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Theoretical analysis of synchronization in delayed complex dynamical networks with discontinuous coupling

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Abstract

In this paper, we study the synchronization problem of time-delayed complex networks with discontinuous coupling. Sufficient conditions for the synchronization are obtained based on the stability theory of differential equations. The theoretical results show that the time-delayed networks can achieve synchronization even if the coupling is switched off sometimes. We find analytically that the speed of synchronization is proportional to the on–off rate of the coupling and the algebraic connectivity of networks. Finally, the analytical results are confirmed by numerical simulations.

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References

  1. Arenas, A., Daz-Guilera, A., Kurths, J., Moreno, Y., Zhou, C.: Synchronization in complex networks. Phys. Rep. 469, 93–153 (2008)

    Article  MathSciNet  Google Scholar 

  2. Pecora, L.M., Carroll, T.L.: Master stability functions for synchronized coupled systems. Phys. Rev. Lett. 80, 2109 (1998)

    Article  Google Scholar 

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

    Article  Google Scholar 

  4. Zhao, M., Zhou, T., Wang, B.-H., Wang, W.-X.: Enhanced synchronizability by structural perturbations. Phys. Rev. E 72, 057102 (2005)

    Article  Google Scholar 

  5. Zhou, C., Motter, A.E., Kurths, J.: Universality in the synchronization of weighted random networks. Phys. Rev. Lett. 96, 034101 (2006)

    Article  Google Scholar 

  6. Duan, Z., Chen, G., Huang, L.: Complex network synchronizability: analysis and control. Phys. Rev. E 76, 056103 (2007)

    Article  Google Scholar 

  7. Nishikawa, T., Motter, A.E., Lai, Y.-C., Hoppensteadt, F.C.: Heterogeneity in oscillator networks: are smaller worlds easier to synchronize? Phys. Rev. Lett. 91, 014101 (2003)

    Article  Google Scholar 

  8. Yan, G., Ren, J., Lai, Y.-C., Lai, C.-H., Li, B.: Controlling complex networks: how much energy is needed? Phys. Rev. Lett. 108, 218703 (2012)

    Article  Google Scholar 

  9. Nagail, K.H., Kori, H.: Noise-induced synchronization of a large population of globally coupled nonidentical oscillators. Phys. Rev. E 81, 065202(R) (2010)

    Article  Google Scholar 

  10. Sun, Y., Zhao, D.: Effects of noise on the outer synchronization of two unidirectionally coupled complex dynamical networks. Chaos 22, 023131 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Sun, Y., Shi, H., Bakare, E.A., Meng, Q.: Noise-induced outer synchronization between two different complex dynamical networks. Nonlinear Dyn. 76, 519–528 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lu, J., Ho, D.W.C., Cao, J., Kurths, J.: Single impulsive controller for globally exponential synchronization of dynamical networks. Nonlinear Anal. Real World Appl. 14, 581–593 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Sun, W., Chen, Z., Lü, J., Chen, S.: Outer synchronization of complex networks with delay via impulse. Nonlinear Dyn. 69, 1751–1764 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, X., Wang, X., Chen, G.: Pinning a complex dynamical network to its equilibrium. IEEE Trans. Circ. Syst. I 51, 2074–2074 (2004)

    Article  MathSciNet  Google Scholar 

  15. Yu, W., Chen, G., Lü, J., Kurths, J.: Synchronization via pinning control on general complex networks. SIAM J. Control Optim. 51, 1395–1416 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jia, Z., Fu, X., Deng, G., Li, K.: Group synchronization in complex dynamical networks with different types of oscillators and adaptive coupling schemes. Commun. Nonlinear Sci. Numer. Simul. 18, 2752–2760 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zheng, S.: Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling. Nonlinear Dyn. 67, 2621–2630 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Xu, C., Sun, W., Xu, J., Kurths, J.: Outer synchronization of coupled discrete-time networks. Chaos 19, 013106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Asheghan, M.M., Miguez, J., Hamidi-Beheshti, M.T., Tavazoei, M.S.: Robust outer synchronization between two complex networks with fractional order dynamics. Chaos 21, 033121 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sun, Y., Li, W., Zhao, D.: Finite-time stochastic outer synchronization between two complex dynamical networks with different topologies. Chaos 22, 023152 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yang, X., Wu, Z., Cao, J.: Finite-time synchronization of complex networks with nonidentical discontinuous nodes. Nonlinear Dyn. 73, 2313–2327 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sun, Y., Li, W., Zhao, D.: Convergence time and speed of multi-agent systems in noisy environments. Chaos 22, 043126 (2012)

    Article  MathSciNet  Google Scholar 

  23. Xu, S., Lam, J.: A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39, 1095–1113 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin, W., Pu, Y., Guo, Y., Kurths, J.: Oscillation suppression and synchronization: frequencies determine the role of control with time delays. Europhys. Lett. 102, 20003 (2013)

    Article  Google Scholar 

  25. Sun, Y., Lin, W., Erban, R.: Time delay can facilitate coherence in self-driven interacting-particle systems. Phys. Rev. E 90, 062708 (2014)

    Article  Google Scholar 

  26. Wang, Q., Chen, G., Perc, M.: Synchronization bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS ONE 6, e15851 (2011)

  27. Li, C., Chen, G.: Synchronization in general complex dynamical networks with coupling delays. Phys. A 343, 263–278 (2004)

    Article  MathSciNet  Google Scholar 

  28. Lu, W., Chen, T.: Synchronization of coupled connected neural networks with delays. IEEE Trans. Circuits Syst. I(51), 2491–2503 (2004)

    Article  MathSciNet  Google Scholar 

  29. Zhou, J., Chen, T.: Synchronization in general complex delayed dynamical networks. IEEE Trans. Circuits Syst. I(53), 733–744 (2006)

    Article  MathSciNet  Google Scholar 

  30. Guan, Z., Liu, Z., Feng, G., Wang, Y.: Synchronization of complex dynamical networks with time-varying delays via impulsive distributed control. IEEE Trans. Circuits Syst. I(57), 2182–2195 (2010)

    Article  MathSciNet  Google Scholar 

  31. Yang, X., Cao, J., Lu, J.: Synchronization of randomly coupled neural networks with Markovian jumping and time-delay. IEEE Trans. Circuits Syst. I(60), 363–376 (2013)

    Article  MathSciNet  Google Scholar 

  32. Wang, Y., Wang, Z., Liang, J.: A delay fractioning approach to global synchronization of delayed complex networks with stochastic disturbances. Phys. Lett. A 372, 6066–6073 (2008)

    Article  MATH  Google Scholar 

  33. Cao, J., Li, P., Wang, W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353, 318–325 (2006)

    Article  Google Scholar 

  34. Yu, W., Cao, J., Lü, J.: Global synchronization of linearly hybrid coupled networks with time-varying delay. SIAM J. Appl. Dyn. Syst. 7, 108–133 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wang, Y., Bian, T., Xiao, J.-W., Huang, Y.: Robust synchronization of complex switched networks with parametric uncertainties and two types of delays. Int. J. Robust Nonlinear Control 23, 190–207 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chen, L., Qiu, C., Huang, H.: Synchronization with on-off coupling: role of time scales in network dynamics. Phys. Rev. E 79, 045101(R) (2009)

    Article  Google Scholar 

  37. Chen, L., Qiu, C., Huang, H., Qi, G., Wang, H.: Facilitated synchronization of complex networks through a discontinuous coupling strategy. Eur. Phys. J. B 76, 625 (2010)

    Article  MATH  Google Scholar 

  38. Stilwell, D.J., Bollt, E.M., Roberson, D.G.: Sufficient conditions for fast switching synchronization in time-varying network topologies. SIAM J. Appl. Dyn. Syst. 5, 140 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (1985)

    Book  MATH  Google Scholar 

  40. Hmamed, A.: Further results on the delay-independent asymptotic stability of linear systems. Int. J. Syst. Sci. 22, 1127–1132 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  41. Ikeda, K., Matsumoto, K.: High-dimensional chaotic behavior in systems with time-delayed feedback. Phys. D 29, 223 (1987)

    Article  MATH  Google Scholar 

  42. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287 (1977)

    Article  Google Scholar 

  43. Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities. Academic Press, New York (1969)

    MATH  Google Scholar 

Download references

Acknowledgments

We thank the anonymous referees for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11226150 and 61403393), the Fundamental Research Funds for the Central Universities (Grant No. 2015XKMS076), and the China Scholarship Council (Grant No. 201308320087).

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Authors and Affiliations

  1. School of Sciences, China University of Mining and Technology, Xuzhou, 221008, People’s Republic of China

    Yongzheng Sun, Zhicai Ma, Feng Liu & Jie Wu

  2. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Yongzheng Sun

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  1. Yongzheng Sun
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  2. Zhicai Ma
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  3. Feng Liu
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  4. Jie Wu
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Correspondence to Yongzheng Sun.

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Sun, Y., Ma, Z., Liu, F. et al. Theoretical analysis of synchronization in delayed complex dynamical networks with discontinuous coupling. Nonlinear Dyn 86, 489–499 (2016). https://doi.org/10.1007/s11071-016-2902-2

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