Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 845–854 | Cite as

Multi-switching combination synchronization of chaotic systems

  • U. E. Vincent
  • A. O. Saseyi
  • P. V. E. McClintock
Original Paper


A novel synchronization scheme is proposed for a class of chaotic systems, extending the concept of multi-switching synchronization to combination synchronization such that the state variables of two or more driving systems synchronize with different state variables of the response system, simultaneously. The new scheme, multi-switching combination synchronization (MSCS), represents a significant extension of earlier multi-switching schemes in which two chaotic systems, in a driver-response configuration, are multi-switched to synchronize up to a scaling factor. In MSCS, the chaotic driving systems multi-switch a response chaotic system in combination synchronization. For certain choices of the scaling factors, MSCS reduces to multi-switching synchronization, implying that the latter is a special case of MSCS. A theoretical approach to control design, based on backstepping, is presented and validated using numerical simulations.


Multi-switching Combination Synchronization Chaos Backstepping 



UEV is supported by the Royal Society of London through their Newton International Fellowship Alumni scheme. We acknowledge and thank all the reviewers for their constructive and critical comments that were very useful for improving the quality of this paper.


  1. 1.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64, 821–824 (1990)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization: A Universal Concept in Non-Linear Sciences. Cambridge University Press, United Kingdom (2001)CrossRefGoogle Scholar
  3. 3.
    Eisencraft, M., Fanganiello, R.D., Grzybowski, J.M.V., Soriano, D.C., Attux, R., Batista, A.M., Macau, E.E.N., Monteiro, L.H.A., Romano, J.M.T., Suyama, R., Yoneyama, T.: Chaos-based communication systems in non-ideal channels. Commun. Nonlinear Sci. Numer. Simul. 17(12), 4707–4718 (2012)CrossRefGoogle Scholar
  4. 4.
    Ren, H.-P., Baptista, M.S., Grebogi, C.: Wireless communication with chaos. Phys. Rev. Lett. 110, 184101 (2013)CrossRefGoogle Scholar
  5. 5.
    Aguilar-López, R., Martnez-Guerra, R., Perez-Pinacho, C.: Nonlinear observer for synchronization of chaotic systems with application to secure data transmission. Er. Phys. J. Spec. Top. 223, 1541–1548 (2014)CrossRefGoogle Scholar
  6. 6.
    Filali, R.L., Benrejeb, M., Borne, P.: On observer-based secure communication design using discrete-time hyperchaotic systems. Commun. Nonlinear Sci. Numer. Sim. 19(5), 1424–1432 (2014)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Mainieri, R., Rehacek, J.: Projective synchronization in three-dimensional chaotic systems. Phys. Rev. Lett. 82, 3042–3045 (1999)CrossRefGoogle Scholar
  8. 8.
    Li, G.-H.: Projective synchronization of chaotic system using backstepping control. Chaos Solitons & Fractals 29, 490–494 (2006)CrossRefMATHGoogle Scholar
  9. 9.
    Shi, X., Wang, Z.: Projective synchronization of chaotic systems with different dimensions via backstepping design. Int. J. Nonlinear Sci. 7(3), 301–306 (2009)MATHMathSciNetGoogle Scholar
  10. 10.
    Park, J.H.: Further results on functional projective synchronization of genesio-tesi chaotic system. Mod. Phys. Lett. B 23, 1889–1895 (2009)CrossRefMATHGoogle Scholar
  11. 11.
    Zhou, P., Zhu, W.: Function projective synchronization for fractional-order chaotic systems. Nonlinear Anal. Real World Appl. 12(2), 811–816 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Farivar, F., Shoorehdeli, M.A., Nekoui, M.A., Teshnehlab, M.: Generalized projective synchronization of uncertain chaotic systems with external disturbance. Expert Syst. Appl. 38(5), 4714–4726 (2011)CrossRefGoogle Scholar
  13. 13.
    Wu, Z., Duan, J., Fu, X.: Complex projective synchronization in coupled chaotic complex dynamical systems. Nonlinear Dyn. 69, 771–779 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Si, G., Sun, Z., Zhang, Y., Chen, W.: Projective synchronization of different fractional-order chaotic systems with non-identical orders. Nonlinear Anal. Real World Appl. 13(4), 1761–1771 (2012)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Farivar, F., Shoorehdeli, M.A., Nekoui, M.A., Teshnehlab, M.: Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via gaussian radial basis adaptive variable structure control. Chaos Solitons & Fractals 45(1), 80–97 (2012)CrossRefMATHGoogle Scholar
  16. 16.
    Dai, H., Si, G., Jia, L., Zhang, Y.: Adaptive generalized function matrix projective lag synchronization between fractional-order and integer-order complex networks with delayed coupling and different dimensions. Phys. Scr. 055006(88), 1–9 (2013)Google Scholar
  17. 17.
    Wu, X., Nie, Z.: Complex projective synchronization in drive-response stochastic complex networks by impulsive pinning control. Discrete Dyn. Nat. Soc. 965297, 1–8 (2014)MathSciNetGoogle Scholar
  18. 18.
    Kuetche-Mbe, E.S., Fotsin, H.B., Kengne, J., Woafo, P.: Parameters estimation based adaptive generalized projective synchronization (GPS) of chaotic Chua’s circuit with application to chaos communication by parametric modulation. Chaos Solitons & Fractals 61, 27–37 (2014)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Wang, S., Yu, Y.G., Wen, G.G.: Hybrid projective synchronization of time-delayed fractional order chaotic systems. Nonlinear Anal. Hybrid Syst. 11, 129–138 (2014)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Luo, R.Z., Wang, Y.L., Deng, S.C.: Combination synchronization of three classic chaotic systems using active backstepping design. Chaos 21(4), 043114 (2011)CrossRefGoogle Scholar
  21. 21.
    Wu, A.: Hyperchaos synchronization of memristor oscillator system via combination scheme. Adv. Differ. Equ. 2014, 86–96 (2014)CrossRefGoogle Scholar
  22. 22.
    Runzi, L., Yinglan, W.: Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication. Chaos 22, 023109 (2013)CrossRefGoogle Scholar
  23. 23.
    Wu, Z., Fu, X.: Combination synchronization of three different order nonlinear systems using active backstepping design. Nonlinear Dyn. 73, 1863–1872 (2013)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Sun, J.W., Shen, Y., Zhang, G.D., Xu, C.J., Cui, G.Z.: Combination-combination synchronization among four identical or different chaotic systems. Nonlinear Dyn. 73(3), 1211–1222 (2013)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Lin, H., Cai, J., Wang, J.: Finite-time combination-combination synchronization for hyperchaotic systems. J. Chaos 304643, 1–7 (2013)CrossRefGoogle Scholar
  26. 26.
    Zhou, X., Xiong, L., Cai, X.: Combination-combination synchronization of four nonlinear complex chaotic systems. Abstr. Appl. Anal. 953265, 1–14 (2014)Google Scholar
  27. 27.
    Sun, J., Shen, Y., Wang, X., Chen, J.: Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control. Nonlinear Dyn. 76, 383–397 (2014)Google Scholar
  28. 28.
    Sun, J., Shen, Y., Yi, Q., Xu, C.: Compound synchronization of four memristor chaotic oscillator systems and secure communication. Chaos 23, 013140 (2013)CrossRefGoogle Scholar
  29. 29.
    Wu, A., Zhang, J.: Compound synchronization of fourth-order memristor oscillator. Adv. Differ. Equ. 2014, 100–106 (2014)CrossRefGoogle Scholar
  30. 30.
    Zhang, B., Deng, F.: Double-compound synchronization of six memristor-based Lorenz systems. Nonlinear Dyn., In press:1–12, (2014)Google Scholar
  31. 31.
    Ojo, K.S., Njah, A.N., Olusola, O.I., Omeike, M.O.: Reduced order projective and hybrid projective combination-combination synchronization of four chaotic Josephson junctions. J. Chaos 282407, 1–9 (2014)CrossRefGoogle Scholar
  32. 32.
    Ojo, K.S., Njah, A.N., Olusola, O.I., Omeike, M.O.: Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique. Nonlinear Dyn. 77, 583–595 (2014)CrossRefMathSciNetGoogle Scholar
  33. 33.
    Ucar, A., Lonngren, K.E., Bai, E.W.: Multi-switching synchronization of chaotic systems with active controllers. Chaos Solitons & Fractals 38, 254–262 (2008)CrossRefGoogle Scholar
  34. 34.
    Sebastian Sudheer, K., Sabir, M.: Switched modified function projective synchronization of hyperchaotic Qi system with uncertain parameters. Commun. Nonlinear Sci. Numer. Simul. 15, 4058–4064 (2010)CrossRefGoogle Scholar
  35. 35.
    Wang, X.-Y., Sun, P.: Multi-switching synchronization of chaotic system with adaptive controllers and unknown parameters. Nonlinear Dyn. 63(4), 599–609 (2011)CrossRefGoogle Scholar
  36. 36.
    Li, H.-M., Li, C.-L.: Switched generalized function projective synchronization of two identical/different hyperchaotic systems with uncertain parameters. Phys. Scr. 045008(86), 1–8 (2012)Google Scholar
  37. 37.
    Yu, F., Wang, C.H., Wan, Q.Z., Hu, Y.: Complete switched modified function projective synchronization of a five-term chaotic system with uncertain parameters and disturbances. Pramana 80(2), 223–235 (2013)CrossRefGoogle Scholar
  38. 38.
    Zhou, X., Xiong, L., Cai, X.: Adaptive switched generalized function projective synchronization between two hyperchaotic systems with unknown parameters. Entropy 16, 377–388 (2014)CrossRefMathSciNetGoogle Scholar
  39. 39.
    Ajayi, A.A., Ojo, K.S., Vincent, U.E., Njah, A.N.: Multiswitching synchronization of a driven hyperchaotic circuit using active backstepping. J. Nonlinear Dyn. 918586, 1–10 (2014)CrossRefGoogle Scholar
  40. 40.
    Radwan, A.G., Moaddy, K., Salama, K.N., Momani, S., Hashim, I.: Control and switching synchronization of fractional order chaotic systems using active control technique. J. Adv. Res. 5(1), 125–132 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • U. E. Vincent
    • 1
    • 2
  • A. O. Saseyi
    • 1
  • P. V. E. McClintock
    • 2
  1. 1.Department of Physical SciencesRedeemer’s UniversityRedemption CityNigeria
  2. 2.Department of PhysicsLancaster UniversityLancasterUK

Personalised recommendations