)

    Article  MATH  Google Scholar 

  49. Dedieu, H., Ogorzalek, M.: Identifiability and identification of chaotic systems based on adaptive synchronization. IEEE Trans. Circuits Syst. I 44(10), 948–962 (1997)

    Article  MathSciNet  Google Scholar 

  50. Liu, H., Lu, J.-A., Lü, J., Hill, D.J.: Structure identification of uncertain general complex dynamical networks with time delay. Automatica 45(8), 1799–1807 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  51. Ma, H.-f., Lin, W.: Nonlinear adaptive synchronization rule for identification of a large amount of parameters in dynamical models. Phys. Lett. A 374(2), 161–168 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  52. Peng, H., Li, L., Sun, F., Yang, Y., Li, X.: Parameter identification and synchronization of dynamical system by introducing an auxiliary subsystem. Adv. Differ. Equ. 2010, 808403 (2010)

    Article  MathSciNet  Google Scholar 

  53. Uçar, A., Lonngren, K.E., Bai, E.W.: Multi-switching synchronization of chaotic systems with active controllers. Chaos Solitons Fractals 38(1), 254–262 (2008)

    Article  Google Scholar 

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Correspondence to Gangquan Si.

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Sun, Z., Zhu, W., Si, G. et al. Adaptive synchronization design for uncertain chaotic systems in the presence of unknown system parameters: a revisit. Nonlinear Dyn 72, 729–749 (2013). https://doi.org/10.1007/s11071-013-0749-3

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  • DOI: https://doi.org/10.1007/s11071-013-0749-3

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