Arabian Journal for Science and Engineering

, Volume 44, Issue 8, pp 7335–7350 | Cite as

Multi-Switching Combination Anti-synchronization of Unknown Hyperchaotic Systems

  • Muhammad Shafiq
  • Israr AhmadEmail author
Research Article - Systems Engineering


The security of the information signal in digital communication systems is an active research problem. This paper proposes a new robust adaptive anti-synchronization control (RAASC) technique and studies the multi-switching combination anti-synchronization (MSCAS) of different unknown hyperchaotic (UH) systems. This controller reduces the overshoots in the transient phase and establishes the anti-synchronization convergence quickly. The proposed MSCAS scheme anti-synchronizes two master UH systems with a slave UH system using the multi-switching mechanism that increases the complexity and enhances the security of the information signal in digital communications. The Lyapunov direct method assures the robust stability of the closed-loop dynamical system. This work provides adaptive laws that estimate the true values of unknown parameters. A numerical example of three different hyperchaotic Pang, Lu and Lorenz–Stenflo systems is simulated. The simulation results verify the theoretical findings. The proposed MSCAS approach is used for the secret communication of digital messages using the chaotic masking technique. The receiver retrieves the information signal with high security and good precession qualitatively, under the effect of different types of disturbances. This paper also discusses the implications of controller constituents of the proposed RAASC algorithm that establishes fast convergence. The article suggests some future research problems related to this work.


Multi-switched combined anti-synchronization Robust adaptive controller Lyapunov stability theory Hyperchaotic systems Unknown parameters 

List of Symbols


n-dimensional Euclidean space


Transpose of a vector (or matrix)

\(i,j,k,r,l\in Z^{+}\)

Positive numbers for subscripts


The first master hyperchaotic system


The second master hyperchaotic system


Stands for the slave hyperchaotic system

\({{\varvec{u}}}=( {u_1 ,u_2 ,\ldots ,u_n } )^\mathrm{T}\in R^{n}\)

State variables of \(M_1\)

\({{\varvec{v}}}=\left( {v_1 ,v_2 ,\ldots ,v_n } \right) ^\mathrm{T}\in R^{n}\)

State variables of \(M_2\)

\({{\varvec{w}}}=\left( {w_1 ,w_2 ,\ldots ,w_n } \right) ^\mathrm{T}\in R^{n}\)

State variables of S

\(G_1 ,G_2 ,G_3 :R^{n}\rightarrow R^{n}\)

Continuous nonlinear vector functions in \(M_1,M_2\), and S, respectively

\(G_{ir} (.)\in R^{1\times n}\)

rth element of the vector \(G_i(.)\in R^{n\times 1} \, ({i=1,2,3})\)

\(a=( {a_1 ,a_2 ,\ldots ,a_n})^\mathrm{T}\in R^{n}\)

Unknown parameter vector of \(M_1\)


An \(n\times n\) matric containing the unknown parameter vector a of \(M_1\)


An ith row of the matrix \(A_a \in R^{n\times n}\)

\(b=\left( {b_1 ,b_2 ,\ldots ,b_n } \right) ^\mathrm{T}\in R^{n}\)

Unknown parameter vector of \(M_2\)


\(n\times n\) matrix containing the unknown parameter vector b of \(M_2\)


jth row of the matrix \(B_b \in R^{n\times n}\)

\(c=\left( {c_1 ,c_2 ,\ldots ,c_n } \right) ^\mathrm{T}\in R^{n}\)

Unknown parameter vector of S


\(n\times n\) matrix containing the unknown parameter vector c of S


kth row of the matrix \(C_c \in R^{n\times n}\)

\(Q_i \in R^{n\times 1}\)

ith row of the vector \(Q\in R^{n\times n}\)

\(f({{\varvec{u}}})=( f_1 ( {u_1 }),f_2({u_2}),\ldots ,\)

Unknown model

   \(f_n ( {u_n }))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in \(M_1\)

\(g({{\varvec{v}}})=(g_1 ({v_1}),g_2({v_2}),\ldots ,\)

Unknown model

   \(g_n ({v_n}))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in \(M_2 \)

\(h({{\varvec{v}}})=( h_1 ( {w_1 } ),h_2( {w_2 } ),\ldots ,\)

Unknown model

   \(h_n ( {w_n } ))^\mathrm{T}\in R^{n\times 1}\)

   uncertainty in S

\({\varvec{\rho }} =( {\rho _1 ,\rho _2 ,\ldots ,\rho _n })^\mathrm{T}\in R^{n\times 1},\)

Unknown external disturbance vector in \(M_1\)

\({\varvec{\sigma }} =( {\sigma _1 ,\sigma _2 ,\ldots ,\sigma _n })^\mathrm{T}\in R^{n\times 1}\)

Unknown external disturbance vector in \(M_2\)

\({\varvec{\delta }} =( {\delta _1 ,\delta _2 ,\ldots ,\delta _n })^\mathrm{T}\in R^{n\times 1}\)

Unknown external disturbance vector in S

\({\varvec{\mathcal{F}}}{=}( {\mathcal{F}_1 ,\mathcal{F}_2 ,\ldots ,\mathcal{F}_n \,})^\mathrm{T}{\in } R^{n\times 1}\)

Control input vector

\({{\varvec{e}}}\in R^{n\times n\times n}\)

MSCAS error vector

\(e\in R^{n\times 1}\)

A column vector, whose elements are chosen arbitrarily from \({{\varvec{e}}}\in R^{n\times n\times n}\)

\({{\varvec{e}}}_{ijk} \in R\)

An ith element of \({{\varvec{e}}}\in R^{n\times n\times n}\), chosen arbitrarily

\(X{=}\mathrm{diag}{[} {\alpha _1 ,\alpha _2 ,\ldots ,\alpha _n }{]}{\in } R^{n\times n}\)

Scaling matrix

\(Y{=}\mathrm{diag}{[} {\beta _1 ,\beta _2 ,\ldots ,\beta _n }{]}{\in } R^{n\times n}\)

Scaling matrix

\(Z{=}\mathrm{diag}{[} {\gamma _1 ,\gamma _2 ,\ldots ,\gamma _n }{]}{\in } R^{n\times n}\)

Scaling matrix


\(n\times n\) identity matrix

\(H^{M_1 M_2 S}( {{{\varvec{u,v,w}}}} )\in R^{1\times n}\)

A residual matrix of the nonlinear terms associated with \(M_1 ,M_2\), and S

\(H_i^{M_1 M_2 S} ( {{{\varvec{u,v,w}}}} )\)

An ith element of   \(H^{M_1 M_2 S}( {{{\varvec{u,v,w}}}} )\in R^{1\times n}\)


The exponential function


The signum function

\(\circ \)

Hadamard product operator

\(D_{\hat{d}e} \in R^{n\times n}\)

\(D_{\hat{d}e}\) is a diagonal matrix, whose diagonal elements are the elements of column vector \(\hat{d}\circ \mathrm{exp}\,({-p| e |} )\)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Cao, L.Y.; Lai, Y.C.: Antiphase synchronism in chaotic systems. Phy. Rev. E. 58(1), 382–386 (1998)CrossRefGoogle Scholar
  2. 2.
    Lynnyk,; Celikovsk, S.: On the anti–synchronization detection for the generalized Lorenz system and its applications to secure encryption. Kybernetika 46(1), 1–18 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Shahzad, M.; Ahmad, I.: Experimental study of synchronization and anti-synchronization for spin orbit problem of enceladus. Int. J. Control Sci. Eng. 3(2), 41–47 (2013)Google Scholar
  4. 4.
    Wang, L.; Chen, T.: Finite-time anti-synchronization of neural networks with time-varying delays. Neurocomputing 275(3), 1595–1600 (2018)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Astakhov, V.; Shabunin, A.; Klimshin, A.; Anishchenko, V.: In-phase and antiphase complete chaotic synchronization in symmetrically coupled discrete maps. Dis. Dyn. Nat. Soc. 7(4), 215–229 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Zhang, R.F.; Chen, D.; Yang, J.G.; Wang, J.: Anti-synchronization for a class of multi-dimensional autonomous and non-autonomous chaotic systems on the basis of the sliding mode with noise. Phy. Scr. 85, 065006 (2012)CrossRefzbMATHGoogle Scholar
  7. 7.
    Choon, K.: An \(\text{ H }-\infty \) approach to anti-synchronization for chaotic systems. Phys. Lett. A. 373, 1729–1733 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hu, J.; Chen, S.H.; Chen, L.: Adaptive control for anti-synchronization of Chua’s chaotic system. Phys. Lett. A 339(6), 455–460 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Alsawalha, A.: Chaos anti-synchronization of two non-identical chaotic systems with known or fully unknown parameters. Chaos Solitons Fract. 42(3), 1926–1932 (2009)CrossRefzbMATHGoogle Scholar
  10. 10.
    Wang, Z.: Anti-synchronization in two non-identical hyperchaotic systems with known or unknown parameters. Commun. Nonlinear Sci. Numer. Simul. 14(5), 2366–2373 (2009)CrossRefGoogle Scholar
  11. 11.
    Shi, X.R.; Wang, Z.L.: Adaptive added-order anti-synchronization of chaotic systems with fully unknown parameters. Appl. Math. Comp. 215(5), 1711–1717 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Liu, S.; Liu, P.: Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters. Nonlinear Anal. Real World Appl. 12, 3046–3055 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Alsawalha, M.M.; Noorani, M.S.M.: Adaptive increasing-order synchronization and anti-synchronization of chaotic systems with uncertain parameters. Chin. Phys. Lett. 28(11), 110507-1-3 (2011)Google Scholar
  14. 14.
    Sun, J.; Shen, Y.: Adaptive anti-synchronization of chaotic complex systems and chaotic real systems with unknown parameters. J. Vib. Cont. 22(13), 1–12 (2014)MathSciNetGoogle Scholar
  15. 15.
    Hu, T.; Sun, W.: Controlling anti-synchronization between two weighted dynamical networks. Phy. Scr. 87(1), 015001 (2013)CrossRefzbMATHGoogle Scholar
  16. 16.
    Sun, J.; Shen, Y.: Compound combination anti-synchronization of five simplest memristor chaotic systems. Optik 127(20), 9192–9200 (2016)CrossRefGoogle Scholar
  17. 17.
    Khan, A.; Khattar, D.; Prajapati, N.: Multiswitching compound anti-synchronization of four chaotic systems. Pranama 89, 90 (2017). CrossRefGoogle Scholar
  18. 18.
    Yu, F.; Wang, C.: Secure communication based on a four-wing chaotic system subject to disturbance inputs. Optik 125(20), 5920–5925 (2014)CrossRefGoogle Scholar
  19. 19.
    Slotine, K.; Li, W.: Applied Nonlinear Control. Prentice-Hall, Upper Saddle River (1990)zbMATHGoogle Scholar
  20. 20.
    Pang, S.; Liu, Y.: A new hyperchaotic system from the Lu system and its control. J. Comput. Appl. Math. 235, 2775–2789 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yang, C.: Adaptive single input control for synchronization of a 4D Lorenz–Stenflo chaotic system. Arab. J. Sci. Eng. 39(3), 2413–2426 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Elabbasy, E.; Agiza, H.; EI-Dessoky, M.: Adaptive synchronization of a hyperchaotic system with uncertain parameters. Chaos Solitons Fract. 30(5), 1133–1142 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Shi, L.; Yang, X.; Li, Y.; Feng, Z.: Finite-time synchronization of nonidentical chaotic systems with multiple time-varying delays and bounded perturbations. Nonlinear Dyn. 83(1–2), 75–87 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Aghababa, M.P.; Aghababa, H.P.: A novel finite-time sliding mode controller for synchronization of chaotic systems with input nonlinearity. Arab. J. Sci. Eng. 3(11), 3221–3232 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Steele, J.M.: The Cauchy–Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, Cambridge (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Khalil, H.K.: Non-linear Systems, pp. 159–170. Prentice-Hall, Upper Saddle River (2002)Google Scholar
  27. 27.
    Ying-ying, M.; Yun-gang, L.: Barbalat’s Lemma and its application in analysis of system stability. J. Shandong Univ. Tech. 37(1), 51–55 (2007)Google Scholar
  28. 28.
    Liu, Y.: Circuit implementation and finite-time synchronization of 4D Rabinovich hyperchaotic system. Nonlinear Dyn. 67(1), 89–96 (2012)CrossRefzbMATHGoogle Scholar
  29. 29.
    Njah, A.: Synchronization via active control of parametrically and externally excited \(\Phi 6\) Van der Pol and Duffing oscillators and application to secure communications. J. Vib. Control 17(4), 493–504 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Yau, H.T.; Pu, Y.C.; Li, S.C.: Application of a chaotic synchronization system to secure communication. Inf. Technol. Control 41(3), 274–282 (2012)Google Scholar
  31. 31.
    Zhang, L.; An, X.; Zhang, J.: A new chaos synchronization scheme and its application to secure communications. Nonlinear Dyn. 73(1–2), 705–722 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Cuomo, C.M.; Oppenheim, A.V.: Circuit implementation of synchronized chaos with applications to communications. Phys. Rev. Lett. 71(1), 65–68 (1993)CrossRefGoogle Scholar
  33. 33.
    Parlitz, U.; Chua, L.; Kocarev, L.; Halle, K.; Shang, A.: Transmission of digital signals by chaotic synchronization. Int. J. Bifur. Chaos. 2(4), 973–977 (1992)CrossRefzbMATHGoogle Scholar
  34. 34.
    Chen, Y.A.: A new secure communication scheme based on synchronization of chaotic system. Adv. Nat. Comput. 4222, 452–455 (2006)Google Scholar
  35. 35.
    Yeh, J.P.; Wu, K.L.: A simple method to synchronize chaotic systems and its application to secure communications. Math. Comput. Model. 47, 894–902 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Mittal, A.K.; Dwivedi, A.; Dwivedi, S.: Parameter adaptation technique for rapid synchronization and secure communication. Eur. J. Phys. ST 223(8), 1549–1560 (2014)CrossRefGoogle Scholar
  37. 37.
    Sun, J.; Shen, Y.; Yin, Q.; Xu, C.: Compound synchronization of four memristor chaotic oscillator oscillators and secure communication. Chaos 23(1), 013140–013149 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wu, X.; Zhu, C.; Kan, H.: An improved secure communication scheme based passive synchronization of hyperchaotic complex nonlinear system. Appl. Math. Comput. 252(1), 201–214 (2015)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Zhang, M.; Wang, X.; Wei, G.; La, C.-H.: Complete synchronization and generalized synchronization of one-way coupled time-delay systems. Phys. Rev E. 68, 036208 (2003)CrossRefGoogle Scholar
  40. 40.
    Jawaada, W.; Alsawalha, M.M.; Noorani, M.S.M.: Robust active sliding mode anti-synchronization of hyperchaotic systems with uncertainties and external disturbances. Nonlinear Ana RWA 13, 2403–2413 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Electrical and computer EngineeringSultan Qaboos UniversityMasqatOman
  2. 2.Department of General RequirementsCollege of Applied SciencesNizwaOman

Personalised recommendations