, 90:50 | Cite as

Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors

  • Zhouchao Wei
  • Karthikeyan Rajagopal
  • Wei Zhang
  • Sifeu Takougang Kingni
  • Akif Akgül


Hidden hyperchaotic attractors can be generated with three positive Lyapunov exponents in the proposed 5D hyperchaotic Burke–Shaw system with only one stable equilibrium. To the best of our knowledge, this feature has rarely been previously reported in any other higher-dimensional systems. Unidirectional linear error feedback coupling scheme is used to achieve hyperchaos synchronisation, which will be estimated by using two indicators: the normalised average root-mean squared synchronisation error and the maximum cross-correlation coefficient. The 5D hyperchaotic system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integration. In addition, fractional-order hidden hyperchaotic system will be considered from the following three aspects: stability, bifurcation analysis and FPGA implementation. Such implementations in real time represent hidden hyperchaotic attractors with important consequences for engineering applications.


Hyperchaos hidden attractor synchronisation circuit implementation fractional-order analysis 


02.30.Oz 02.30.Hq 



This work was supported by the Natural Science Foundation of China (Nos 11772306 and 11401543), the Open Foundation for Guangxi Colleges and Universities Key Lab of Complex System Optimization and Big Data Processing (No. 2016CSOBDP0202), Scientific Research Program of Hubei Provincial Department of Education (No. B2017599), the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (No. CUGL150419) and Sakarya University Scientific Research Projects Unit (No. 201609-00-008).


  1. 1.
    J C Sprott, Phys. Rev. E 50, 543 (1994)CrossRefGoogle Scholar
  2. 2.
    L P Shil’nikov, Math USSR-Shornik 10, 91 (1970)CrossRefGoogle Scholar
  3. 3.
    C P Silva, IEEE Trans. Circuits Syst. I 40, 657 (1993)CrossRefGoogle Scholar
  4. 4.
    G A Leonov and N V Kuznetsov, Int. J. Bifurc. Chaos 23, 1330002 (2013)CrossRefGoogle Scholar
  5. 5.
    G A Leonov, N V Kuznetsov and V I Vagaitsev, Phys. Lett. A 375, 2230 (2011)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    G A Leonov, N V Kuznetsov and V I Vagaitsev, Physica D 241, 1482 (2012)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Z C Wei and Q G Yang, Nonlinear Dyn. 68, 543 (2012)CrossRefGoogle Scholar
  8. 8.
    X Wang and G R Chen, Commun. Nonlinear. Sci. Numer. Simul. 17, 1264 (2012)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Q G Yang, Z C Wei and G R Chen, Int. J. Bifurc. Chaos 20, 1061 (2010)CrossRefGoogle Scholar
  10. 10.
    M Molaie, S Jafari, J C Sprott and S Mohammad, Int. J. Bifurc. Chaos 23, 1350188 (2013)CrossRefGoogle Scholar
  11. 11.
    Z C Wei, I Moroz, Z Wang, J C Sprott and T Kapitaniak, Int. J. Bifurc. Chaos 26, 1650125 (2016)CrossRefGoogle Scholar
  12. 12.
    B C Bao, Q D Li, N Wang and Q Xu, Chaos 26, 043111 (2016)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    Z C Wei, I Moroz, J C Sprott, Z Wang and W Zhang, Int. J. Bifurc. Chaos 27(2), 1730008 (2017)CrossRefGoogle Scholar
  14. 14.
    Z C Wei, P Yu, W Zhang and M H Yao, Nonlinear Dyn. 82, 131 (2015)CrossRefGoogle Scholar
  15. 15.
    Z C Wei and W Zhang, Int. J. Bifurc. Chaos 24, 1450127 (2014)CrossRefGoogle Scholar
  16. 16.
    T Kapitaniak and G A Leonov, Eur. Phys. J. Special Topics 224, 1405 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    D Dudkowski, S Jafari, T Kapitaniak, N V Kuznetsov, G A Leonov and A Prasad, Phys. Rep. 637, 1 (2016)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Y Feng and W Q Pan, Pramana – J. Phys. 88, 62 (2017)ADSCrossRefGoogle Scholar
  19. 19.
    C H Wang, H Xia and L Zhou, Pramana – J. Phys. 88, 34 (2017)ADSCrossRefGoogle Scholar
  20. 20.
    K Vishal, S K Agrawal and S Das, Pramana – J. Phys. 86(1), 59 (2016)ADSCrossRefGoogle Scholar
  21. 21.
    Z C Wei, I Moroz, J C Sprott, A Akgul and W Zhang, Chaos 27(3), 033101 (2017)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    R Shaw, Z. Naturforsch A 36, 80 (1981)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    G A Leonov and N V Kuznetsov, Int. J. Bifurc. Chaos 17, 1079 (2007)CrossRefGoogle Scholar
  24. 24.
    A Wolf, J B Swift, H L Swinney and J A Vastano, Physica D 16, 285 (1985)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A Akgul, I Moroz, I Pehlivan and V Sundarapandian, Optik 127(13), 5491 (2016)ADSCrossRefGoogle Scholar
  26. 26.
    A Akgul, H Shafqat and I Pehlivan, Optik 127(18), 7062 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    D Baleanu, K Diethelm, E Scalas and J J Trujillo, Fractional calculus: Models and numerical methods (World Scientific, Singapore, 2014)zbMATHGoogle Scholar
  28. 28.
    Y Zhou, Basic theory of fractional differential equations (World Scientific, Singapore, 2014)CrossRefzbMATHGoogle Scholar
  29. 29.
    K Diethelm, The Analysis of fractional differential equations (Springer, Berlin, 2010)CrossRefzbMATHGoogle Scholar
  30. 30.
    K Rajagopal, L Guessas, S Vaidyanathan, A Karthikeyan and A Srinivasan, Math. Probl. Eng. 2017, 7307452 (2017)CrossRefGoogle Scholar
  31. 31.
    K Rajagopal, A Karthikeyan and P Duraisamy, Hyperchaotic chameleon: Fractional order FPGA implementation, Complexity, in press.
  32. 32.
    Karthikeyan Rajagopal, Laarem Guessas, Anitha Karthikeyan, Ashokkumar Srinivasan and Girma Adam, Complexity 2017, 1892618 (2017)Google Scholar
  33. 33.
    K Rajagopal, A Karthikeyan and A Srinivasan, Nonlinear Dyn. 87(4), 2281 (2017)CrossRefGoogle Scholar
  34. 34.
    A Charef, H H Sun and Y Y Tsao, IEEE Trans. Auto. Contr. 37, 14651470 (1992)CrossRefGoogle Scholar
  35. 35.
    G A Adomian, Math. Comp. Model. 13, 17 (1990)CrossRefGoogle Scholar
  36. 36.
    H H Sun, A A Abdelwahab and B Onaral, IEEE Trans. Auto. Contr. 29, 441 (1984)CrossRefGoogle Scholar
  37. 37.
    M S Tavazoei and M Haeri, IET Sign. Proc. 1, 171 (2007)CrossRefGoogle Scholar
  38. 38.
    S B He, K H Sun and H H Wang, Acta Phys. Sin. 63, 030502 (2014)Google Scholar
  39. 39.
    R Caponetto and S Fazzino, Int. J. Bifurc. Chaos 23, 1350050 (2013)CrossRefGoogle Scholar
  40. 40.
    S B He, Kehui Sun and Huihai Wang, Entropy 17, 8299 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Indian Academy of Sciences 2018

Authors and Affiliations

  1. 1.School of Mathematics and PhysicsChina University of GeosciencesWuhanChina
  2. 2.Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data ProcessingYulin Normal UniversityYulinChina
  3. 3.Center for Nonlinear Dynamics, Department of Electrical and Communication EngineeringThe PNG University of TechnologyLaePapua New Guinea
  4. 4.College of Mechanical EngineeringBeijing University of TechnologyBeijingChina
  5. 5.Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum IndustriesUniversity of MarouaMarouaCameroon
  6. 6.Department of Electrical and Electronics Engineering, Faculty of TechnologySakarya UniversityAdapazarıTurkey

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