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Coexistence of attractors in a simple chaotic oscillator with fractional-order-memristor component: analysis, FPGA implementation, chaos control and synchronization

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Abstract

In this paper, the simplest chaotic oscillator with fractional-order-memristor component (SCOF) is proposed. Dynamical characteristics of the proposed chaotic oscillator are investigated both analytically and numerically. The results indicate that the proposed chaotic oscillator possesses novel dynamical characteristics: double-scroll chaotic attractor, four-scroll chaotic attractor and coexisting attractors. The proposed SCOF system is implemented in field programmable gate arrays (FPGA) using the Adomian decomposition method to verify the numerical simulation results. Finally, control and synchronization of the simple chaotic oscillator with fractional-order-memristor component is studied using the linear feedback control and the unidirectional linear error feedback coupling scheme, respectively.

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References

  1. Y. Feng, J. Pu, Z. Wei, Eur. Phys. J. Special Topics 224, 1593 (2015)

    Article  ADS  Google Scholar 

  2. Z. Wei, W. Zhang, Z. Wang, M. Yao, Int. J. Bifurc. Chaos 25, 1550028 (2015)

    Article  Google Scholar 

  3. N. Fataf, S.K. Palit, S. Mukherjee, M. Said, D.H. Son, S. Banerjee, Eur. Phys. J. Plus 132, 492 (2017)

    Article  Google Scholar 

  4. J.C. Sprott, Phys. Rev. E 50, R647 (1994)

    Article  ADS  Google Scholar 

  5. A. Bayani, K. Rajagopal, A.J.M. Khalaf, S. Jafari, G. Leutcho, J. Kengne, Phys. Lett. A 383, 1450 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  6. Ü. Çavusoǧlu, S. Panahi, A. Akgül, S. Jafari, S. Kaçar, Analog Integr. Circ. Sig. Process. 98, 85 (2019)

    Article  Google Scholar 

  7. X. Wang, S.T. Kingni, C. Volos, V.T. Pham, D. Vo Hoang, S. Jafari, Int. J. Electron. 106, 109 (2019)

    Article  Google Scholar 

  8. G. Chen, T. Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999)

    Article  Google Scholar 

  9. O.E. Rössler, Phys. Lett. A 57, 397 (1976)

    Article  ADS  Google Scholar 

  10. S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 79 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  11. S. Jafari, J.C. Sprott, S.M.R. Hashemi Golpayegani, Phys. Lett. A 377, 699 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  12. M. Molaie, S. Jafari, J.C. Sprott, S.M.R. Hashemi Golpayegani, Int. J. Bifurc. Chaos 23, 1350188 (2013)

    Article  Google Scholar 

  13. X. Wang, G. Chen, Commun. Nonlinear Sci. Numer. Simul. 17, 1264 (2012)

    Article  ADS  MathSciNet  Google Scholar 

  14. Z. Wei, Phys. Lett. A 376, 102 (2011)

    Article  ADS  MathSciNet  Google Scholar 

  15. K. Barati, S. Jafari, J.C. Sprott, V.-T. Pham, Int. J. Bifurc. Chaos 26, 1630034 (2016)

    Article  Google Scholar 

  16. T. Gotthans, J.C. Sprott, J. Petrzela, Int. J. Bifurc. Chaos 26, 1650137 (2016)

    Article  Google Scholar 

  17. T. Gotthans, J. Petržela, Nonlinear Dyn. 81, 1143 (2015)

    Article  Google Scholar 

  18. R.C. Hilborn, Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (Oxford University Press, Oxford, 2000)

  19. M. Kiseleva, N. Kuznetsov, G. Leonov, Hidden and self-excited attractors in electromechanical systems with and without equilibria, https://arXiv:1601.06909 (2016)

  20. N. Kuznetsov, O. Kuznetsova, G. Leonov, V. Vagaitsev, Analytical-numerical localization of hidden attractor in electrical Chua’s circuit, in Informatics in Control, Automation and Robotics (Springer, 2013), pp. 149–158

  21. N. Kuznetsov, G. Leonov, T. Mokaev, S. Seledzhi, T. Simos, C. Tsitouras, Hidden attractor in the Rabinovich system, Chua circuits and PLL, in AIP Conference Proceedings (2016), p. 210008

  22. S. Jafari, F. Nazarimehr, J.C. Sprott, S.M.R. Hashemi Golpayegani, Int. J. Bifurc. Chaos 25, 1550182 (2015)

    Article  Google Scholar 

  23. S. Jafari, J.C. Sprott, F. Nazarimehr, Eur. Phys. J. Special Topics 224, 1469 (2015)

    Article  ADS  Google Scholar 

  24. F. Nazarimehr, S. Jafari, S.M.R.H. Golpayegani, J. Sprott, Int. J. Bifurc. Chaos 27, 1750023 (2017)

    Article  Google Scholar 

  25. B. Bao, A. Hu, H. Bao, Q. Xu, M. Chen, H. Wu, Complexity 2018, 3872573 (2018)

    Google Scholar 

  26. B. Bao, P. Wu, H. Bao, H. Wu, X. Zhang, M. Chen, Chaos Solitons Fractals 109, 146 (2018)

    Article  ADS  Google Scholar 

  27. B. Bao, P. Wu, H. Bao, Q. Xu, M. Chen, Chaos Solitons Fractals 106, 161 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  28. D.B. Strukov, G.S. Snider, D.R. Stewart, R.S. Williams, Nature 453, 80 (2008)

    Article  ADS  Google Scholar 

  29. L. Chua, I.E.E.E. Trans, Circuit Theory 18, 507 (1971)

    Article  Google Scholar 

  30. L. Wang, Z. Zeng, M.-F. Ge, J. Hu, Neural Networks 105, 65 (2018)

    Article  Google Scholar 

  31. Q. Xu, Q. Zhang, B. Bao, Y. Hu, IEEE Access 5, 21039 (2017)

    Article  Google Scholar 

  32. Q. Xu, Q. Zhang, H. Qian, H. Wu, B. Bao, Int. J. Circuit Theory Appl. 46, 1917 (2018)

    Article  Google Scholar 

  33. D. Biolek, Z. Biolek, V. Biolková, AEU Int. J. Electron. Commun. 97, 263 (2018)

    Article  Google Scholar 

  34. D. Biolek, Z. Biolek, V. Biolkova, Radioengineering 20, 228 (2011)

    Google Scholar 

  35. Z. Biolek, D. Biolek, V. Biolkova, Radioengineering 18, 210 (2009)

    Google Scholar 

  36. Q. Lai, A. Akgul, X.-W. Zhao, H. Pei, Int. J. Bifurc. Chaos 27, 1750142 (2017)

    Article  Google Scholar 

  37. Z. Wang, A. Akgul, V.-T. Pham, S. Jafari, Nonlinear Dyn. 89, 1877 (2017)

    Article  Google Scholar 

  38. Y. Tang, H.R. Abdolmohammadi, A.J.M. Khalaf, Y. Tian, T. Kapitaniak, Pramana 91, 11 (2018)

    Article  ADS  Google Scholar 

  39. Z. Wang, H.R. Abdolmohammadi, F.E. Alsaadi, T. Hayat, V.-T. Pham, Chaos Solitons Fractals 110, 252 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  40. Z. Wei, V.-T. Pham, A.J.M. Khalaf, J. Kengne, S. Jafari, Int. J. Bifurc. Chaos 28, 1850085 (2018)

    Article  Google Scholar 

  41. A.J.M.K. Yan-Xia Tang, K. Rajagopal, V.-T. Pham, S. Jafari, Y. Tian, Chin. Phys. B 27, 40502 (2018)

    Article  Google Scholar 

  42. C. Li, J.C. Sprott, Phys. Lett. A 382, 581 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  43. C. Li, J.C. Sprott, W. Hu, Y. Xu, Int. J. Bifurc. Chaos 27, 1750160 (2017)

    Article  Google Scholar 

  44. V. Lakshmikantham, A. Vatsala, Nonlinear Anal.: Theory Methods Appl. 69, 2677 (2008)

    Article  Google Scholar 

  45. C. Li, K. Su, L. Wu, J. Comput. Nonlinear Dyn. 8, 031005 (2013)

    Article  Google Scholar 

  46. C. Li, J. Xiong, W. Li, Y. Tong, Y. Zeng, Indian J. Phys. 87, 673 (2013)

    Article  ADS  Google Scholar 

  47. C. Li, J. Zhang, Int. J. Syst. Sci. 47, 2440 (2016)

    Article  Google Scholar 

  48. Q. Xu, M. Shi, Z. Wang, Chaos: Interdisciplinary, J. Nonlinear Sci. 26, 084301 (2016)

    Google Scholar 

  49. M. Borah, B.K. Roy, ISA Trans. 82, 2 (2018)

    Article  Google Scholar 

  50. A. Radwan, K. Moaddy, I. Hashim, Amplitude modulation and synchronization of fractional-order memristor-based Chua’s circuit, in Abstract and Applied Analysis (Hindawi, 2013), Vol. 2013

    Article  MathSciNet  MATH  Google Scholar 

  51. Y. Yu, Z. Wang, Int. J. Bifurc. Chaos 28, 1850091 (2018)

    Article  Google Scholar 

  52. K. Rajagopal, S. Jafari, A. Karthikeyan, A. Srinivasan, B. Ayele, Circuits Syst, Signal Process. 39, 3702 (2018)

    Google Scholar 

  53. K. Rajagopal, A. Akgul, S. Jafari, B. Aricioglu, Nonlinear Dyn. 91, 957 (2018)

    Google Scholar 

  54. B. Muthuswamy, S. Banerjee, A Route to Chaos Using FPGAs (Springer, 2015)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  57. E. Tlelo-Cuautle, J. Rangel-Magdaleno, A. Pano-Azucena, P. Obeso-Rodelo, J.C. Nuñez-Perez, Commun. Nonlinear Sci. Numer. Simul. 27, 66 (2015)

    Article  ADS  MathSciNet  Google Scholar 

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

    Google Scholar 

  59. D. Valli, B. Muthuswamy, S. Banerjee, M. Ariffin, A. Wahab, K. Ganesan, C.K. Subramaniam, J. Kurths, Eur. Phys. J. Special Topics 223, 1465 (2014)

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  61. M.F. Tolba, L.A. Said, A.H. Madian, A.G. Radwan, FPGA implementation of fractional-order integrator and differentiator based on Grünwald Letnikov’s definition, in 2017 29th International Conference on Microelectronics (ICM) (2017), pp. 1–4

  62. M. Fouda, A. Radwan, J. Fractional Calculus Appl. 4, 1 (2013)

    Google Scholar 

  63. H.M. Baskonus, H. Bulut, Open Math. 13, 547 (2015)

    Article  MathSciNet  Google Scholar 

  64. C. Sánchez-López, V.H. Carbajal-Gómez, M. Carrasco-Aguilar, I. Carro-Perez, Complexity 2018, 2806976 (2018)

    Article  Google Scholar 

  65. L. Teng, H.H. Iu, X. Wang, X. Wang, Nonlinear Dyn. 77, 231 (2014)

    Article  Google Scholar 

  66. K. Diethelm, N.J. Ford, J. Math. Anal. Appl. 265, 229 (2002)

    Article  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  68. K. Diethelm, A.D. Freed, Forschung und wissenschaftliches Rechnen 1999, 57 (1998)

    Google Scholar 

  69. R. Garrappa, Predictor-corrector PECE method for fractional differential equations, MATLAB Central File Exchange [File ID: 32918] (2011)

  70. H. Jahanshahi, K. Rajagopal, A. Akgul, N.N. Sari, H. Namazi, S. Jafari, Int. J. Non-Linear Mech. 107, 126 (2018)

    Article  ADS  Google Scholar 

  71. A. Karthikeyan, K. Rajagopal, Complexity 2017, 3815146 (2017)

    Article  MathSciNet  Google Scholar 

  72. K. Rajagopal, F. Nazarimehr, A. Karthikeyan, A. Srinivasan, S. Jafari, Asian J. Control 20, 1979 (2018)

    Article  MathSciNet  Google Scholar 

  73. K. Rajagopal, S. Jafari, G. Laarem, Pramana 89, 92 (2017)

    Article  ADS  Google Scholar 

  74. G. Adomian, Math. Comput. Modell. 13, 17 (1990)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  77. S.-B. He, K.-H. Sun, H.-H. Wang, Acta Phys. Sin. 63, 030502 (2014)

    Google Scholar 

  78. T. Wang, N. Jia, Appl. Math. Comput. 218, 7231 (2012)

    MathSciNet  Google Scholar 

  79. Y. Luo, Y. Chen, H.-S. Ahn, Y. Pi, Control Eng. Pract. 18, 1022 (2010)

    Article  Google Scholar 

  80. A. Razminia, V.J. Majd, D. Baleanu, Adv. Difference Equ. 2011, 15 (2011)

    Article  Google Scholar 

  81. A. Nourian, S. Balochian, Pramana 86, 1401 (2016)

    Article  ADS  Google Scholar 

  82. S. Bowong, F.M. Kakmeni, Chaos Solitons Fractals 21, 999 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  83. M.S. Couceiro, N.F. Ferreira, J.T. Machado, Commun. Nonlinear Sci. Numer. Simul. 15, 895 (2010)

    Article  ADS  Google Scholar 

  84. H. Delavari, R. Ghaderi, A. Ranjbar, S. Momani, Commun. Nonlinear Sci. Numer. Simul. 15, 963 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  85. Z.M. Odibat, N. Corson, M. Aziz-Alaoui, C. Bertelle, Int. J. Bifurc. Chaos 20, 81 (2010)

    Article  Google Scholar 

  86. D. Chen, R. Zhang, J.C. Sprott, X. Ma, Nonlinear Dyn. 70, 1549 (2012)

    Article  Google Scholar 

  87. D. Cafagna, G. Grassi, Commun. Nonlinear Sci. Numer. Simul. 19, 2919 (2014)

    Article  ADS  MathSciNet  Google Scholar 

  88. D. Cafagna, G. Grassi, Int. J. Bifurc. Chaos 21, 955 (2011)

    Article  Google Scholar 

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Authors and Affiliations

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

    Karthikeyan Rajagopal

  2. Institute of Energy, Mekelle University, Mek’ele, Ethiopia

    Karthikeyan Rajagopal

  3. Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  4. Ministry of Higher Education and Scientific Research, Baghdad, Iraq

    Abdul Jalil M. Khalaf

  5. Faculty of Computer Science and Engineering, Shahid Beheshti University, Velenjak, 19839-69411, Tehran, Iran

    Yasser Shekofteh

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

    Fahimeh Nazarimehr

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  1. Karthikeyan Rajagopal
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  2. Sifeu Takougang Kingni
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  3. Abdul Jalil M. Khalaf
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  4. Yasser Shekofteh
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  5. Fahimeh Nazarimehr
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Correspondence to Fahimeh Nazarimehr.

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Rajagopal, K., Kingni, S.T., Khalaf, A.J.M. et al. Coexistence of attractors in a simple chaotic oscillator with fractional-order-memristor component: analysis, FPGA implementation, chaos control and synchronization. Eur. Phys. J. Spec. Top. 228, 2035–2051 (2019). https://doi.org/10.1140/epjst/e2019-900001-8

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  • DOI: https://doi.org/10.1140/epjst/e2019-900001-8

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