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Fractional and non-fractional chaotic amphibian attractors with self-excited and hidden properties: numerical dynamics, circuit realization and FPGA-based application

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Abstract

In this paper we announce a novel chaotic system which can have self-excited or hidden attractor depending on the parameters. The system shows same Lyapunov exponents for both self-excited and hidden attractors. Fractional order model of the proposed novel chaotic system is derived and investigated. Bifurcation diagrams of the fractional order chaotic system with respect to change in the fractional orders are investigated and it is proved that the system possess its largest Lyapunov exponent in the fractional order. Using the Adomian decomposition method, the proposed fractional order chaotic systems are implemented in FPGA and power efficiency analysis for various fractional orders is done.

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References

  1. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Phys. Lett. A 375, 2230 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  2. G.A. Leonov, N.V. Kuznetsov, V.I. Vagaitsev, Physica. D 241, 1482 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  3. G.A. Leonov, N.V. Kuznetsov, Int. J. Bifurc. Chaos 23, 1330002 (2013)

    Article  Google Scholar 

  4. S. Brezetskyi, D. Dudkowski, T. Kapitaniak, Eur. Phys. J. Special Topics 224, 1459 (2015)

    Article  ADS  Google Scholar 

  5. T. Kapitaniak, G. Leonov, Eur. Phys. J. Special Topics 224, 1405 (2015)

    Article  ADS  Google Scholar 

  6. E. Tlelo-Cuautle, L. Gerardo de la Fraga, V.-T. Pham, C. Volos, S. Jafari, A. de Jesus Quintas-Valles, Nonlinear Dyn. 1 (2017)

  7. G.A. Leonov, N.V. Kuznetsov, M.A. Kiseleva, E.P. Solovyeva, A.M. Zaretskiy, Nonlinear Dyn. 77, 277 (2014)

    Article  Google Scholar 

  8. G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, Commun. Nonlinear Sci. Numer. Simul. 28, 166 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  9. G.A. Leonov, N.V. Kuznetsov, T.N. Mokaev, Eur. Phys. J. Special Topics 224, 1421 (2015)

    Article  ADS  Google Scholar 

  10. F. Nazarimehr, B. Saedi, S. Jafari, J. Clinton Sprott, Int. J. Bifurc. Chaos 27, 1750037 (2017)

    Article  Google Scholar 

  11. F. Nazarimehr, S. Jafari, S. Mohammad Reza Hashemi Golpayegani, J. Clinton Sprott, Int. J. Bifurc. Chaos 27, 1750023 (2017)

    Article  Google Scholar 

  12. V.-T. Pham, C. Volos, S. Jafari, Z. Wei, X. Wang, Int. J. Bifurc. Chaos 24, 1450073 (2014)

    Article  Google Scholar 

  13. S. Jafari, J.C. Sprott, V.-T. Pham, S. Mohammad Reza Hashemi Golpayegani, A. Homayoun Jafari, Int. J. Bifurc. Chaos 24, 1450134 (2014)

    Article  Google Scholar 

  14. F. Rahma Tahir, S. Jafari, V.-T. Pham, C. Volos, X. Wang, Int. J. Bifurc. Chaos 25, 1550056 (2015)

    Article  Google Scholar 

  15. S. Jafari, V.-T. Pham, T. Kapitaniak, Int. J. Bifurc. Chaos 26, 1650031 (2016)

    Article  Google Scholar 

  16. S.-K. Lao, Y. Shekofteh, S. Jafari, J. Clinton Sprott, Int. J. Bifurc. Chaos 24, 1450010 (2014)

    Article  Google Scholar 

  17. V.-T. Pham, S. Vaidyanathan, C. Volos, S. Jafari, S. Takougang Kingni, Optik 127, 3259 (2016)

    Article  ADS  Google Scholar 

  18. V.-T. Pham, S. Vaidyanathan, C.K. Volos, S. Jafari, N.V. Kuznetsov, T.M. Hoang, Eur. Phys. J. Special Topics 225, 127 (2016)

    Article  ADS  Google Scholar 

  19. A. Akgul, I. Pehlivan, Tech. Gaz.(Croatia) 23, 209 (2016)

    Google Scholar 

  20. M. Molaie, S. Jafari, J. Clinton Sprott, S. Mohammad Reza Hashemi Golpayegani, Int. J. Bifurc. Chaos 23, 1350188 (2013)

    Article  Google Scholar 

  21. S.T. Kingni, S. Jafari, H. Simo, P. Woafo, Eur. Phys. J. Plus 129, 1 (2014)

    Article  Google Scholar 

  22. V.-T. Pham, S. Jafari, C. Volos, A. Giakoumis, S. Vaidyanathan, T. Kapitaniak, IEEE Trans. Circuits Syst. II: Express Briefs 63, 878 (2016)

    Article  Google Scholar 

  23. V.-T. Pham, S. Jafari, C. Volos, Optik 131, 343 (2017)

    Article  ADS  Google Scholar 

  24. V.-T. Pham, S. Jafari, C. Volos, T. Gotthans, X. Wang, D. Vo Hoang, Optik 130, 365 (2017)

    Article  ADS  Google Scholar 

  25. V.-T. Pham, S. Jafari, C. Volos, S. Vaidyanathan, T. Kapitaniak, Optik 127, 9111 (2016)

    Article  ADS  Google Scholar 

  26. V.-T. Pham, S. Jafari, X. Wang, J. Ma, Int. J. Bifurc. Chaos 26, 1650069 (2016)

    Article  Google Scholar 

  27. Y. Maistrenko, T. Kapitaniak, P. Szuminski, Phys. Rev. E 56, 6393 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  28. B. Blażejczyk-Okolewska, T. Kapitaniak, Chaos Solitons Fractals 9, 1439 (1998)

    Article  ADS  Google Scholar 

  29. A. Chudzik, P. Perlikowski, A. Stefanski, T. Kapitaniak, Int. J. Bifurc. Chaos 21, 1907 (2011)

    Article  Google Scholar 

  30. P.R. Sharma, M.D. Shrimali, A. Prasad, N.V. Kuznetsov, G.A. Leonov, Eur. Phys. J. Special Topics 224, 1485 (2015)

    Article  ADS  Google Scholar 

  31. P. Rani Sharma, M. Dev Shrimali, A. Prasad, N.V. Kuznetsov, G.A. Leonov, Int. J. Bifurc. Chaos 25, 1550061 (2015)

    Article  Google Scholar 

  32. Q. Lai, S. Chen, Int. J. Bifurc. Chaos 26, 1650177 (2016)

    Article  Google Scholar 

  33. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, in Fractional calculus: models and numerical methods (World Scientific Publishing Co. Pte. Ltd., Singapore, 2016), Vol. 5

  34. Y. Zhou, J. Wang, L. Zhang, Basic theory of fractional differential equations (World Scientific Publishing Co. Pte. Ltd., Singapore, 2016)

  35. K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type (Springer-Verlag, Berlin, Heidelberg, 2010)

  36. M. Pourmahmood Aghababa, J. Comput. Nonlinear Dyn. 7, 021010 (2012)

    Article  Google Scholar 

  37. E. Amini Boroujeni, H. Reza Momeni, Signal Process. 92, 2365 (2012)

    Article  Google Scholar 

  38. R. Zhang, J. Gong, Syst. Sci. Control Eng. 2, 751 (2014)

    Article  Google Scholar 

  39. M.-F. Danca, W.K.S. Tang, G. Chen, Chaos Solitons Fractals 84, 31 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  40. R.H. Li, W.S. Chen, Chin. Phys. B 22, 040503 (2013)

    Article  ADS  Google Scholar 

  41. I. Petráš, Acta Montanistica Slovaca 11, 273 (2006)

    Google Scholar 

  42. Z. Trzaska, Matlab solutions of chaotic fractional order circuits, in Engineering education and research using MATLAB, edited by A. Assi (Intech, Rijeka, 2011), Chap. 19

  43. V. Rashtchi, M. Nourazar, Circuits Syst. Signal Process. 34, 3101 (2015)

    Article  Google Scholar 

  44. K. Rajagopal, L. Guessas, A. Karthikeyan, A. Srinivasan, G. Adam, Complexity 2017, 1892619 (2017)

    Google Scholar 

  45. E. Tlelo-Cuautle, V.H. Carbajal-Gomez, P.J. Obeso-Rodelo, J.J. Rangel-Magdaleno, J.C. Nuñez-Perez, Nonlinear Dyn. 82, 1879 (2015)

    Article  Google Scholar 

  46. E. Tlelo-Cuautle, A.D. Pano-Azucena, J.J. Rangel-Magdaleno, V.H. Carbajal-Gomez, G. Rodriguez-Gomez, Nonlinear Dyn. 85, 2143 (2016)

    Article  Google Scholar 

  47. E. Dong, Z. Liang, S. Du, Z. Chen, Nonlinear Dyn. 83, 623 (2016)

    Article  Google Scholar 

  48. Q. Wang, S. Yu, C. Li, J. Lü, X. Fang, C. Guyeux, J.M. Bahi, IEEE Trans. Circuits Syst. I: Regul. Pap. 63, 401 (2016)

    Article  MathSciNet  Google Scholar 

  49. X. Ya-Ming, W. Li-Dan, D. Shu-Kai, Acta Phys. Sin. 65, 120503 (2016)

    Google Scholar 

  50. R. Karthikeyan, A. Prasina, R. Babu, S. Raghavendran, Indian J. Sci. Technol. 8, 2 (2015)

    Google Scholar 

  51. M. Ali Jafari, E. Mliki, A. Akgul, V.-T. Pham, S. Takougang Kingni, X. Wang, S. Jafari, Nonlinear Dyn. 88, 1 (2017)

    Article  Google Scholar 

  52. G. Adomian, Comput. Math. Appl. 21, 101 (1991)

    Article  MathSciNet  Google Scholar 

  53. R.C. Hilborn, Chaos and nonlinear dynamics: an introduction for scientists and engineers (Oxford University Press Inc., New York, United States, 2000)

  54. J. Clinton Sprott, J.C. Sprott, Chaos and time-series analysis (Oxford University Press, Oxford, 2003), Vol. 69

  55. N.V. Kuznetsov, G. Leonov, On stability by the first approximation for discrete systems, in International Conference on Proceedings of Physics and Control (IEEE, 2005)

  56. N.V. Kuznetsov, T.A. Alexeeva, G.A. Leonov, arXiv:1410.2016 (2014)

  57. N.V. Kuznetsov, T.N. Mokaev, P.A. Vasilyev, Commun. Nonlinear Sci. Numer. Simul. 19, 1027 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  58. A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  59. D.M.W. Leenaerts, Int. J. Bifurc. Chaos 7, 1431 (1997)

    Article  Google Scholar 

  60. C. Pradhan, S.K. Jena, S.R. Nadar, N. Pradhan, Comput. Math. Methods Med. 2012, 206857 (2012)

    Article  Google Scholar 

  61. K. Rajagopal, L. Guessas, S. Vaidyanathan, A. Karthikeyan, A. Srinivasan, Math. Prob. Eng. https://doi.org/10.1016/j.aej.2017.01.044 (2017)

  62. C. Li, I. Pehlivan, J. Clinton Sprott, A. Akgul, IEICE Electron. Express 12, 20141116 (2015)

    Article  Google Scholar 

  63. A. Akgul, I. Moroz, I. Pehlivan, S. Vaidyanathan, Optik 127, 5491 (2016)

    Article  ADS  Google Scholar 

  64. S. Kaçar, Optik 127, 9551 (2016)

    Article  ADS  Google Scholar 

  65. Z. Wei, I. Moroz, J.C. Sprott, A. Akgul, W. Zhang, Chaos: Interdiscip, J. Nonlinear Sci. 27, 033101 (2017)

    Article  Google Scholar 

  66. G. Kai, W. Zhang, Z.C. Wei, J.F. Wang, A. Akgul, Math. Prob. Eng. 2017, 2490580 (2017)

    Article  Google Scholar 

  67. Ü. Çavuşoğlu, A. Akgül, S. Kaçar, İ. Pehlivan, A. Zengin, Security and Communication Networks (2016)

  68. C. Li, J. Clinton Sprott, and Y. Mei, Nonlinear Dyn. 89, 1 (2017)

    Article  Google Scholar 

  69. K. Rajagopal, A. Akgul, S. Jafari, A. Karthikeyan, I. Koyuncu, Chaos Solitons Fractals 103, 476 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  70. W. Hu, A. Akgul, C. Li, T. Zheng, P. Li, J. Circuits Syst. Comput. 26, 1750158 (2017)

    Article  Google Scholar 

  71. D. Dudkowski, S. Jafari, T. Kapitaniak, N.V. Kuznetsov, G.A. Leonov, A. Prasad, Phys. Rep. 637, 1 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  72. K. Rajagopal, A. Karthikeyan, A. Srinivasan, Nonlinear Dyn. 87, 2281 (2017)

    Article  Google Scholar 

  73. A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, IEEE Trans. Autom. Control 37, 1465 (1992)

    Article  Google Scholar 

  74. H.H. Sun, A. Abdelwahab, B. Onaral, IEEE Trans. Autom. Control 29, 441 (1984)

    Article  Google Scholar 

  75. K. Rajagopal, A. Karthikeyan, P. Duraisamy, Complexity 2017, 8979408 (2017)

    Google Scholar 

  76. R. Caponetto, S. Fazzino, Int. J. Bifurc. Chaos 23, 1350050 (2013)

    Article  Google Scholar 

  77. H.H. Wang, K.H. Sun, S.B. He, Phys. Scr. 90, 015206 (2014)

    Article  Google Scholar 

  78. S. He, K. Sun, H. Wang, Entropy 17, 8299 (2015)

    Article  ADS  Google Scholar 

Download references

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Authors and Affiliations

  1. Centre for Non-Linear Dynamics, Defense University, Bishoftu, Ethiopia

    Karthikeyan Rajagopal & Anitha Karthikeyan

  2. Biomedical Engineering Department, Amirkabir University of Technology, Tehran, Iran

    Fahimeh Nazarimehr & Sajad Jafari

Authors
  1. Karthikeyan Rajagopal
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  2. Fahimeh Nazarimehr
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  3. Sajad Jafari
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  4. Anitha Karthikeyan
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Correspondence to Sajad Jafari.

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Rajagopal, K., Nazarimehr, F., Jafari, S. et al. Fractional and non-fractional chaotic amphibian attractors with self-excited and hidden properties: numerical dynamics, circuit realization and FPGA-based application. Eur. Phys. J. Spec. Top. 226, 3827–3850 (2017). https://doi.org/10.1140/epjst/e2018-00013-1

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