A class of higher-dimensional hyperchaotic maps

  • Chen Chen
  • Kehui SunEmail author
  • Shaobo He
Regular Article


Based on the hybrid modulation coupling (HMC) pattern, a class of higher-dimensional (HD) hyperchaotic maps is proposed using three one-dimensional (1D) seed maps. The seed maps are chaotic maps or the combination of chaotic maps and non-chaotic maps. Taking the HMC of iterative chaotic map with infinite collapse (ICMIC), Sine map and a linear map (ISL-HMC) as an example, the equilibrium points are mathematically analyzed. The dynamical performance of the 3D ISL-HMC map is evaluated by phase diagram, Lyapunov exponents (LEs), bifurcation diagram and chaos diagram. Furthermore, compared with existing chaotic maps, complexity and distribution characteristic are analyzed. As application of the ISL-HMC map, a pseudorandom number generator (PRNG) is designed and tested by NIST SP 800-22 and TestU01. Experimental results show that the ISL-HMC map has rich dynamical behaviors and good randomness. So this class of HD hyperchaotic maps is a potential model for cryptography and other applications.


  1. 1.
    M. Sciamanna, K.A. Shore, Nat. Photon. 9, 151 (2017)ADSCrossRefGoogle Scholar
  2. 2.
    J.A. Marusich, J.L. Wiley, T.W. Lefever, P.R. Patel, B.F. Thomas, Neuropharmacology 134, 73 (2018)CrossRefGoogle Scholar
  3. 3.
    Q. Jiang, J. Ma, X. Lu, Y. Tian, J. Med. Syst. 38, 1 (2014)CrossRefGoogle Scholar
  4. 4.
    S. Kim, Y. Kim, J. Lee, H.S. Kim, Adv. Meteorol. 2015, 1 (2015)Google Scholar
  5. 5.
    Z. Lin, S. Yu, J. Lü, S. Cai, G. Chen, IEEE T Circ. Syst. Vid. 25, 1203 (2015)CrossRefGoogle Scholar
  6. 6.
    Z. Lin, S. Yu, C. Li, J. Lü, Q. Wang, Int. J. Bifurcat. Chaos 26, 1650158 (2016)CrossRefGoogle Scholar
  7. 7.
    L. Liu, H. Song, Electron. Design Eng. 13, 123 (2014)Google Scholar
  8. 8.
    Z. Hua, Y. Zhou, C. Chen, Digital Signal Processing and Signal Processing Education Meeting (IEEE, 2013) pp. 118--123Google Scholar
  9. 9.
    G. Wang, F. Yuan, Acta. Phys. Sin. 62, 020506 (2013)Google Scholar
  10. 10.
    Y. Zhou, L. Bao, C. Chen, Signal Process. 97, 172 (2012)CrossRefGoogle Scholar
  11. 11.
    H. Natiq, N. Al-Saidi, M. Said, A. Kilicman, Eur. Phys. J. Plus 133, 6 (2018)CrossRefGoogle Scholar
  12. 12.
    C. Pak, L. Huang, Signal Process. 138, 129 (2017)CrossRefGoogle Scholar
  13. 13.
    J. Li, H. Liu, IET. Inform. Secur. 7, 265 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Y. Wu, J. Electron. Imaging 21, 1 (2012)Google Scholar
  15. 15.
    Z. Hua, Y. Zhou, Inform. Sci. 339, 237 (2018)CrossRefGoogle Scholar
  16. 16.
    Z. Hua, Y. Shuang, Y. Zhou, C. Li, Y. Wu, IEEE T. Cybernetics 48, 463 (2018)CrossRefGoogle Scholar
  17. 17.
    M. Yu, K. Sun, W. Liu, S. He, Chaos, Solitons Fractals 106, 107 (2018)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    W. Liu, K. Sun, S. He, Nonlinear Dyn. 89, 2521 (2017)CrossRefGoogle Scholar
  19. 19.
    Z. Hua, Y. Zhou, C. Pun, C. Chen, Inform. Sci. 297, 80 (2015)CrossRefGoogle Scholar
  20. 20.
    A. Alamodi, K. Sun, W. Ai, C. Chen, D. Peng, Chin. Phys. B 28, 020503 (2019)ADSCrossRefGoogle Scholar
  21. 21.
    J. Munkhammar, Fract. Calc. Appl. Anal. 16, 511 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Z. Liu, T. Xia, Appl. Comput. Inform. 14, 177 (2017)CrossRefGoogle Scholar
  23. 23.
    Z. Liu, T. Xia, J. Wang, Chin. Phys. B 27, 030502 (2018)ADSCrossRefGoogle Scholar
  24. 24.
    D. He, C. He, L. Jiang, H. Zhu, G. Hu, Proc. Tencon 3, 95 (2000)Google Scholar
  25. 25.
    H. Natiq, S. Banerjee, S. He, M. Said, A. Kilicman, Chaos, Solitons Fractals 114, 506 (2018)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    H. Natiq, S. Banerjee, M. Ariffin, M. Said, Chaos 29, 011103 (2019)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    S. He, K. Sun, H. Wang, Physica A 461, 812 (2016)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    S. Pincus, Chaos 5, 110 (1995)ADSCrossRefGoogle Scholar
  29. 29.
    S. Chang, Acta Phys. Sin. 62, 709 (2013)Google Scholar
  30. 30.
    K. Sun, S. He, L. Yin, L. Duo, Acta Phys. Sin. 61, 130507 (2012)Google Scholar
  31. 31.
    H. Hu, Y. Deng, L. Liu, Commun. Nonlinear Sci. 19, 1970 (2014)CrossRefGoogle Scholar
  32. 32.
    C. Chen, K. Sun, Y. Peng, A. Alamodi, Eur. Phys. J. Plus 134, 31 (2019)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.School of Physics and ElectronicsCentral South UniversityChangshaChina

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