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The European Physical Journal B

, Volume 77, Issue 2, pp 257–264 | Cite as

Cluster synchronization in networks of distinct groups of maps

  • W. L. LuEmail author
  • B. Liu
  • T. Chen
Interdisciplinary Physics

Abstract.

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.

Keywords

Couple System Graph Topology Chaos Synchronization Complete Synchronization Maximum Lyapunov Exponent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. Pikovsky, M. Roseblum, J. Kurths, Synchronization: A universal concept in nonlinear sciences (Cambridge University Press, 2001) Google Scholar
  2. 2.
    S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D.-U. Hwang, Phys. Rep. 424, 175 (2006) CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    X.F. Wang, G. Chen, IEEE Circ. Syst. Mag. 3, 6 (2003) CrossRefGoogle Scholar
  4. 4.
    H. Fujisaka, T. Yamada, Prog. Theor. Phys. 69, 32 (1983) zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    H. Fujisaka, T. Yamada, Prog. Theor. Phys. 72, 885 (1984) CrossRefADSGoogle Scholar
  6. 6.
    V.S. Afraimovich, N.N. Verichev, M.I. Rabinovich, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 29, 795 (1986) MathSciNetADSGoogle Scholar
  7. 7.
    S.H. Strogatz, I. Stewart, Sci. Amer. 269, 102 (1993) CrossRefGoogle Scholar
  8. 8.
    L.M. Pecora, T.L. Carroll, Phys. Rev. Lett. 64, 821 (1990) CrossRefMathSciNetADSGoogle Scholar
  9. 9.
    J.F. Heagy, T.L. Carroll, L.M. Pecora, Phys. Rev. E. 50, 1874 (1994) CrossRefADSGoogle Scholar
  10. 10.
    J. Jost, M.P. Joy, Phys. Rev. E 65, 016201 (2001) CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    X.F. Wang, G. Chen, IEEE Trans. Circ. Syst. I 49, 54 (2002) CrossRefGoogle Scholar
  12. 12.
    G. Rangarajan, M. Ding: Phys. Lett. A 296, 204 (2002) zbMATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Y.H. Chen, G. Rangarajan, M. Ding: Phys. Rev. E. 67, 026209 (2003) CrossRefADSGoogle Scholar
  14. 14.
    C.W. Wu, L.O. Chua, IEEE Trans. Circuits Syst. I 42, 430 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    I.V. Belykh, V.N. Belykh, M. Hasler, Physica D 195, 159 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  16. 16.
    I.V. Belykh, V.N. Belykh, M. Hasler, Physica D 195, 188 (2004) zbMATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    J. Cao, P. Li, W. Wang, Phys. Lett. A 353, 318 (2006) CrossRefADSGoogle Scholar
  18. 18.
    W. Lu, T. Chen, Physica D 213, 214 (2006) zbMATHCrossRefMathSciNetADSGoogle Scholar
  19. 19.
    A. Schnitzler, J. Gross, Nat. Rev. Neurosci 6, 285 (2005) CrossRefGoogle Scholar
  20. 20.
    P.R. Chandler, M. Patcher, S. Rasmussen, Proceedings of the American Control Society, 20 (2001) Google Scholar
  21. 21.
    K.M. Passino, IEEE Control Syst. Mag. 22, 52 (2002) CrossRefGoogle Scholar
  22. 22.
    J. Finke, K. Passino, A.G. Sparks, IEEE Control Syst. Mag. 14, 789 (2006) Google Scholar
  23. 23.
    B. Blasius, A. Huppert, L. Stone, Nature (London) 399, 354 (1999) CrossRefADSGoogle Scholar
  24. 24.
    E. Montbrió, J. Kurths, B. Blasius, Phys. Rev. E 70, 056125 (2004) CrossRefADSGoogle Scholar
  25. 25.
    N.F. Rulkov, Chaos 6, 262 (1996) CrossRefMathSciNetADSGoogle Scholar
  26. 26.
    L. Stone, R. Olinky, B. Blasius, A. Huppert, B. Cazelles, Proceedings of the Sixth Experimental Chaos Conference, AIP Conf. Proc. No. 662, (2002), p. 476 Google Scholar
  27. 27.
    E. Jones, B. Browning, M.B. Dias, B. Argall, M. Veloso, A. Stentz, Proceedings IEEE International Conference on Robotics and Automation, Orlando, 570 (2006) Google Scholar
  28. 28.
    K.-S, Hwang, S.-W. Tan, C.-C. Chen, IEEE Trans. Fuzzy Syst. 12, 569 (2004) CrossRefGoogle Scholar
  29. 29.
    V.N. Belykh, I.V. Belykh, M. Hasler, Phys. Rev. E 62, 6332 (2000) CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    V.N. Belykh, I.V. Belykh, E. Mosekilde, Phys. Rev. E 63, 036216 (2001) CrossRefADSGoogle Scholar
  31. 31.
    Z. Ma, Z. Liu, G. Zhang, Chaos 16, 023103 (2006) CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    W. Wu, T. Chen, Physica D 238, 355 (2009) zbMATHCrossRefMathSciNetADSGoogle Scholar
  33. 33.
    W. Wu, W. Zhou, T. Chen, IEEE Trans. Circuits Syst. -I, in press (2008) Google Scholar
  34. 34.
    S. Jalan, R.E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003) CrossRefADSGoogle Scholar
  35. 35.
    S. Jalan, R.E. Amritkar, C.-K. Hu, Phys. Rev. E 72, 016211 (2005) CrossRefADSGoogle Scholar
  36. 36.
    S. Jalan, R.E. Amritkar, C.-K. Hu, Phys. Rev. E 72, 016212 (2005) CrossRefADSGoogle Scholar
  37. 37.
    X. Liu, T. Chen, Physica D 237, 630 (2008) zbMATHCrossRefMathSciNetADSGoogle Scholar
  38. 38.
    F. Sorrentino, E. Ott, Phys. Rev. E 76, 056114 (2007) CrossRefMathSciNetADSGoogle Scholar
  39. 39.
    L. Chen, J. Lu, J. Syst. Sci. Complexity 20, 21 (2008) Google Scholar
  40. 40.
    I.V. Belykh, V.N. Belykh, M. Hasler, Chaos 13, 165 (2003) zbMATHCrossRefMathSciNetADSGoogle Scholar
  41. 41.
    W. Lu, B. Liu, T. Chen, Chaos 20, 013120 (2010) CrossRefADSGoogle Scholar
  42. 42.
    The sense of transverse stability is diverse according to the ergodic measure by which the MLE is computed. For instance, Milnor stability, essential stability, or Lyapunov stability. These can correspond to the diversity of the senses of cluster synchronization we discuss in this paper. To avoid rigorous mathematics, we do not present the details. For interesting readers, we refer to [44] for the details Google Scholar
  43. 43.
    A.-L. Barabási, R. Albert, Science 286, 509 (1999) CrossRefMathSciNetGoogle Scholar
  44. 44.
    P. Ashwin, J. Buescu, I. Stewart, Nonlinearity 9, 703 (1996) zbMATHCrossRefMathSciNetADSGoogle Scholar
  45. 45.
    P.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, New York, 1985) Google Scholar
  46. 46.
    Q.-C. Pham, J.-J. Slotine, Neural Netw. 20, 62 (2007) zbMATHCrossRefGoogle Scholar
  47. 47.
    W. Lohmiller, J.-J. Slotine, Automatica 34, 671 (1998) CrossRefMathSciNetGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Centre for Computational Systems Biology, Fudan UniversityShanghaiP.R. China
  2. 2.Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan UniversityShanghaiP.R. China

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