A new 4-D chaotic hyperjerk system, its synchronization, circuit design and applications in RNG, image encryption and chaos-based steganography

  • S. Vaidyanathan
  • A. Akgul
  • S. Kaçar
  • U. Çavuşoğlu
Regular Article
Part of the following topical collections:
  1. Focus Point on Systems and Security: Advanced Methods with Chaos and Complexity


Hyperjerk systems have received significant interest in the literature because of their simple structure and complex dynamical properties. This work presents a new chaotic hyperjerk system having two exponential nonlinearities. Dynamical properties of the chaotic hyperjerk system are discovered through equilibrium point analysis, bifurcation diagram, dissipativity and Lyapunov exponents. Moreover, an adaptive backstepping controller is designed for the synchronization of the chaotic hyperjerk system. Also, a real circuit of the chaotic hyperjerk system has been carried out to show the feasibility of the theoretical hyperjerk model. The chaotic hyperjerk system can also be useful in scientific fields such as Random Number Generators (RNGs), data security, data hiding, etc. In this work, three implementations of the chaotic hyperjerk system, viz. RNG, image encryption and sound steganography have been performed by using complex dynamics characteristics of the system.


  1. 1.
    S. Vaidyanathan, C. Volos, Advances and Applications in Chaotic Systems (Springer, Berlin, Germany, 2016)Google Scholar
  2. 2.
    A.T. Azar, S. Vaidyanathan, Advances in Chaos Theory and Intelligent Control (Springer, Berlin, Germany, 2016)Google Scholar
  3. 3.
    S. Winnings, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, Berlin, German, 2003)Google Scholar
  4. 4.
    E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)ADSCrossRefGoogle Scholar
  5. 5.
    O.E. Rössler, Phys. Lett. A 57, 397 (1976)ADSCrossRefGoogle Scholar
  6. 6.
    J.C. Sprott, Phys. Rev. 50, 647 (1994)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    G. Chen, T. Ueda, Int. J. Bifurcat. Chaos 9, 1465 (1999)CrossRefGoogle Scholar
  8. 8.
    H.P.W. Gottlieb, Am. J. Phys. 64, 525 (1996)ADSCrossRefGoogle Scholar
  9. 9.
    R. Tchitnga, T. Nguazon, P.H.L. Fotso, J.A.C. Gallas, IEEE Trans. Circuits Syst. II: Express Briefs 63, 239 (2016)CrossRefGoogle Scholar
  10. 10.
    J. Heidel, J. Zhang, Int. J. Bifurcat. Chaos 17, 2049 (2007)CrossRefGoogle Scholar
  11. 11.
    S. Ghorui, S.N. Sahasrabudhe, P.S.S. Muryt, A.K. Das, N. Venkatramani, IEEE Trans. Plasma Sci. 28, 235 (2000)CrossRefGoogle Scholar
  12. 12.
    H.G. Enjieu Kadji, J.B. Chabi Orou, R. Yamapi, P. Woafo, Chaos Solitons Fractals 32, 862 (2007)ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    P. Coullet, C. Tresser, A. Arneodo, Phys. Lett. A 72, 268 (1979)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    J.C. Sprott, Phys. Lett. A 228, 271 (1997)ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    Z. Elhadj, J.C. Sprott, Palest. J. Math. 2, 38 (2013)MathSciNetGoogle Scholar
  16. 16.
    K.E. Chlouverakis, J.C. Sprott, Chaos Solitons Fractals 28, 739 (2006)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    S.J. Linz, Chaos Solitons Fractals 37, 741 (2008)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    S. Vaidyanathan, C. Volos, V.T. Pham, K. Madhavan, Arch. Control Sci. 25, 135 (2015)MathSciNetGoogle Scholar
  19. 19.
    S. Vaidyanathan, Int. J. Control Theory Appl. 9, 257 (2016)Google Scholar
  20. 20.
    S. Vaidyanathan, Arch. Control Sci. 26, 311 (2016)Google Scholar
  21. 21.
    S. Vaidyanathan, A. Sambas, M. Mamat, M. Sanjaya WS, Int. J. Model. Identif. Control 26, 153 (2017)CrossRefGoogle Scholar
  22. 22.
    X. Wang, S. Vaidyanathan, C. Volos, V.T. Pham, T. Kapitaniak, Nonlinear Dyn. 89, 1673 (2017)CrossRefGoogle Scholar
  23. 23.
    A. Sambas, S. Vaidyanathan, M. Mamat, M. Sanjaya WS, R.P. Prastio, Int. J. Control Theory Appl. 9, 141 (2016)Google Scholar
  24. 24.
    Z.T. Njitacke, J. Kengne, L. Kamdjeu Kengne, Chaos Solitons Fractals 105, 77 (2017)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    A.S. Mansingka, M. Affan Zidan, M.L. Barakat, A.G. Radwan, K.N. Salama, Microelectron. J. 44, 744 (2013)CrossRefGoogle Scholar
  26. 26.
    R.A. El-Nabulsi, Int. J. Non-Linear Mech. 93, 65 (2017)ADSCrossRefGoogle Scholar
  27. 27.
    H. Bao, N. Wang, B. Bao, M. Chen, P. Jin, G. Wang, Commun. Nonlinear Sci. Numer. Simul. 57, 264 (2018)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    F.Y. Dalkiran, J.C. Sprott, Int. J. Bifurcat. Chaos 26, 1650189 (2016)CrossRefGoogle Scholar
  29. 29.
    A. Wolf, J.B. Swift, H.L. Swinney, J.A. Vastano, Physica D 16, 285 (1985)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    P. Frederickson, J. Kaplan, E. Yorke, J. Yorke, J. Differ. Equ. 49, 185 (1983)ADSCrossRefGoogle Scholar
  31. 31.
    H.K. Khalil, Nonlinear Systems (Prentice Hall, New Jersey, USA, 2002)Google Scholar
  32. 32.
    A statistical test suite for random and pseudo RNGs for cryptographic applications, NIST-800-22 (National Institute of Standards and Techniques, 2001)
  33. 33.
    P.K. Narendra, P. Vinod, K.S. Krishan, Image Vision Comput. 24, 926 (2006)CrossRefGoogle Scholar
  34. 34.
    E. Biham, A. Shamir, J. Cryptol. 4, 3 (1991)CrossRefGoogle Scholar
  35. 35.
    Y. Wang, K.W. Wong, X. Liao, T. Xiang, G. Chen, Chaos Solitons Fractals 41, 1773 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    C.E. Shannon, Bell Syst. Tech. J. 28, 656 (1948)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Vel Tech UniversityResearch and Development CentreTamil NaduIndia
  2. 2.Department of Electrical and Electronics Engineering, Faculty of TechnologySakarya UniversitySakaryaTurkey
  3. 3.Department of Computer Engineering, Faculty of TechnologySakarya UniversitySakaryaTurkey

Personalised recommendations