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A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems

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Abstract

In this paper, we discuss and investigate the impulsive synchronization of fractional-order discrete-time chaotic systems. The proposed method is based on the impulsive synchronization theory used in the integer-order case on the one hand and the mathematical analysis of the fractional-order discrete-time systems on the other hand. Sufficient conditions for the stability of synchronization error system are given, and application example with numerical simulations is illustrated in order to verify that the proposed method is applicable and effective. Furthermore, in order to validate the proposed synchronization approach, we have also provided the experimental implementation results using Arduino Mega boards.

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References

  1. Chua, L.O., Koracev, I., Eckert, K., Itoh, M.: Experimental chaos synchronization in chua’s circuit. Int. J. Bifurc. Chaos 2, 705–708 (1992)

    Article  MATH  Google Scholar 

  2. Vaidyanathan, S.: Hybrid synchronization of Liu and Lü chaotic systems via adaptative control. Int. J. Adv. Inf. Technol. (IJAIT) 1(6), 13 (2011)

    Google Scholar 

  3. Maggio, G.M., Kennedy, M.P.: Experimental manifestations of chaos in the Colpitts oscillator. In: ICECS Proceeding, pp. 194–204 (1997)

  4. Li, C., Chen, G.: Chaos in the fractional order Chen system and its control. Chaos Soliton Fractals 22(3), 549–554 (2004)

    Article  MATH  Google Scholar 

  5. Yu, Y., Li, H., Wang, S., Yu, J.: Dynamic analysis of a fractional-order Lorenz chaotic system. Chaos Soliton Fractals 42(2), 1181–1189 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Luo, C., Wang, X.: Chaos in the fractional-order complex Lorenz system and its synchronization. Nonlinear Dyn. 71(01), 241–257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cui, Z., Yu, P., Wen, Z.: Dynamical behaviors and chaos in a new fractional-order financial system. In: 5th International Workshop (IWCFTA) on Chaos Fractals Theories and Applications, pp. 109–113 (2012)

  8. Liu, Y.: Chaotic synchronization between linearly coupled discrete fractional Hénon maps. Indian J. Phys. 90(03), 313–317 (2016)

    Article  Google Scholar 

  9. Hu, T.: Discrete chaos in fractional Hénon map. Appl. Math. 5(15), 2243–2248 (2014)

    Article  Google Scholar 

  10. Wua, G.C., Baleanu, D.: Chaos synchronization of the discrete fractional logistic map. Signal Process. 102, 96–99 (2014)

    Article  Google Scholar 

  11. Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hamiche, H., Guermah, S., Saddaoui, R., Hannoun, K., Laghrouche, M., Djennoune, S.: Analysis and implementation of a novel robust transmission scheme for private digital communications using Arduino Uno board. Nonlinear Dyn. 81(4), 1921–1932 (2015)

    Article  MathSciNet  Google Scholar 

  13. Hamiche, H., Ghanes, M., Barbot, J.P., Kemih, K., Djennoune, S.: Hybrid dynamical systems for private digital communications. Int. J. Model. Ident. Contr. 20(02), 99–113 (2013)

    Article  Google Scholar 

  14. Mainieri, R., Rehace, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82(15), 3042–3045 (1999)

    Article  Google Scholar 

  15. 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 (1995)

    Article  Google Scholar 

  16. Loria, A., Panteley, E., Zavala, A.: Adaptive observers with persistency of excitation for synchronization of chaotic systems. IEEE Trans. Circ. Syst. I Regul. Pap. 56(11), 2703–2716 (2009)

    Article  MathSciNet  Google Scholar 

  17. Nijmeijer, H., Mareels, I.M.Y.: An observer looks at synchronization. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44, 882–890 (1997)

    Article  MathSciNet  Google Scholar 

  18. Hamiche, H., Guermah, S., Djennoune, S., Kemih, K., Ghanes, M., Barbot, J.P.: Chaotic synchronization and secure communication via sliding-mode and impulsive observers. Int. J. Model. Identif. Control. 20(4), 305–318 (2013)

    Article  Google Scholar 

  19. Feng, Y., Zheng, J., Sun, L.: Chaos synchronization based on sliding mode observer. In: Systems and Control in Aerospace and Astronautics, pp. 1366–1373 (2006)

  20. Yang, T., Chua, L.O.: Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 44(10), 976–988 (1997)

    Article  MathSciNet  Google Scholar 

  21. Megherbi, O., Guermah, S., Hamiche, H., Djennoune, S., Ghanes, M.: A novel transmission scheme based on impulsive synchronization of two Colpitts chaotic systems. In: 3rd International Conference on Systems and Control (ICSC’13), Algiers, Algeria (2013)

  22. Hamiche, H., Kemih, K., Ghanes, M., Zhang, G., Djennoune, S.: Passive and impulsive synchronization of a new four-dimensional chaotic system. Nonlinear Anal. Theory Methods Appl. 74(04), 1146–1154 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, Z., Fang, J., Zhang, W., Wang, X.: Delayed impulsive synchronization of discrete-time complex networks with distributed delays. Int. J. Nonlinear Dyn. Chaos Eng. Syst. 82(04), 2081–2096 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ji, Y., Liu, X., Ding, F.: New criteria for the robust impulsive synchronization of uncertain chaotic delayed nonlinear systems. Int. J. Nonlinear Dyn. Chaos Eng. Syst. 79(01), 1–9 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kadir, A., Aili, M., Sattar, M.: Color image encryption scheme using coupled hyper chaotic system with multiple impulse injections. Int. J. Light Electron Opt. 129, 231–238 (2017)

    Article  Google Scholar 

  26. Jie, F., Miao, Y., Tie-Dong, M.: Modified impulsive synchronization of fractional order hyperchaotic systems. Chin. Phys. B 20(12), 120508 (2011)

    Article  Google Scholar 

  27. Liu, J.G.: A novel study on the impulsive synchronization of fractional-order chaotic systems. Chin. Phys. B 22(6), 060510 (2013)

    Article  Google Scholar 

  28. Andrew, L.Y.T., Li, X.F., Chu, Y.D., Hui, Z.: A novel adaptive-impulsive synchronization of fractional-order chaotic systems. Chin. Phys. B 24(10), 100502 (2015)

    Article  Google Scholar 

  29. Muthukumar, P., Balasubramaniam, P., Ratnavelu, K.: Sliding mode control design for synchronization of fractional order chaotic systems and its application to a new cryptosystem. Int. J. Dyn. Control 5(1), 115–123 (2017)

  30. Wang, Q., Qi, D.L.: Synchronization for fractional order chaotic systems with uncertain parameters. Int. J. Control Autom. Syst. 14(1), 211–216 (2016)

    Article  Google Scholar 

  31. N’Doye, I., Darouach, M., Voos, H.: Observer-based approach for fractional-order chaotic synchronization and communication. In: European Control Conference (ECC), Zurich, Switzerland (2013)

  32. Bhalekar, S., Daftardar-Gejji, V.: Synchronization of different fractional-order chaotic systems using active control. Commun. Nonlinear Sci. Numer. Simulat. 15(11), 3536–3546 (2010)

  33. Tuntas, R.: A new intelligent hardware implementation based on field programmable gate array for chaotic systems. Appl. Soft Comput. 35, 237–246 (2015)

    Article  Google Scholar 

  34. Tuntas, R.: The modelling and analysis of nonlinear systems using a new expert system approach. Iran. J. Sci. Technol. Trans. A Sci. 38, 365–372 (2014)

    MathSciNet  Google Scholar 

  35. Tang, Y., Wang, Z., Fang, J.: Pinning control of fractional-order weighted complex networks. Chaos 19, 013112 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wu, X., Lai, D., Lu, H.: Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes. Nonlinear Dyn. 69, 667–683 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Kassim, S., Hamiche, H., Djennoune, S., Bettayeb, M.: A novel secure image transmission scheme based on synchronization of fractional-order discrete-time hyperchaotic systems. Nolinear Dyn. 88(4), 2473–2489 (2017)

  38. Miller, K.S., Ross, B.: Fractional difference calculus. In: Proceedings of the International Symposium of the Univalent Functions, Fractional Calculus and their Applications, Nihon University Koriyama, pp. 139–152 (1988)

  39. Guermah, S., Bettayeb, M., Djennoune, S.: Controllability and the observability of linear discrete-time fractional-order systems. Int. J. Appl. Math. Comput. Sci. 18, 213–222 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  41. Atici, F.M., Eloe, P.W.: Linear systems of fractional nabla difference equations. Rocky Mt. J. Math. 41(02), 353–370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Mozyrska, D.: Multiparameter fractional difference linear control systems. Discrete Dyn. Nat. Soc. 2014, Article ID 183782, 8 pp (2014). doi:10.1155/2014/183782

  43. Dzielinski, A., Sierociuk, D.: Stability of discrete fractional order state–space systems. J. Vib. Control. 14(09–10), 1543–1556 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Feki, M., Robert, B., Gelle, G., Colas, M.: Secure digital communication using discrete-time chaos synchronization. Chaos Solitons Fractals 18(4), 881–890 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. Mozyrska, D., Pawluszewicz, E.: Local controllability of nonlinear discrete-time fractional order systems. Bull. Pol. Acad. Tech. 61(01), 251–256 (2013)

    MATH  Google Scholar 

  46. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and application of fractional differential equations. In: van Mill, J. (ed.) North Holland Mathematics Studies. Elsevier, Amsterdam (2006)

    Google Scholar 

  47. Hamiche, H., Kassim, S., Djennoune, S., Guermah, S., Lahdir, M., Bettayeb, M.: Secure data transmission scheme based on fractional-order discrete chaotic system. In: International Conference on Control, Engineering and Information Technology (CEIT’2015), Tlemcen, Algeria (2015)

  48. Liao, X., Gao, Z., Huang, H.: Synchronization control of fractional-order discrete-time chaotic systems. In: European Control Conference (ECC), Zürich, Switzerland (2013)

  49. Bettayeb, M., Djennoune, S., Guermah, S., Ghanes, M.: Structural properties of linear discrete-time fractional-order systems. In: Proceedings of the International Federation of Automatic Control (IFAC), Seoul, Korea (July 2008)

  50. Megherbi, O., Kassim,S., Hamiche, H., Djennoune,S.: A New robust hybrid transmission scheme based on the synchronization of discrete-time chaotic systems. In: International Workshop on Cryptography and its Applications (IWCA’16), Oran, Algeria (2016)

  51. Zheng, Y.A., Nyan, Y.B., Liu, Z.R.: Impulsive synchronization of discrete chaotic systems. Chin. Phys. Lett. 20(2), 199–201 (2003)

  52. Zheng, Y.I., Nian, Y.B., Liu, Z.R.: Impulsive control for the stabilization of discrete chaotic system. Chin. Phys. B 19(9), 1251–1253 (2002)

    Google Scholar 

  53. Dzielinski, A., Sierociuk, D.: Adaptive feedback control of fractional order discrete-time state-space systems. In: Proceedings of the 2005 International Conference on Computational Intelligence for Modelling, Control and Automation, and International Conference on Intelligent Agents, Web Technologies and Internet Commerce, (CIMCA-IAWTIC’05), Vienna, Austria (2005)

  54. Liu, B., Marquez, H.J.: Uniform stability of discrete delay systems and synchronization of discrete delay dynamical networks via Razumikhin technique. IEEE Trans. Circuits Syst. I Regul. Pap. 55(9), 2795–2805 (2008)

    Article  MathSciNet  Google Scholar 

  55. Hsien, T.L., Lee, C.H.: Exponential stability of discrete time uncertain systems with time-varying delay. J. Frankl. Inst. 332(4), 479–489 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  56. Hamiche, H., Megherbi, O., Kara, R., Saddaoui, R., Laghrouche, M., Djennoune, S.: A new implementation of an impulsive synchronization of two discrete-time hyperchaotic systems using Arduino-Uno boards. Int. J. Model. Identif. Control (2017). doi:10.1504/IJMIC.2017.10006362

  57. Pano-Azucena, A.D., Rangel-Magdaleno, J.J., Tlelo-Cuautle, E., Quintas-Valles, A.J.: Arduino-based chaotic secure communication system using multi-directional multi-scroll chaotic oscillators. Nonlinear Dyn. 87(4), 2203–2217 (2017)

  58. Arduino: www.arduino.cc

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Authors and Affiliations

  1. Laboratoire de Conception et Conduite des Systèmes de Production (L2CSP), UMMTO, BP 17 RP, 15000, Tizi-Ouzou, Algeria

    Ouerdia Megherbi, Hamid Hamiche & Saïd Djennoune

  2. University of Sharjah, Sharjah, UAE

    Maamar Bettayeb

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  1. Ouerdia Megherbi
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  2. Hamid Hamiche
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  3. Saïd Djennoune
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  4. Maamar Bettayeb
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Correspondence to Ouerdia Megherbi.

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Megherbi, O., Hamiche, H., Djennoune, S. et al. A new contribution for the impulsive synchronization of fractional-order discrete-time chaotic systems. Nonlinear Dyn 90, 1519–1533 (2017). https://doi.org/10.1007/s11071-017-3743-3

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  • DOI: https://doi.org/10.1007/s11071-017-3743-3

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