Nonlinear Dynamics

, Volume 92, Issue 3, pp 905–921 | Cite as

Adaptive exponential cluster synchronization in colored community networks via aperiodically intermittent pinning control

  • Peipei Zhou
  • Shuiming Cai
  • Jianwei Shen
  • Zengrong Liu
Original Paper

Abstract

This paper investigates the problem of pinning cluster synchronization for colored community networks via adaptive aperiodically intermittent control. Firstly, a general colored community network model is proposed, where the isolated nodes can interact through different kinds of connections in different communities and the interactions between different pair of communities can also be different, and moreover, the nodes in different communities can have different state dimensions. Then, an adaptive aperiodically intermittent control strategy combined with pinning scheme is developed to realize cluster synchronization of such colored community network. By introducing a novel piecewise continuous auxiliary function, some globally exponential cluster synchronization criteria are rigorously derived according to Lyapunov stability theory and piecewise analysis approach. Based on the derived criteria, a guideline to illustrate which nodes in each community should be preferentially pinned is given. It is noted that the adaptive intermittent pinning control is aperiodic, in which both control width and control period are allowed to be variable. Finally, a numerical example is provided to show the effectiveness of the theoretical results obtained.

Keywords

Exponential cluster synchronization Colored community network Nodes of different state dimensions Adaptive aperiodically intermittent control Pinning scheme 

Notes

Acknowledgements

The authors thank the editor and anonymous reviewers for their valuable comments and suggestions which have helped to improve the quality of the paper.

References

  1. 1.
    Girvan, M., Newman, M.E.J.: Community structure in social and biological networks. Proc. Natl. Acad. Sci. USA 99, 7821–7826 (2002)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Zhang, Y., Friend, A.J., Traud, A.L., Porter, M.A., Fowler, J.H., Mucha, P.J.: Community structure in Congressional cosponsorship networks. Physica A 387, 1705–1712 (2008)CrossRefGoogle Scholar
  3. 3.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486, 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Wan, X., Cai, S., Zhou, J., Liu, Z.: Emergence of modularity and disassortativity in protein–protein interaction networks. Chaos 20, 045113 (2010)CrossRefGoogle Scholar
  5. 5.
    Newman, M.E.J.: The structure and function of complex networks. SIAM Rev. 45, 167–256 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Wang, K., Fu, X., Li, K.: Cluster synchronization in community networks with nonidentical nodes. Chaos 19, 023106 (2009)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Wu, Z.: Cluster synchronization in colored community network with different order node dynamics. Commun. Nonlinear Sci. Numer. Simul. 19, 1079–108 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yang, L., Jiang, J., Liu, X.: Cluster synchronization in community network with hybrid coupling. Chaos Solitons Fract. 86, 82–91 (2016)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Boccaletti, S., Kurths, J., Osipov, G., Valladares, D.L., Zhou, C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Arenas, A., Diaz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chen, G., Wang, X., Li, X.: Introduction to Complex Networks: Models, Structure and Dynamics. High Education Press, Beijing (2012)Google Scholar
  12. 12.
    Li, C., Chen, G.: Phase synchronization in small-world networks of chaotic oscillators. Physica A 341, 73–79 (2004)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Liu, H., Sun, W., Al-mahbashi, G.: Parameter identification based on lag synchronization via hybrid feedback control in uncertain drive-response dynamical networks. Adv. Differ. Eq. 2017, 122 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hu, A., Xu, Z., Guo, L.: The existence of generalized synchronization of chaotic systems in complex networks. Chaos 20, 013112 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Zheng, S., Bi, Q., Cai, G.: Adaptive projective synchronization in complex networks with time-varying coupling delay. Phys. Lett. A 373, 1553–1559 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Ma, Z., Liu, Z., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos 16, 023103 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Schnitzler, A., Gross, J.: Normal and pathological oscillatory communication in the brain. Nat. Rev. Neurosci. 6, 285–296 (2005)CrossRefGoogle Scholar
  18. 18.
    Rulkov, N.F.: Images of synchronized chaos: experiments with circuits. Chaos 6, 262–279 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kaneko, K.: Relevance of dynamic clustering to biological networks. Physica D 75, 55–73 (1994)CrossRefMATHGoogle Scholar
  20. 20.
    Cao, J., Li, L.: Cluster synchronization in an array of hybrid coupled neural networks with delay. Neural Netw. 22, 335–342 (2009)CrossRefMATHGoogle Scholar
  21. 21.
    Cai, S., Li, X., Jia, Q., Liu, Z.: Exponential cluster synchronization of hybrid-coupled impulsive delayed dynamical networks: average impulsive interval approach. Nonlinear Dyn. 85, 2405–2423 (2016)CrossRefMATHGoogle Scholar
  22. 22.
    Cai, S., He, Q., Hao, J., Liu, Z.: Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes. Phys. Lett. A 374, 2539–2550 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Wang, X., Chen, G.: Pinning control of scale-free dynamical networks. Physica A 310, 521C531 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Chen, T., Liu, X., Lu, W.: Pinning complex networks by a single controller. IEEE Trans. Circuits Syst. I(54), 1317–1326 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lu, J., Ho, D.W.C.: Globally exponential synchronization and synchronizability for general dynamical networks. IEEE Trans. Syst. Man Cybern. B 40, 350–361 (2010)CrossRefGoogle Scholar
  26. 26.
    Song, Q., Cao, J.: On pinning synchronization of directed and undirected complex dynamical networks. IEEE Trans. Circuits Syst. I(57), 672–680 (2010)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Yu, W., Chen, G., Lü, J., Kurths, J.: Synchronization via pinning control on general complex networks. SIAM J. Control Optim. 51, 1395–1416 (2013)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Lu, J., Zhong, J., Huang, C., Cao, J.: On pinning controllability of Boolean control networks. IEEE Trans. Automat. Control 61, 1658–1663 (2016)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zhou, J., Lu, J., Lü, J.: Pinning adaptive synchronization of a general complex dynamical network. Automatica 44, 996–1003 (2008)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhou, J., Wu, Q., Xiang, L.: Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn. 69, 1393–1403 (2012)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lu, J., Kurths, J., Cao, J., Mahdavi, N., Huang, C.: Synchronization control for nonlinear stochastic dynamical networks: pinning impulsive strategy. IEEE Trans. Neural Netw. 23, 285–292 (2012)CrossRefGoogle Scholar
  32. 32.
    Lu, J., Ding, C., Lou, J., Cao, J.: Outer synchronization of partially coupled dynamical networks via pinning impulsive controllers. J. Frankl. Inst. 352, 5024–5041 (2015)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li, Y.: Impulsive synchronization of stochastic neural networks via controlling partial states. Neural Process Lett. 46, 59–69 (2017)CrossRefGoogle Scholar
  34. 34.
    Xia, W., Cao, J.: Pinning synchronization of delayed dynamical networks via periodically intermittent control. Chaos 19, 013120 (2009)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Cai, S., Zhou, P., Liu, Z.: Pinning synchronization of hybrid-coupled directed delayed dynamical network via intermittent control. Chaos 24, 033102 (2014)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Fan, Y., Liu, H., Zhu, Y., Mei, J.: Fast synchronization of complex dynamical networks with time-varying delay via periodically intermittent control. Neurocomputing 205, 182–194 (2016)CrossRefGoogle Scholar
  37. 37.
    Wu, W., Zhou, W., Chen, T.: Cluster synchronization of linearly coupled complex networks under pinning control. IEEE Trans. Circuits Syst. I 56, 829–839 (2009)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Hu, C., Jiang, H.: Cluster synchronization for directed community networks via pinning partial schemes. Chaos Solitons Fract. 45, 1368–1377 (2012)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wang, J., Feng, J., Yu, C., Zhao, Y.: Cluster synchronization of nonlinearly-coupled complex networks with nonidentical nodes and asymmetrical coupling matrix. Nonlinear Dyn. 67, 1635–1646 (2012)MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Su, H., Rong, Z., Chen, M.Z.Q., Wang, X., Chen, G., Wang, H.: Decentralized adaptive pinning control for cluster synchronization of complex dynamical networks. IEEE Trans. Cybern. 43, 394–399 (2013)CrossRefGoogle Scholar
  41. 41.
    Wu, Z., Fu, X.: Cluster synchronization in community networks with nonidentical nodes via edge-based adaptive pinning control. J. Frankl. Inst. 351, 1372–1385 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Cai, S., Hao, J., He, Q., Liu, Z.: New results on synchronization of chaotic systems with time-varying delays via intermittent control. Nonlinear Dyn. 67, 393–402 (2012)CrossRefMATHGoogle Scholar
  43. 43.
    Song, Q., Huang, T.: Stabilization and synchronization of chaotic systems with mixed time-varying delays via intermittent control with non-fixed both control period and control width. Neurocomputing 154, 61–69 (2015)CrossRefGoogle Scholar
  44. 44.
    Liu, X., Chen, T.: Cluster synchronization in directed networks via intermittent pinning control. IEEE Trans. Neural Netw. 22, 1009–1020 (2011)CrossRefGoogle Scholar
  45. 45.
    Hu, C., Jiang, H.: Pinning synchronization for directed networks with node balance via adaptive intermittent control. Nonlinear Dyn. 80, 295–307 (2015)CrossRefMATHGoogle Scholar
  46. 46.
    Cai, S., Jia, Q., Liu, Z.: Cluster synchronization for directed heterogeneous dynamical networks via decentralized adaptive intermittent pinning control. Nonlinear Dyn. 82, 689–702 (2015)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Cai, S., Zhou, P., Liu, Z.: Intermittent pinning control for cluster synchronization of delayed heterogeneous dynamical networks. Nonlinear Anal. Hybrid Syst. 18, 134–155 (2015)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Liu, X., Chen, T.: Synchronization of nonlinear coupled networks via aperiodically intermittent pinning control. IEEE Trans. Neural Netw. Learn. 26, 113–126 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Liu, X., Chen, T.: Synchronization of linearly coupled networks with delays via aperiodically intermittent pinning control. IEEE Trans. Neural Netw. Learn. 26, 2396–2407 (2015)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Liu, M., Jiang, H., Hu, C.: Synchronization of hybrid-coupled delayed dynamical networks via aperiodically intermittent pinning control. J. Frankl. Inst. 353, 2722–2742 (2016)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Cai, S., Lei, X., Liu, Z.: Outer synchronization between two hybrid-coupled delayed dynamical networks via aperiodically adaptive intermittent pinning control. Complexity 21, 593–605 (2016)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Lei, X., Cai, S., Jiang, S., Liu, Z.: Adaptive outer synchronization between two complex delayed dynamical networks via aperiodically intermittent pinning control. Neurocomputing 222, 26–35 (2017)CrossRefGoogle Scholar
  53. 53.
    Zhou, P., Cai, S.: Pinning synchronization of complex directed dynamical networks under decentralized adaptive strategy for aperiodically intermittent control. Nonlinear Dyn. 90, 287–299 (2017)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Stefanovska, A., Haken, H., McClintock, P.V.E., Hoz̆ic̆, M., Bajrović, F., Ribaric̆, S.: Reversible transitions between synchronization states of the cardiorespiratory system. Phys. Rev. Lett. 85, 4831–4834 (2000)CrossRefGoogle Scholar
  55. 55.
    Wu, Z., Xu, X., Chen, G., Fu, X.: Generalized matrix projective synchronization of general colored networks with different-dimensional node dynamics. J. Frankl. Inst. 351, 4584–4595 (2014)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Tan, M., Pan, Q., Zhou, X.: Adaptive stabilization and synchronization of non-diffusively coupled complex networks with nonidentical nodes of different dimensions. Nonlinear Dyn. 85, 303–316 (2016)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Li, Y., Tang, W.K.S., Chen, G.: Generating hyperchaos via state feedback control. Int. J. Bifurc. Chaos 15, 3367–3375 (2005)CrossRefGoogle Scholar
  58. 58.
    Lellis, P., Bernardo, M., Garofalo, F.: Synchronization of complex networks through local adaptive coupling. Chaos 18, 037110 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  • Peipei Zhou
    • 1
  • Shuiming Cai
    • 1
  • Jianwei Shen
    • 2
  • Zengrong Liu
    • 3
  1. 1.Faculty of ScienceJiangsu UniversityZhenjiangChina
  2. 2.Department of MathematicsXuchang UniversityXuchangChina
  3. 3.Department of MathematicsYunnan Normal UniversityKunmingPR China

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