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Effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network

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Abstract

In this paper, effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network is investigated. On the basis of matrix theory, the threshold values of the coupling strength and the weight parameter for cluster synchronization have been obtained by utilizing the attractiveness criteria of the invariant synchronization manifold. It shows that cluster synchronization bifurcation comes forth, which concept is first put forward. That is to say, via changing the weight parameter and the coupling strength, the purpose of controlling the number of clusters is achieved, which provides a new idea for control the number of clusters in a network. Numerical simulations are given to demonstrate the theoretical results. In addition, the theoretical results and the numerical simulations also show that full synchronization may not be realized even if the network is globally coupled when there are some negative couplings.

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References

  1. Wang, X.F., Chen, G.R.: Complex networks: small-world, scale-free and beyond. IEEE Circuits Syst. Mag. 3, 6–20 (2003)

    Article  Google Scholar 

  2. Arenas, A., Guilera, A.D., Kurths, J., Moreno, Y., Zhou, C.S.: Synchronization in complex networks. Phys. Rep. 469, 93–143 (2008)

    Article  MathSciNet  Google Scholar 

  3. Yu, W.W., Chen, G.R., Lü, J.H.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)

    Article  MATH  Google Scholar 

  4. Chen, J.R., Jiao, L.C., Wu, J.S., Wang, X.D.: Projective synchronization with different scale factors in a driven-response complex network and its application in image encryption. Nonlinear Anal., Real World Appl. 11, 3045–3058 (2010)

    Article  MATH  Google Scholar 

  5. Sun, W., Chen, Z., Lü, J.H., Chen, S.H.: Outer synchronization of complex networks with delay via impulse. Nonlinear Dyn. (2012). doi:10.1007/s11071-012-0383-5

    Google Scholar 

  6. Wang, J.L., Wu, H.N.: Synchronization criteria for impulsive complex dynamical networks with time-varying delay. Nonlinear Dyn. (2012). doi:10.1007/s11071-012-0427-x

    Google Scholar 

  7. Belykh, V.N., Osipov, G.V., Petrov, V.S., Suykens, J.A.K., Vandewalle, J.: Cluster synchronization in oscillatory networks. Chaos 18, 037106 (2008)

    Article  MathSciNet  Google Scholar 

  8. Pogromsky, A.Y.: A partial synchronization theorem. Chaos 18, 037107 (2008)

    Article  MathSciNet  Google Scholar 

  9. Ma, Z.J., Liu, Z.R., Zhang, G.: A new method to realize cluster synchronization in connected chaotic networks. Chaos 16, 023103 (2006)

    Article  MathSciNet  Google Scholar 

  10. Ma, Z.J., Zhang, G., Wang, Y., Liu, Z.R.: Cluster synchronization in star-like complex networks. J. Phys. A, Math. Theor. 41, 155101 (2008)

    Article  MathSciNet  Google Scholar 

  11. Huang, L., Park, K., Lai, Y.C., Yang, L., Yang, K.Q.: Abnormal synchronization in complex clustered networks. Phys. Rev. Lett. 97, 164101 (2006)

    Article  Google Scholar 

  12. Zhou, J., Chen, T.P., Xiang, L.: Adaptive synchronization of coupled chaotic systems based on parameters identification and its applications. Int. J. Bifurc. Chaos 16, 2923–2933 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Belykh, V.N., Belykh, I.V., Hasler, M.: Hierarchy and stability of partially synchronous oscillations of diffusively coupled dynamical systems. Phys. Rev. E 62(5), 6332–6345 (2000)

    Article  MathSciNet  Google Scholar 

  14. Belykh, V.N., Belykh, I.V., Hasler, M.: Connected graph stability method for synchronized coupled chaotic systems. Physica D 195, 159–187 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, B., Lu, W.L., Chen, T.P.: New conditions on synchronization of networks of linearly coupled dynamical systems with non-Lipschitz right-hand sides. Neural Netw. 25, 5–13 (2012)

    Article  Google Scholar 

  16. Wu, W., Chen, T.P.: Partial synchronization in linearly and symmetrically coupled ordinary differential systems. Physica D 238, 355–364 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Belykh, V.N., Belykh, I.V., Nevidin, K., Hasler, M.: Persistent clusters in lattices of coupled nonidentical chaotic systems. Chaos 13, 165–178 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Belykh, V.N., Belykh, I.V., Hasler, M., Nevidin, K.V.: Cluster synchronization in three-dimensional lattices of diffusively coupled oscillators. Int. J. Bifurc. Chaos 13, 755–779 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wang, X.F., Chen, G.R.: Synchronization in small-world dynamical networks. Int. J. Bifurc. Chaos 12, 187–192 (2002)

    Article  Google Scholar 

  20. Wang, X.F., Chen, G.R.: Synchronization in scale-free dynamical networks: robustness and fragility. IEEE Trans. Circuits Syst. I 49, 54–62 (2002)

    Article  Google Scholar 

  21. Lü, J.H., Yu, X.H., Chen, G.R.: Chaos synchronization of general complex dynamical networks. Physica A 334, 281–302 (2004)

    Article  MathSciNet  Google Scholar 

  22. Golubitsky, M., Stewart, I., Török, A.: Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4(1), 78–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Golubitsky, M., Stewart, I.: Nonlinear dynamics of networks: the groupoid formalism. Bull. Am. Math. Soc. 43(3), 305–364 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Shil’nikov, L.P.: Chua’s circuit: rigorous results and future problems. Int. J. Bifurc. Chaos 4, 784–786 (1993)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 60964006, 61004101, and 11162004), the Fund of Guangxi Experiment Center of Information Science and the Science Foundation of Guangxi Province, China (No. 2013GXNSFAA019006). The authors would also like to thank the anonymous referees for their invaluable comments and suggestions.

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Authors and Affiliations

  1. Guangxi Experiment Center of Information Science, Guilin University of Electronic Technology, Guilin, 541004, China

    Zhongjun Ma, Guirong Jiang & Kezan Li

  2. School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, 541004, China

    Zhongjun Ma, Suzhang Zhang, Guirong Jiang & Kezan Li

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  1. Zhongjun Ma
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  2. Suzhang Zhang
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  3. Guirong Jiang
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Correspondence to Zhongjun Ma.

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Ma, Z., Zhang, S., Jiang, G. et al. Effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network. Nonlinear Dyn 74, 55–64 (2013). https://doi.org/10.1007/s11071-013-0948-y

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