Combination-Combination Anti-Synchronization of Four Fractional Order Identical Hyperchaotic Systems

  • Ayub Khan
  • Shikha Singh
  • Ahmad Taher AzarEmail author
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 921)


In this manuscript, we investigate the methodology of combination-combination anti-synchronization of four identical fractional order hyperchaotic system. The methodology is implemented by considering a 4D fractional order hyperchaotic system. The controllers are constructed using adaptive control technique to ensure the combination-combination anti - synchronization. The synchronization schemes such as chaos control problem, projective anti-synchronization, combination anti-synchronization becomes the special cases of combination-combination anti-synchronization. The combination - combination scheme can additionally enhances the security of transmission of message signals. The theoretical results and numerical simulations are given to justify the validity and feasibility of the proposed control technique.


Fractional-order hyperchaotic system Combination-combination anti-synchronization Stability theory Active control 


  1. 1.
    Azar, A.T., Serrano, F.E.: Fractional order sliding mode PID controller/observer for continuous nonlinear switched systems with PSO parameter tuning. In: International Conference on Advanced Machine Learning Technologies and Applications, pp. 13–22. Springer (2018)Google Scholar
  2. 2.
    Gao, Y., Liang, C., Wu, Q., Yuan, H.: A new fractional-order hyperchaotic system and its modified projective synchronization. Chaos Soliton. Fract. 76, 190–204 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Khan, A., Pal, R.: Adaptive hybrid function projective synchronization of chaotic space-tether system. Nonlinear Dyn. Syst. Theor. 14(1), 44–57 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Khan, A., Shikha, S.: Increased and reduced order synchronisations between 5D and 6D hyperchaotic systems. Indian J. Ind. Appl. Math. 8(1), 118–131 (2017)CrossRefGoogle Scholar
  5. 5.
    Khan, A., Shikha, S.: Mixed tracking and projective synchronization of 6D hyperchaotic system using active control. Int. J. Nonlinear Sci. 22(1), 44–53 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Khan, A., Singh, S.: Chaotic analysis and combination-combination synchronization of a novel hyperchaotic system without any equilibria. Chinese J. Phys. (2017)Google Scholar
  7. 7.
    Khan, A., Singh, S.: Generalization of combination-combination synchronization of n-dimensional time-delay chaotic system via robust adaptive sliding mode control. Math. Method. Appl. Sci. (2018)Google Scholar
  8. 8.
    Khan, A., et al.: Hybrid function projective synchronization of chaotic systems via adaptive control. Int. J. Dyn. Control 5(4), 1114–1121 (2017)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Koeller, R.: Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. 51, 299–307 (1984). (ISSN 0021-8936)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, C., Liao, X., Yu, J.: Synchronization of fractional order chaotic systems. Phys. Rev. E 68(6), 067203 (2003)CrossRefGoogle Scholar
  11. 11.
    Lu, J.G.: Chaotic dynamics of the fractional-order lü system and its synchronization. Phys. Lett. A 354(4), 305–311 (2006)CrossRefGoogle Scholar
  12. 12.
    Mahmoud, G.M., Mahmoud, E.E.: Complete synchronization of chaotic complex nonlinear systems with uncertain parameters. Nonlinear Dyn. 62(4), 875–882 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82(15), 3042 (1999)CrossRefGoogle Scholar
  14. 14.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821 (1990)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and Some of their Applications, vol. 198. Academic press (1998)Google Scholar
  16. 16.
    Podlubny, I.: Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers. IEEE Trans. Autom. Control 44(1), 208–214 (1999)CrossRefGoogle Scholar
  17. 17.
    Rosenblum, M.G., Pikovsky, A.S., Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators. Phys. Rev. Lett. 78(22), 4193 (1997)CrossRefGoogle Scholar
  18. 18.
    Singh, S., Azar, A.T., Zhu, Q.: Multi-switching master-slave synchronization of non-identical chaotic systems. In: Innovative Techniques and Applications of Modelling, Identification and Control, pp. 321–330. Springer (2018)Google Scholar
  19. 19.
    Singh, S., Azar, A.T., Ouannas, A., Zhu, Q., Zhang, W., Na, J.: Sliding mode control technique for multi-switching synchronization of chaotic systems. In: 9th International Conference on Modelling, Identification and Control (ICMIC) 2017, pp. 880–885. IEEE (2017)Google Scholar
  20. 20.
    Singh, S.V.S., Serrano, F.E., Sambas, A.: A novel hyperchaotic system with adaptive control, synchronization, and circuit simulation. In: Advances in System Dynamics and Control, p. 382 (2018)Google Scholar
  21. 21.
    Vaidyanathan, S., Azar, A.T.: Hybrid synchronization of identical chaotic systems using sliding mode control and an application to vaidyanathan chaotic systems. In: Advances and Applications in Sliding Mode Control Systems, pp. 549–569. Springer (2015)Google Scholar
  22. 22.
    Zheng, Z., Hu, G.: Generalized synchronization versus phase synchronization. Phys. Rev. E 62(6), 7882 (2000)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNew DelhiIndia
  2. 2.Department of Mathematics, Jesus and Mary CollegeUniversity of DelhiNew DelhiIndia
  3. 3.Faculty of Computers and InformationBenha UniversityBenhaEgypt
  4. 4.School of Engineering and Applied SciencesNile University Campus6th of October City, GizaEgypt

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