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Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems

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Abstract

This paper introduces an observer-based approach to achieve projective synchronization in fractional-order chaotic systems using a scalar synchronizing signal. The proposed method, which enables a linear fractional error system to be obtained, exploits the Kalman decomposition and a proper stability criterion in order to stabilize the error dynamics at the origin. The approach combines three desirable features, that is, the theoretical foundation of the method, the adoption of a scalar synchronizing signal, and the exact analytical solution of the fractional error system written in terms of Mittag-Leffler function. Finally, the projective synchronization of the fractional-order hyperchaotic Rössler systems is illustrated in detail.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Peng, J.H., Ding, E.J., Ding, M., Yang, W.: Synchronizing hyperchaos with a scalar transmitted signal. Phys. Rev. Lett. 76, 904–907 (1996)

    Article  Google Scholar 

  3. Duan, C.K., Yang, S.S.: Synchronizing hyperchaos with a scalar signal by parameter controlling. Phys. Lett. A 229, 151–155 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. Grassi, G., Mascolo, S.: Nonlinear observer design to synchronize hyperchaotic systems via a scalar signal. IEEE Trans. Circuits Syst. I 44, 1011–1014 (1997)

    Article  Google Scholar 

  6. Miller, D.A., Grassi, G.: Experimental realization of observer-based hyperchaos synchronization. IEEE Trans. Circuits Syst. I 48, 366–374 (2001)

    Article  Google Scholar 

  7. Grassi, G., Mascolo, S.: Synchronization of hyperchaotic oscillators using a scalar signal. IEE Electron. Lett. 34, 424–425 (1998)

    Article  Google Scholar 

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

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grassi, G., Miller, D.A.: Projective synchronization via a linear observer: application to time-delay, continuous-time and discrete-time systems. Int. J. Bifurc. Chaos 17, 1337–1342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Grassi, G., Miller, D.A.: Arbitrary observer scaling of all chaotic drive system states via a scalar synchronizing signal. Chaos Solitons Fractals 39, 1246–1252 (2009)

    Article  MATH  Google Scholar 

  12. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 34101 (2003)

    Article  Google Scholar 

  13. Li, C.G., Chen, G.R.: Chaos and hyperchaos in the fractional-order Rössler equations. Physica A 341, 55–61 (2004)

    Article  MathSciNet  Google Scholar 

  14. Cafagna, D., Grassi, G.: Hyperchaos in the fractional-order Rössler system with lowest-order. Int. J. Bifurc. Chaos 19, 339–347 (2009)

    Article  MATH  Google Scholar 

  15. Hartley, T., Lorenzo, C., Qammer, H.: Chaos in a fractional order Chua’s system. IEEE Trans. Circuits Syst. I 42, 485–490 (1995)

    Article  Google Scholar 

  16. Cafagna, D., Grassi, G.: Fractional-order Chua’s circuit: time-domain analysis, bifurcation, chaotic behaviour and test for chaos. Int. J. Bifurc. Chaos 18, 615–639 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  18. Cafagna, D., Grassi, G.: Bifurcation and chaos in the fractional-order Chen system via a time-domain approach. Int. J. Bifurc. Chaos 18, 1845–1863 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Deng, W.H., Li, C.P.: Chaos synchronization of the fractional Lu system. Physica A 353, 61–72 (2005)

    Article  Google Scholar 

  20. Li, C.P., Deng, W.H., Xu, D.: Chaos synchronization of the Chua system with a fractional order. Physica A 360, 171–185 (2006)

    Article  MathSciNet  Google Scholar 

  21. Tavazoei, M.S., Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller. Physica A 387, 57–70 (2008)

    Article  MathSciNet  Google Scholar 

  22. Zhu, H., Zhou, S., He, Z.: Chaos synchronization of the fractional-order Chen’s system. Chaos Solitons Fractals 41, 2733–2740 (2009)

    Article  MATH  Google Scholar 

  23. Hu, J., Han, Y., Zhao, L.: Synchronizing chaotic systems using control based on a special matrix structure and extending to fractional chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 15, 115–123 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Peng, G.J., Jiang, Y.L.: Generalized projective synchronization of a class of fractional-order chaotic systems via a scalar transmitted signal. Phys. Lett. A 372, 3963–3970 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wang, J.W., Chen, A.M.: A new scheme to projective synchronization of fractional-order chaotic systems. Chin. Phys. Lett. 27, 110501 (2010)

    Article  Google Scholar 

  26. Shao, S.Q., Gao, X., Liu, X.W.: Projective synchronization in coupled fractional order chaotic Rossler system and its control. Chin. Phys. 16, 2612 (2007)

    Article  Google Scholar 

  27. Niu, Y.J., Wang, X.Y., Nian, F.Z., Wang, M.J.: Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization. Chin. Phys. B 19, 120507 (2010)

    Article  Google Scholar 

  28. Yang, Q., Zeng, C.: Chaos in fractional conjugate Lorenz system and its scaling attractors. Commun. Nonlinear Sci. Numer. Simul. 15, 4041–4051 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Shao, S.: Controlling general projective synchronization of fractional order Rössler systems. Chaos Solitons Fractals 39, 1572–1577 (2009)

    Article  MATH  Google Scholar 

  30. Wu, X., Lu, Y.: Generalized projective synchronization of the fractional-order Chen hyperchaotic system. Nonlinear Dyn. 57, 25–35 (2009)

    Article  MATH  Google Scholar 

  31. Wu, X., Wang, H.: A new chaotic system with fractional order and its projective synchronization. Nonlinear Dyn. 61, 407–417 (2010)

    Article  MATH  Google Scholar 

  32. Chang, C.M., Chen, H.K.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 62, 851–858 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Matignon, D.: Stability results of fractional differential equations with application to control processing. In: IEEE-SMC Proceedings, Computational Engineering in Systems and Application Multi-Conference, Lille, France. IMACS, vol. 2, pp. 963–968 (1996)

    Google Scholar 

  34. Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractal and Fractional Calculus in Continuum Mechanics, pp. 223–276. Springer, Wien (1997)

    Google Scholar 

  35. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  36. Mathai, A.M., Haubold, H.J.: Special Functions for Applied Scientists. Springer, New York (2008)

    Book  MATH  Google Scholar 

  37. Odibat, Z.M.: Analytic study on linear systems of fractional differential equations. Comput. Math. Appl. 59, 1171–1183 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lu, J.G.: Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. Chaos Solitons Fractals 27, 1125–1133 (2005)

    Article  Google Scholar 

  40. Chen, C.T.: Linear System: Theory and Design. Oxford University Press, New York (1999)

    Google Scholar 

  41. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A 71, 155–157 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  42. Feng, Y., Yu, X., Sun, L.: Synchronization of uncertain chaotic systems using a single transmission channel. Chaos Solitons Fractals 35, 755–762 (2008)

    Article  Google Scholar 

  43. Grassi, G., Mascolo, S.: A system theory approach for designing cryptosystems based on hyperchaos. IEEE Trans. Circuits Syst. I 46, 1135–1138 (1999)

    Article  MATH  Google Scholar 

  44. Grassi, G., Mascolo, S.: Synchronization of hyperchaotic oscillators using a scalar signal. IEE Electron. Lett. 34, 424–425 (1998)

    Article  Google Scholar 

  45. Chee, C.Y., Xu, D.: Secure digital communication using controlled projective synchronisation of chaos. Chaos Solitons Fractals 23, 1063–1070 (2005)

    MATH  Google Scholar 

  46. Chee, C.Y., Xu, D.: Chaos-based M-ary digital communication technique using controlled projective synchronization. IEE Proc., Circuits Devices Syst. 153, 357–360 (2006)

    Article  Google Scholar 

  47. Martin, R., Quintana, J.: Modeling of electrochemical double layer capacitors by means of fractional impedance. J. Comput. Nonlinear Dyn. 3, 1303–1309 (2008)

    Google Scholar 

  48. Radwan, A.G., Soliman, A.M., Elwakil, A.S.: Fractional-order sinusoidal oscillators: Four practical circuit design examples. Int. J. Circuit Theory Appl. 36, 473–492 (2008)

    Article  MATH  Google Scholar 

  49. Maundy, B., Elwakil, A., Gift, S.: On a multivibrator that employs a fractional capacitor. J. Analog Integr. Circuits Signal Process. 62, 99–103 (2010)

    Article  Google Scholar 

  50. Elwakil, A.S.: Fractional-order circuits and systems: an emerging interdisciplinary research area. IEEE Circuits Syst. Mag. 4, 40–50 (2010)

    Article  Google Scholar 

  51. Sheu, L.J.: A speech encryption using fractional chaotic systems. Nonlinear Dyn. 65, 103–108 (2011)

    Article  MathSciNet  Google Scholar 

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  1. Dipartimento Ingegneria Innovazione, Università del Salento, Lecce, Italy

    Donato Cafagna & Giuseppe Grassi

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  1. Donato Cafagna
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  2. Giuseppe Grassi
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Correspondence to Donato Cafagna.

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Cafagna, D., Grassi, G. Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic Rössler systems. Nonlinear Dyn 68, 117–128 (2012). https://doi.org/10.1007/s11071-011-0208-y

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