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Pramana

, 90:14 | Cite as

FPGA implementation of fractional-order discrete memristor chaotic system and its commensurate and incommensurate synchronisations

  • Anitha Karthikeyan
  • Karthikeyan Rajagopal
Article

Abstract

A new fourth-order memristor chaotic oscillator is taken to investigate its fractional-order discrete synchronisation. The fractional-order differential model memristor system is transformed to its discrete model and the dynamic properties of the fractional-order discrete system are investigated. A new method for synchronising commensurate and incommensurate fractional discrete chaotic maps are proposed and validated. Numerical results are established to support the proposed methodologies. This method of synchronisation can be applied for any fractional discrete maps. Finally the fractional-order memristor system is implemented in FPGA to show that the chaotic system is hardware realisable.

Keywords

Memristor discretisation fractional order synchronisation field programmable gate arrays 

PACS No

05.45.–a 

References

  1. 1.
    T Geisel, Nature 298, 322 (1982)ADSCrossRefGoogle Scholar
  2. 2.
    O E Rössler, Phys. Lett. A 71, 155 (1979)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    E N Lorentz, J. Atmos. Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  4. 4.
    G Chen and T Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999)CrossRefGoogle Scholar
  5. 5.
    C X Liu, T Liu, L Liu and K Liu, Chaos Solitons Fractals 22, 1031 (2004)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    V Sundarapandian and I Pehlivan, Math. Comput. Model 55, 1904 (2012)CrossRefGoogle Scholar
  7. 7.
    V Sundarapandian, J. Eng. Sci. Technol. Rev.6, 45 (2013)Google Scholar
  8. 8.
    V T Pham, C Volos, S Jafari, Z Wei and X Wang, Int. J. Bifurc. Chaos 24, 1450073 (2014)CrossRefGoogle Scholar
  9. 9.
    V Sundarapandian and C Volos, Arch. Control Sci. 25(3), 333 (2015)MathSciNetGoogle Scholar
  10. 10.
    L O Chua, IEEE Trans. Circuit Theory 18, 507 (1971)CrossRefGoogle Scholar
  11. 11.
    L Chua and S M Kang, Proc. IEEE 64(2), 209 (1976)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Z Biolek, D Biolek and V Biolková, Radioengineering 18(2), 210 (2009)Google Scholar
  13. 13.
    Robinson E Pino and Kristy A Campbell, Proceeding of the International Symposium on Nanoscale Architecture (2010) pp. 1–4Google Scholar
  14. 14.
    Rak and G Cserey, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 29(4), 632 (2010)Google Scholar
  15. 15.
    A Ishaq Ahamed and M Lakshmanan, Int. J. Bifurc. Chaos 23 (2013)Google Scholar
  16. 16.
    Shibing Wang, X Wang, Y Zhou and B Han, Entropy 18, 58 (2016),  https://doi.org/10.3390/e18020058ADSCrossRefGoogle Scholar
  17. 17.
    B A Idowu, U E Vincent and A N Njah, Chaos Solitons Fractals 39, 2322 (2009)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    S Vaidyanathan and K Rajagopal, Int. J. Syst. Signal Control Eng. Appl. 4, 55 (2011)Google Scholar
  19. 19.
    V Sundarapandian and R Karthikeyan, Eur. J. Sci. Res. 64, 94 (2011)Google Scholar
  20. 20.
    V Sundarapandian and R Karthikeyan, J. Eng. Appl. Sci. 7, 45 (2012)Google Scholar
  21. 21.
    Sajjad Shoja Majidabad and Heydar Toosian Shandiz, J. Control Syst. Eng.1(1), 1 (2013)Google Scholar
  22. 22.
    A N Njah, Nonlinear Dyn.61(1–2), 1 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    O S Onma, O I Olusola and A N Njah, Nonlinear Dyn. Article ID 861727, 15 pages, http://dx.doi.org/10.1155/2014/861727 (2014)
  24. 24.
    B Wang, Y Li and D L Zhu, Int. J. Comput. Appl. 8(8), 425 (2015)Google Scholar
  25. 25.
    Chun Yin, Sara Dadras, Shou-ming Zhong and Yang Quan Chen, Appl. Math. Model.37(4), 2469 (2013)Google Scholar
  26. 26.
    Haorui Liu and Juan Yang, Entropy17, 4202 (2015)ADSCrossRefGoogle Scholar
  27. 27.
    S Wang, Y Yu and M Diao, Physica A 389, 4981 (2010)Google Scholar
  28. 28.
    Z Wanga, X Huang and H Shend, Neurocomputing 83, 83 (2012)Google Scholar
  29. 29.
    Shaojuan Ma, Jie Zheng and Yuqin Li, J. Inf. Comput. Sci.11(10), 3469 (2014)CrossRefGoogle Scholar
  30. 30.
    R H Li and W S Chen, Chin. Phys. B22, 040503 (2013)ADSCrossRefGoogle Scholar
  31. 31.
    I Petras, Chaos Solitons Fractals38, 140 (2008)ADSCrossRefGoogle Scholar
  32. 32.
    A M A El-Sayed, Z F El-Raheem and S M Salman, Adv. Differ. Equ. Article ID: 2014-66 (2014)Google Scholar
  33. 33.
    A M A El-Sayed and S M Salman, J. Frac. Calc. Appl. 4(2), 251 (2013)Google Scholar
  34. 34.
    D Baleanu, K Diethelm, E Scalas and J J Trujillo, Fractional calculus: Models and numerical methods (World Scientific, Singapore, 2014)Google Scholar
  35. 35.
    Y Zhou, Basic theory of fractional differential equations (World Scientific, Singapore, 2014)Google Scholar
  36. 36.
    K Diethelm, The analysis of fractional differential equations (Springer, Berlin, 2010)Google Scholar
  37. 37.
    W Trzaska Zdzislaw, Matlab solutions of chaotic fractional-order circuits (Intech Open, 2014)Google Scholar
  38. 38.
    B Bao, P Jiang, H Wu and F Hu, Nonlinear Dyn.79, 2333 (2015)CrossRefGoogle Scholar
  39. 39.
    K Rajagopal, S Vaidhyanathan, A Karthikeyan and P Duraisamy, Electr. Eng.99(2), 721 (2017)CrossRefGoogle Scholar
  40. 40.
    H K Khalil, Nonlinear systems (Prentice Hall, New York, 2002)Google Scholar
  41. 41.
    K Rajagopal, L Guessas, A Karthikeyan, A Srinivasan and G Adam, Complexity https://www.hindawi.com/journals/complexity/2017/1892618/ (2017)
  42. 42.
    K Rajagopal, A Karthikeyan and A K Srinivasan, Nonlinear Dyn.87(4), 2281 (2017)CrossRefGoogle Scholar
  43. 43.
    K Rajagopal, A Karthikeyan and A Srinivasan, Chaos Solitons Fractals103, 347 (2017)Google Scholar
  44. 44.
    K Rajagopal, S Jafari and G Laarem, PramanaJ. Phys. (accepted)Google Scholar
  45. 45.
    C Wang, R Chu and J Ma, Complexity21, 370 (2015)ADSMathSciNetCrossRefGoogle Scholar
  46. 46.
    Y Jin, Y Q Chen and D Xue, IET Control Theory Appl. 5, 164 (2011)CrossRefGoogle Scholar
  47. 47.
    H Zhang, X Y Wang and X H Lin, Asian J. Control19, 106 (2017)MathSciNetCrossRefGoogle Scholar
  48. 48.
    I Petráš, Fractional-order nonlinear systems: Modeling, analysis and simulation (Springer, 2011)CrossRefzbMATHGoogle Scholar
  49. 49.
    U N Katugampola, Bull. Math. Anal. Appl.6, 1 (2014)MathSciNetGoogle Scholar
  50. 50.
    M A Herzallah, J. Fract. Calc. Appl.5, 1 (2014)MathSciNetGoogle Scholar
  51. 51.
    Y Li, Y Chen and I Podlubny, Comput. Math. Appl.59, 1810 (2010)MathSciNetCrossRefGoogle Scholar
  52. 52.
    J A Gallegos and M A Duarte-Mermoud, Appl. Math. Comput.287, 161 (2016)MathSciNetGoogle Scholar
  53. 53.
    M S Tavazoei and M Haeri, Phys. Lett. A372, 798 (2008)ADSCrossRefGoogle Scholar
  54. 54.
    K Konishi, H Kokame and N Hara, Nonlinear Dyn.63, 513 (2011)CrossRefGoogle Scholar
  55. 55.
    D Chen, P Shi and X Ma, Int. J. Innov. Comput. Inf.8, 7237 (2012)Google Scholar
  56. 56.
    Jun Ma, Fuqiang Wu, Guodong Ren and Jun Tang, Phys. Lett. A298, 65 (2017)Google Scholar
  57. 57.
    Y Xu, H Ying, Y Jia, J Ma and T Hayat, Sci. Rep.7, 43452 (2017)ADSCrossRefGoogle Scholar
  58. 58.
    Fuqiang Wu, Chunni Wang, Wuyin Jin and Jun Ma, Physica A469, 81 (2017)ADSMathSciNetCrossRefGoogle Scholar
  59. 59.
    X Wang and Y He, Phys. Lett. A372(4), 435 (2008)ADSCrossRefGoogle Scholar
  60. 60.
    X Wang, X Zhang and C Ma, Nonlinear Dyn.69(1–2), 511 (2012)CrossRefGoogle Scholar
  61. 61.
    X Wang, Y He and M Wang, Nonlinear Anal. Theory Methods Appl.71(12), 6126 (2009)CrossRefGoogle Scholar
  62. 62.
    C Luo and X Wang, Nonlinear Dyn.71(1–2), 241 (2013)CrossRefGoogle Scholar
  63. 63.
    X Wang and J M Song, Commun. Nonlinear Sci. Numer. Simul.14(8), 3351 (2009)ADSCrossRefGoogle Scholar
  64. 64.
    X Wang and M Wang, Chaos 17, 033106 (2007)ADSCrossRefGoogle Scholar
  65. 65.
    Y Zhanga and X Y Wang, Inf. Sci.273, 329 (2014)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2017

Authors and Affiliations

  1. 1.Department of Electrical and Communication EngineeringUniversity of TechnologyLaePapua New Guinea
  2. 2.Center for Nonlinear DynamicsDefence UniversityBishoftuEthiopia

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