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Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography

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Abstract

In this paper, the stability conditions and chaotic behaviors of new different fractional orders of reverse butterfly-shaped dynamical system are analytically and numerically investigated. Designing an appropriate feedback controller, the fractional order chaotic system is synchronized. Applying the synchronized fractional order systems in digital cryptography, a well secured key system is obtained. The numerical simulations are given to validate the correctness of the proposed synchronized fractional order chaotic systems and proposed key system.

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References

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

    Article  Google Scholar 

  2. Alligood, K.T., Sauer, T., Yorke, J.A.: Chaos: an Introduction to Dynamical Systems. Springer, Berlin (1997)

    Book  Google Scholar 

  3. Ling, L., Yan-Chen, S., Chong-Xin, L.: Experimental confirmation of a new reversed butterfly-shaped attractor. Chin. Phys. 16, 1897–1900 (2007)

    Article  Google Scholar 

  4. Mitra, M., Banerjee, S.: Digital cryptography and feedback synchronization of chaotic systems. Int. J. Mod. Phys. B 25, 521–529 (2011)

    Article  MATH  Google Scholar 

  5. Zhang, L.-f., An, X.-l., Zhang, J.-g.: A new chaos synchronization scheme and its application to secure communications. Nonlinear Dyn. doi:10.1007/s11071-013-0824-9

  6. Travazoei, M.S., Haeri, M.: A necessary condition for double scroll attractor existence in fractional-order systems. Phys. Lett. A 367, 102–113 (2007)

    Article  Google Scholar 

  7. Chua, L.O., Komuro, M., Matsumoto, T.: The double-scroll family. IEEE Trans. Circuits Syst. 33, 1072–1118 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cafagna, D., Grassi, G.: New 3D-scroll attractors in hyperchaotic Chua’s circuits forming a ring. Int. J. Bifurc. Chaos Appl. Sci. Eng. 13, 2889–2903 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Lü, J., Chen, G., Yu, X., Leung, H.: Design and analysis of multiscroll chaotic attractors from saturated function series. IEEE Trans. Circuits Syst. I 51, 2476–2490 (2004)

    Article  Google Scholar 

  10. Butzer, P.L., Westphal, U.: An Introduction to Fractional Calculus. World Scientific, Singapore (2000)

    Google Scholar 

  11. Grigorenko, I., Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system. Phys. Rev. Lett. 91, 034101 (2003)

    Article  Google Scholar 

  12. Petras, I.: A note on the fractional-order Chua’s system. Chaos Solitons Fractals 38, 140–147 (2008)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Lu, J.G., Chen, G.: A note on the fractional-order Chen system. Chaos Solitons Fractals 27, 685–688 (2006)

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system. In: Proceedings of ECCTD, Budapest, pp. 1259–1262 (1997)

    Google Scholar 

  18. Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M.: Chaotic dy-namics of the fractionally damped Duffing equation. Chaos Solitons Fractals 31, 1459–1468 (2007)

    Article  Google Scholar 

  19. Ge, Z.M., Ou, C.Y.: Chaos in a fractional order modi-fied Duffing system. Chaos Solitons Fractals 34, 262–291 (2007)

    Article  MATH  Google Scholar 

  20. Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua system. IEEE Trans. Circuits Syst. I 42, 485–490 (1995)

    Article  Google Scholar 

  21. Chen, J.H., Chen, W.C.: Chaotic dynamics of the fractionally damped van der Pol equation. Chaos Solitons Fractals 35, 188–198 (2008)

    Article  Google Scholar 

  22. Ge, Z.M., Zhang, A.R.: Chaos in a modified van der Pol system and in its fractional order systems. Chaos Solitons Fractals 32, 1791–1822 (2007)

    Article  Google Scholar 

  23. Sheu, L.J., Chen, H.K., Chen, J.H., Tam, L.M., Chen, W.C., Lin, K.T., Kang, Y.: Chaos in the Newton–Leipnik system with fractional order. Chaos Solitons Fractals 36, 98–103 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sheu, L.J., Tam, L.M., Lao, S.K., Kang, Y., Lin, K.T., Chen, J.H., Chen, H.K.: Parametric analysis and impulsive synchronization of fractional-order Newton-Leipnik systems. Int. J. Nonlinear Sci. Numer. Simul. 10, 33–44 (2009)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  26. Ching-Ming, C., Hsien-Keng, C.: Chaos and hybrid projective synchronization of commensurate and incommensurate fractional-order Chen–Lee systems. Nonlinear Dyn. 62, 851–858 (2010)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Peng, G.: Synchronization of fractional order chaotic systems. Phys. Lett. A 363, 426–432 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, C., Yan, J.: The synchronization of three fractional differential systems. Chaos Solitons Fractals 32, 751–757 (2007)

    Article  MathSciNet  Google Scholar 

  30. 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 

  31. Wu, X., Li, J., Chen, G.J.: Chaos in the fractional order unified system and its synchronization. J. Franklin Inst. 345, 392–401 (2008)

    Article  MATH  Google Scholar 

  32. Wang, Z., Huang, X., Zhao, Z.: Synchronization of nonidentical chaotic fractional-order systems with different orders of fractional derivatives. Nonlinear Dyn. 69, 999–1007 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  33. Chen, X.R., Liu, C.X.: Chaos synchronization of fractional order unified chaotic system via nonlinear control. Int. J. Mod. Phys. B 25, 407–415 (2011)

    Article  MATH  Google Scholar 

  34. Shi, X., Wang, Z.: Adaptive synchronization of the energy resource systems with mismatched parameters via linear feedback control. Nonlinear Dyn. 69, 993–997 (2012)

    Article  MATH  Google Scholar 

  35. Agrawal, S.K., Srivastava, M., Das, S.: Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems. Nonlinear Dyn. 69, 2277–2288 (2012)

    Article  MathSciNet  Google Scholar 

  36. Zhao, L.-D., Hu, J.-B., Fang, J.-A., Zhang, W.-B.: Studying on the stability of fractional-order nonlinear system. Nonlinear Dyn. 70, 475–479 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Zhang, f.-d.: The synchronization of a fractional-order chaotic system. Adv. Mater. Res. 655–657, 1488–1491 (2013)

    Google Scholar 

  38. Li, C., Tong, Y.: Adaptive control and synchronization of a fractional-order chaotic system. PRAMANA J. Phys. 80, 583–592 (2013)

    Article  Google Scholar 

  39. Huang, L., Liu, A.: Analysis and synchronization for a new fractional-order chaotic system with absolute value term. Nonlinear Dyn. doi:10.1007/s11071-012-0480-5

  40. Liang, H., Wang, Z., Yue, Z., Lu, R.: Generalized synchronization and control for incommensurate fractional unified chaotic system and applications in secure communication. Kybernetika 48, 190–205 (2012)

    MathSciNet  MATH  Google Scholar 

  41. Zhen, W., Xia, H., Yu-Xia, Li., Xiao-Na, S.: A new image encryption algorithm based on the fractional-order hyperchaotic Lorenz system. Chin. Phys. B. 22, 010504 (2013)

    Article  Google Scholar 

  42. Ahmed, E., El-sayed, A.M.A., El-Saka, H.A.A.: Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models. J. Math. Anal. Appl. 325, 542–553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  43. Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Proceedings of the Computational Engineering in Systems and Application Multiconference, France, pp. 963–968 (1996)

    Google Scholar 

  44. Naschie, M.S.El: Statistical geometry of a Cantor diocretum and semiconductors. Comput. Math. Appl. 29, 103–110 (1995)

    Article  MATH  Google Scholar 

  45. Hoggat, V.E.El: Fibonacci and Lucas Numbers. Houghton-Mifflin, Palo Alto (1969)

    Google Scholar 

  46. Gould, H.W.: A history of the Fibonacci Q-matrix and a higher-dimensional problem. Fibonacci Q. 19, 250–257 (1981)

    MATH  Google Scholar 

  47. Silvester, J.R.: Fibonacci properties by matrix methods. Math. Gaz. 63, 188–191 (1979)

    Article  MATH  Google Scholar 

  48. Stakhov, O.P.: A generalization of the Fibonacci Q-matrix. Rep. Natl. Acad. Sci. 9, 46–49 (1999)

    MathSciNet  Google Scholar 

  49. Barnett, S.: Polynomials and Linear Control Systems. Marcel Dekker, New York (1983)

    MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the University Grants Commission-Basic Science Research, Government of India, New Delhi. The authors are very thankful to the editors and anonymous reviewers for their careful reading, constructive comments, and fruitful suggestions to improve this manuscript.

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  1. Department of Mathematics, Gandhigram Rural Institute, Deemed University Gandhigram, Dindigul, Tamil Nadu, India

    P. Muthukumar & P. Balasubramaniam

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  1. P. Muthukumar
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  2. P. Balasubramaniam
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Correspondence to P. Balasubramaniam.

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Muthukumar, P., Balasubramaniam, P. Feedback synchronization of the fractional order reverse butterfly-shaped chaotic system and its application to digital cryptography. Nonlinear Dyn 74, 1169–1181 (2013). https://doi.org/10.1007/s11071-013-1032-3

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