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Coexisting attractors, crisis route to chaos in a novel 4D fractional-order system and variable-order circuit implementation

  • Chengyi Zhou
  • Zhijun LiEmail author
  • Fei Xie
Regular Article
  • 41 Downloads

Abstract.

In this paper, a novel 4D fractional-order chaotic system is proposed, and the corresponding dynamics are systematically investigated by considering both fractional-order and traditional system parameters as bifurcation parameters. When varying the traditional system parameters, this system exhibits some conspicuous characteristics. For example, four separate single-wing chaotic attractors coexist, and they will pairwise combine, resulting in a pair of double-wing attractors. More distinctively, by choosing the specific control parameters, transitions from a four-wing attractor to a pair of double-wing attractors to four coexisting single-wing attractors are observed, which means that the novel fractional-order system experiences an unusual and striking double-dip symmetry recovering crisis. However, numerous studies have shown that the fractional differential order has an important effect on the dynamical behavior of a fractional-order system. However, these studies are based only on numerical simulations. Thus, the design of a variable fractional-order circuit to investigate the influence of the order on the dynamical behavior of the fractional-order chaotic circuit is urgently needed. Varying with the order, coexisting period-doubling bifurcation modes appear, which suggests that the orbits have transitions from a coexisting periodic state to a coexisting chaotic state. A variable fractional-order circuit is designed, and the experimental observations are found to be in good agreement with the numerical simulations.

References

  1. 1.
    C. Luo, X. Wang, Nonlinear Dyn. 71, 241 (2012)CrossRefGoogle Scholar
  2. 2.
    D. Cafagna, G. Grassi, Int. J. Bifurc. Chaos 19, 339 (2009)CrossRefGoogle Scholar
  3. 3.
    I. Petras, Chaos, Solitons Fractals 38, 140 (2008)CrossRefGoogle Scholar
  4. 4.
    P. Arena, R. Caponetto, L. Fortuna, D. Porto, in Proceedings ECCTD, Budapest, September, 1259 (1997)Google Scholar
  5. 5.
    W. Deng, C. Li, Physica A 353, 61 (2005)CrossRefGoogle Scholar
  6. 6.
    J. Lü, Chaos, Solitons Fractals 27, 685 (2006)CrossRefGoogle Scholar
  7. 7.
    S. Zhang, Y. Zeng, Z. Li, Int. J. Nonlinear Mech. 106, 1 (2018)CrossRefGoogle Scholar
  8. 8.
    A. Charef, H.H. Sun, Y.Y. Tsao, B. Onaral, IEEE Trans. Autom. Control 29, 441 (1984)CrossRefGoogle Scholar
  9. 9.
    K. Diethelm, Electron, Trans. Numer. Anal. 5, 1 (1997)MathSciNetGoogle Scholar
  10. 10.
    S. He, K. Sun, S. Banerjee, Eur. Phys. J. Plus 131, 254 (2016)CrossRefGoogle Scholar
  11. 11.
    J. Ruan, K. Sun, J. Mou, S. He, L. Zhang, Eur. Phys. J. Plus 133, 3 (2018)CrossRefGoogle Scholar
  12. 12.
    S. Zhang, Y. Zeng, Z. Li, Chaos 28, 013113 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Q. Lai, B. Norouzi, F. Liu, Chaos, Solitons Fractals 114, 230 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Shahzad, V.-T. Pham, M.A. Ahmad, S. Jafari et al., Eur. Phys. J. ST 224, 1637 (2015)CrossRefGoogle Scholar
  15. 15.
    L. Zhang, K. Sun, S. He, H. Wang, Y. Xu, Eur. Phys. J. Plus 132, 31 (2017)CrossRefGoogle Scholar
  16. 16.
    S. Zhang, Y. Zeng, Z. Li, C. Zhou, Int. J. Bifurc. Chaos 28, 1850167 (2018)CrossRefGoogle Scholar
  17. 17.
    C. Li, J.C. Sprott, Optik 127, 10389 (2016)CrossRefGoogle Scholar
  18. 18.
    H. Jia, Z. Guo, G. Qi, Z. Chen, Optik 155, 233 (2018)CrossRefGoogle Scholar
  19. 19.
    Y. Wang, K. Sun, S. He, H. Wang, Eur. Phys. J. ST 223, 1591 (2014)CrossRefGoogle Scholar
  20. 20.
    X. Zhang, Z. Li, D. Chang, AEU-Int. J. Electron. Commun. 82, 435 (2017)CrossRefGoogle Scholar
  21. 21.
    Z. Li, M. Ma, M. Wang. Yi. Zeng, AEU-Int. J. Electron. Commun. 71, 21 (2016)CrossRefGoogle Scholar
  22. 22.
    D. Chang, Z. Li, M. Wang. Yi. Zeng, AEU-Int. J. Electron. Commun. 88, 20 (2018)CrossRefGoogle Scholar
  23. 23.
    J. Ma, F. Wu, G. Ren, J. Tang, Appl. Math. Comput. 367, 192 (2017)Google Scholar
  24. 24.
    S. Zhang, Z. Yi, Z. Li, Chin. J. Phys. 56, 793 (2018)CrossRefGoogle Scholar
  25. 25.
    V.-T. Pham, S.T. Kingni, C. Volos, S. Jafari, T. Kapitaniak, AEU-Int. J. Electron. Commun. 78, 220 (2017)CrossRefGoogle Scholar
  26. 26.
    M. Borah, B.K. Roy, ISA Trans.  https://doi.org/10.1016/j.isatra.2017.02.007 (2017)
  27. 27.
    Y. Xu, K. Sun, S. He et al., Eur. Phys. J. Plus 131, 186 (2016)CrossRefGoogle Scholar
  28. 28.
    J.M. Munoz-Pacheco, E. Zambrano-Serrano, C. Volos et al., Entropy 20, 564 (2018)CrossRefGoogle Scholar
  29. 29.
    K. Rajagopal, V.-T. Pham, F.E. Alssadi et al., Eur. Phys. J. ST 227, 837 (2018)CrossRefGoogle Scholar
  30. 30.
    S. He, S. Banerjee, B. Yan, Complexity 2018, 4140762 (2018)Google Scholar
  31. 31.
    N. Yang, C. Cheng, C. Wu, R. Jia, C. Liu, Int. J. Bifurc. Chaos 27, 1750199 (2017)CrossRefGoogle Scholar
  32. 32.
    S. Cicek, Y. Uyaroglu, I. Pehlivan, J. Circ. Syst. Comput. 22, 1350022 (2013)CrossRefGoogle Scholar
  33. 33.
    A. Akgul, H. Calgan, I. Koyuncu, I. Pehlivan, A. Istanbullu, Y. Uyaroglu, I. Pehlivan, Nonlinear Dyn. 84, 481 (2016)CrossRefGoogle Scholar
  34. 34.
    C.K. Volos, I.M. Kyprianidis, I.N. Stouboulos, Robot. Autom. Syst. 60, 651 (2012)CrossRefGoogle Scholar
  35. 35.
    B. Wang, H. Xu, P. Yang, L. Liu, J. Li, Entropy 17, 2082 (2015)CrossRefGoogle Scholar
  36. 36.
    E. Fatemi-Behbahani, K. Aansari-Asl, E. Farshidi, Circ. Syst. Signal Process. 35, 3830 (2016)CrossRefGoogle Scholar
  37. 37.
    A. Oustaloup, F. Levron, B. Mathieu, F.M. Nanot, IEEE Trans. Circ. Syst. I 47, 25 (2000)CrossRefGoogle Scholar
  38. 38.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego, 1999)Google Scholar
  39. 39.
    B. Liu, G. Chen, Int. J. Bifurc. Chaos 21, 261 (2003)CrossRefGoogle Scholar
  40. 40.
    H.F.v. Bremen, F.E. Udwadia, W. Proskurowski, Physica D 101, 1 (1997)MathSciNetCrossRefGoogle Scholar
  41. 41.
    B. Christoph, P. Bernd, Phys. Rev. Lett. 88, 174102 (2002)CrossRefGoogle Scholar
  42. 42.
    H.A. Larrondo, C.M. González, M.T. Martín et al., Physica A 356, 133 (2005)CrossRefGoogle Scholar
  43. 43.
    S. He, K. Sun, S. Banerjee, Eur. Phys. J. Plus 131, 254 (2016)CrossRefGoogle Scholar
  44. 44.
    Z. Cai, J. Sun, Int. J. Bifurc. Chaos 19, 977 (2011)CrossRefGoogle Scholar
  45. 45.
    S. He, K. Sun, H. Wang, Entropy 17, 8299 (2015)CrossRefGoogle Scholar
  46. 46.
    C. Sun, Z. Chen, Q. Xu, Int. J. Bifurc. Chaos 27, 1750197 (2017)CrossRefGoogle Scholar
  47. 47.
    S. Yu, J. Lü, G. Chen, IEEE Trans. Circ. Syst. I 54, 2087 (2007)CrossRefGoogle Scholar
  48. 48.
    F. Wang, C. Liu, Acta Phys. Sin. 55, 3922 (2006)Google Scholar
  49. 49.
    Y. Shao, H. Min, M. Ma et al., Acta Phys. Sin. 62, 130504 (2013)Google Scholar
  50. 50.
    V.R. Folifack Signing, J. Kengne, L.K. Kana, Chaos, Solitons Fractals 113, 263 (2018)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Z.T. Njitacke, J. Kengne, AEU-Int. J. Electron. Commun. 93, 242 (2018)CrossRefGoogle Scholar
  52. 52.
    B. Bao, H. Wu, L. Xu, M. Chen, Int. J. Bifurc. Chaos 28, 1850019 (2018)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Information EngineeringXiangtan UniversityXiangtanChina

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