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

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Abstract

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.

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References

  1. J C Sprott, Phys. Rev. E 50, 543 (1994)

    Article  Google Scholar 

  2. L P Shil’nikov, Math USSR-Shornik 10, 91 (1970)

    Article  Google Scholar 

  3. C P Silva, IEEE Trans. Circuits Syst. I 40, 657 (1993)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  7. Z C Wei and Q G Yang, Nonlinear Dyn. 68, 543 (2012)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  9. Q G Yang, Z C Wei and G R Chen, Int. J. Bifurc. Chaos 20, 1061 (2010)

    Article  Google Scholar 

  10. M Molaie, S Jafari, J C Sprott and S Mohammad, Int. J. Bifurc. Chaos 23, 1350188 (2013)

    Article  Google Scholar 

  11. Z C Wei, I Moroz, Z Wang, J C Sprott and T Kapitaniak, Int. J. Bifurc. Chaos 26, 1650125 (2016)

    Article  Google Scholar 

  12. B C Bao, Q D Li, N Wang and Q Xu, Chaos 26, 043111 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  13. Z C Wei, I Moroz, J C Sprott, Z Wang and W Zhang, Int. J. Bifurc. Chaos 27(2), 1730008 (2017)

    Article  Google Scholar 

  14. Z C Wei, P Yu, W Zhang and M H Yao, Nonlinear Dyn. 82, 131 (2015)

    Article  Google Scholar 

  15. Z C Wei and W Zhang, Int. J. Bifurc. Chaos 24, 1450127 (2014)

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  18. Y Feng and W Q Pan, Pramana – J. Phys. 88, 62 (2017)

    Article  ADS  Google Scholar 

  19. C H Wang, H Xia and L Zhou, Pramana – J. Phys. 88, 34 (2017)

    Article  ADS  Google Scholar 

  20. K Vishal, S K Agrawal and S Das, Pramana – J. Phys. 86(1), 59 (2016)

    Article  ADS  Google Scholar 

  21. Z C Wei, I Moroz, J C Sprott, A Akgul and W Zhang, Chaos 27(3), 033101 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  22. R Shaw, Z. Naturforsch A 36, 80 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  23. G A Leonov and N V Kuznetsov, Int. J. Bifurc. Chaos 17, 1079 (2007)

    Article  Google Scholar 

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

    Article  ADS  MathSciNet  Google Scholar 

  25. A Akgul, I Moroz, I Pehlivan and V Sundarapandian, Optik 127(13), 5491 (2016)

    Article  ADS  Google Scholar 

  26. A Akgul, H Shafqat and I Pehlivan, Optik 127(18), 7062 (2016)

    Article  ADS  Google Scholar 

  27. D Baleanu, K Diethelm, E Scalas and J J Trujillo, Fractional calculus: Models and numerical methods (World Scientific, Singapore, 2014)

    MATH  Google Scholar 

  28. Y Zhou, Basic theory of fractional differential equations (World Scientific, Singapore, 2014)

    Book  MATH  Google Scholar 

  29. K Diethelm, The Analysis of fractional differential equations (Springer, Berlin, 2010)

    Book  MATH  Google Scholar 

  30. K Rajagopal, L Guessas, S Vaidyanathan, A Karthikeyan and A Srinivasan, Math. Probl. Eng. 2017, 7307452 (2017)

    Article  Google Scholar 

  31. K Rajagopal, A Karthikeyan and P Duraisamy, Hyperchaotic chameleon: Fractional order FPGA implementation, Complexity, in press. https://www.hindawi.com/journals/complexity/aip/8979408/

  32. Karthikeyan Rajagopal, Laarem Guessas, Anitha Karthikeyan, Ashokkumar Srinivasan and Girma Adam, Complexity 2017, 1892618 (2017)

    Google Scholar 

  33. K Rajagopal, A Karthikeyan and A Srinivasan, Nonlinear Dyn. 87(4), 2281 (2017)

    Article  Google Scholar 

  34. A Charef, H H Sun and Y Y Tsao, IEEE Trans. Auto. Contr. 37, 14651470 (1992)

    Article  Google Scholar 

  35. G A Adomian, Math. Comp. Model. 13, 17 (1990)

    Article  Google Scholar 

  36. H H Sun, A A Abdelwahab and B Onaral, IEEE Trans. Auto. Contr. 29, 441 (1984)

    Article  Google Scholar 

  37. M S Tavazoei and M Haeri, IET Sign. Proc. 1, 171 (2007)

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  40. S B He, Kehui Sun and Huihai Wang, Entropy 17, 8299 (2015)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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).

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Authors and Affiliations

  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, China

    Zhouchao Wei

  2. Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin, 537000, China

    Zhouchao Wei

  3. Center for Nonlinear Dynamics, Department of Electrical and Communication Engineering, The PNG University of Technology, Lae, Papua New Guinea

    Karthikeyan Rajagopal

  4. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    Wei Zhang

  5. Department of Mechanical and Electrical Engineering, Institute of Mines and Petroleum Industries, University of Maroua, P.O. Box 46, Maroua, Cameroon

    Sifeu Takougang Kingni

  6. Department of Electrical and Electronics Engineering, Faculty of Technology, Sakarya University, Adapazarı, Sakarya, Turkey

    Akif Akgül

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  1. Zhouchao Wei
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  2. Karthikeyan Rajagopal
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  3. Wei Zhang
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  5. Akif Akgül
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Correspondence to Zhouchao Wei.

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Wei, Z., Rajagopal, K., Zhang, W. et al. Synchronisation, electronic circuit implementation, and fractional-order analysis of 5D ordinary differential equations with hidden hyperchaotic attractors. Pramana - J Phys 90, 50 (2018). https://doi.org/10.1007/s12043-018-1540-2

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  • DOI: https://doi.org/10.1007/s12043-018-1540-2

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