Nonlinear Dynamics

, Volume 87, Issue 1, pp 271–279 | Cite as

Spatiotemporal combination synchronization of different nonlinear objects

  • Yuan Chai
  • Shuang Liu
  • Qingyun Wang
Original Paper


In this paper, a kind of spatiotemporal combination synchronization between two drive systems (networks) and one spatiotemporal response system is investigated by means of active sliding mode design. In particular, we divide the spatiotemporal spaces into two parts, which are loaded in networks and systems, respectively. According to active sliding mode design method, we design the controllers, which are added to different regions of the spatiotemporal spaces. Then, the analytic result of spatiotemporal combination synchronization is obtained. Finally, some numerical results are established to illustrate the obtained analytic results.


Spatiotemporal combination synchronization Active sliding mode control Spatiotemporal chaos Complex network 



This research was supported by the National Science Foundation of China (Grants11502139) and the Shanghai Universities Young Teacher Training Scheme (GrantsZZsdl15116).


  1. 1.
    Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Zhong, J., Lu, J., Liu, Y., Cao, J.: Synchronization in an array of output-coupled Boolean networks with time delay. IEEE Trans. Neural Netw. Learn. Syst. 25(12), 2288–2294 (2014)CrossRefGoogle Scholar
  3. 3.
    Lu, R., Yu, W., Lu, J., Xue, A.: Synchronization on complex networks of networks. IEEE Trans. Neural Netw. Learn. Syst. 25(11), 2110–2118 (2014)CrossRefGoogle Scholar
  4. 4.
    Fang, C.M.: Synchronization for complex dynamical networks with time delay and discrete-time information. Appl. Math. Comput. 258, 1–11 (2015)MathSciNetMATHGoogle Scholar
  5. 5.
    Xu, Y., Lu, R., Peng, H., Xie, K., Xue, A. (2015) Asynchronous dissipative state estimation for stochastic complex networks with quantized jumping coupling and uncertain measurements. IEEE Trans. Neural Netw. Learn. Syst. doi: 10.1109/TNNLS.2015.2503772
  6. 6.
    Lu, J.Q., Ding, C.D., Lou, J.G., Cao, J.D.: Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. J. Frankl. Inst. 352(11), 5024–5041 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Landsman, A.S., Schwartz, I.B.: Complete chaotic synchronization in mutually coupled time-delay systems. Phys. Rev. E 75(2), 026201 (2007)CrossRefGoogle Scholar
  8. 8.
    Li, C., Chen, G.: Phase synchronization in small-world networks of chaotic oscillators. Phys. A 341(1), 73–79 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kacarev, L., Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76(11), 1816–1819 (1996)CrossRefGoogle Scholar
  10. 10.
    Wang, X., Wang, M.: Projective synchronization of nonlinear-coupled spatiotemporal chaotic systems. Nonlinear Dyn. 62(3), 567–571 (2010)CrossRefGoogle Scholar
  11. 11.
    Du, H., Zeng, Q., Lü, N.: A general method for modified function projective lag synchronization in chaotic systems. Phys. Lett. A 374(13–14), 1493–1496 (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Bhalekara, S., Daftardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control. Commun. Nonlinear Sci. Numer. Simul. 15(11), 3536–3546 (2010)CrossRefMATHGoogle Scholar
  13. 13.
    Lin, D., Wang, X., Nian, F., Zhang, Y.: Dynamic fuzzy neural networks modeling and adaptive backstepping tracking control of uncertain chaotic systems. Neurocomputing 73(16–18), 2873–2881 (2010)CrossRefGoogle Scholar
  14. 14.
    Lu, J., Cao, J.: Adaptive synchronization of uncertain dynamical networks with delayed coupling. Nonlinear Dyn. 53(1), 107–115 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Bowong, S., Kakmeni, F.M.M., Tchawoua, C.: Controlled synchronization of chaotic systems with uncertainties via a sliding mode control design. Phys. Rev. E 70(6), 066217 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Liu, S., Chen, L.Q.: Second-order terminal sliding mode control for networks synchronization. Nonlinear Dyn. 79(1), 205–213 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sun, J., Shen, Y., Zhang, G., Xu, C., Cui, G.: Combination-combination synchronization among four identical or different chaotic systems. Nonlinear Dyn. 73(3), 1211–1222 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Wu, Z., Fu, X.: Combination synchronization of three different order nonlinear systems using active backstepping design. Nonlinear Dyn. 73(3), 1863–1872 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Luo, R., Wang, Y., Deng, S.: Combination synchronization of three classic chaotic systems using active backstepping design. Chaos 21(4), 043114 (2011)CrossRefMATHGoogle Scholar
  20. 20.
    Zhang, Y.Q., Wang, X.Y.: A symmetric image encryption algorithm based on mixed linear-nonlinear coupled map lattice. Inf. Sci. 273, 329–351 (2014)CrossRefGoogle Scholar
  21. 21.
    Zhang, Y.Q., Wang, X.Y.: A new image encryption algorithm based on non-adjacent coupled map lattices. Appl. Soft Comput. 26, 10–20 (2015)CrossRefGoogle Scholar
  22. 22.
    Wang, Y., Zhang, H., Wang, X.: Networked synchronization control of coupled dynamic networks with time-varying delay. IEEE Trans. Syst. Man Cybern. B Cybern. 40(6), 1468–1479 (2010)CrossRefGoogle Scholar
  23. 23.
    Lin, D., Wang, X.: Observer-based decentralized fuzzy neural sliding mode control for interconnected unknown chaotic systems via network structure adaptation. Fuzzy Sets Syst. 161(15), 2066–2080 (2010)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Grossberg, S.: From brain synapses to systems for learning and memory: object recognition, spatial navigation, timed conditioning, and movement control. Brain Res. 1621(24), 270–293 (2015)CrossRefGoogle Scholar
  25. 25.
    Hayakawa, H., Samura, T., Kamijo, T.C., Sakai, Y., Aihara, T.: Spatial information enhanced by non-spatial information in hippocampal granule cells. Cogn. Neurodyn. 9(1), 1–12 (2015)Google Scholar
  26. 26.
    Smaoui, N., Karouma, A., Zribi, M.: Secure communications based on the synchronization of the hyperchaotic Chen and the unified chaotic systems. Commun. Nonlinear Sci. Numer. Simul. 16(8), 3279–3293 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wang, X.Y., Song, J.M.: Synchronization of the fractional order hyperchaos Lorenz systems with activation feedback control. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3351–3357 (2009)CrossRefMATHGoogle Scholar
  28. 28.
    Wang, X., He, Y.: Projective synchronization of fractional order chaotic system based on linear separation. Phys. Lett. A 372(4), 435–441 (2008)CrossRefMATHGoogle Scholar
  29. 29.
    Wang, X., Zhang, X., Ma, C.: Modified projective synchronization of fractional-order chaotic systems via active sliding mode control. Nonlinear Dyn. 69(1), 511–517 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Cai, N., Jing, Y., Zhang, S.: Modified projective synchronization of chaotic systems with disturbances via active sliding mode control. Commun. Nonlinear Sci. Numer. Simul. 15(6), 1613–1620 (2010)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lynch, D.T.: Chaotic behavior of reaction systems: mixed cubic and quadratic autocatalysis. Chem. Eng. Sci. 47(17–18), 4435–4444 (1992)CrossRefGoogle Scholar
  32. 32.
    Lorenz, E.: Deterministic nonperiodic flow. J. Atmos. Sci. 20(2), 130–141 (1963)CrossRefGoogle Scholar
  33. 33.
    Genesio, R., Tesi, A.: Harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 28(3), 531–548 (1992)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Dynamics and ControlBeihang UniversityBeijingChina
  2. 2.School of Mathematics and PhysicsShanghai University of Electric PowerShanghaiChina
  3. 3.School of Mechanical EngineeringShanghai Institute of TechnologyShanghaiChina

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