Abstract
This paper investigates the output synchronization of a class of impulsive complex dynamical networks with time-varying delay. By constructing suitable Lyapunov functionals, some new and useful conditions are obtained to guarantee the local and global exponential output synchronization of the impulsive complex networks. Finally, numerical examples are given to demonstrate the effectiveness of the theoretical results.
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References
Wang, J.L., Yang, Z.C., Wu, H.N.: Passivity analysis of complex dynamical networks with multiple time-varying delays. J. Eng. Math. (2011). Available online. doi:10.1007/s10665-011-9480-6
Wang, J.L., Yang, Z.C., Huang, T.W., **ao, M.Q.: Synchronization criteria in complex dynamical networks with nonsymmetric coupling and multiple time-varying delays. Appl. Anal. 91, 923–935 (2012)
Wang, J.L., Wu, H.N.: Local and global exponential output synchronization of complex delayed dynamical networks. Nonlinear Dyn. 67, 497–504 (2012)
Wang, J.L., Yang, Z.C., Huang, T.W., **ao, M.Q.: Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete Contin. Dyn. Syst., Ser. B 16, 393–408 (2011)
Yu, W., Chen, G., Lü, J.: On pinning synchronization of complex dynamical networks. Automatica 45, 429–435 (2009)
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)
**ang, L., Chen, Z., Liu, Z., Chen, F., Yuan, Z.: Pinning control of complex dynamical networks with heterogeneous delays. Comput. Math. Appl. 56, 1423–1433 (2008)
**ang, L., Liu, Z., Chen, Z., Yuan, Z.: Pinning weighted complex networks with heterogeneous delays by a small number of feedback controllers. Sci. China Ser. F 51, 511–523 (2008)
Lu, W., Atay, F.M., Jost, J.: Synchronization of discrete-time dynamical networks with time-varying couplings. SIAM J. Math. Anal. 39, 1231–1259 (2007)
Zhou, J., Lu, J.A., Lü, J.: Pinning adaptive synchronization of a general complex dynamical network. Automatica 44, 996–1003 (2008)
Liu, B., Hill, D.J.: Impulsive consensus for complex dynamical networks with nonidentical nodes and coupling time-delays. SIAM J. Control Optim. 49, 315–338 (2011)
Yao, J., Guan, Z.H., Hill, D.J.: Passivity-based control and synchronization of general complex dynamical networks. Automatica 45, 2107–2113 (2009)
Yao, J., Wang, H.O., Guan, Z.H., Xu, W.: Passive stability and synchronization of complex spatio-temporal switching networks with time delays. Automatica 45, 1721–1728 (2009)
Li, Z., Jiao, L., Lee, J.: Robust adaptive global synchronization of complex dynamical networks by adjusting time-varying coupling strength. Physica A 387, 1369–1380 (2008)
Huang, T., Chen, G., Kurth, J.: Synchronization of chaotic systems with time-varying coupling delays. Discrete Contin. Dyn. Syst., Ser. B 16, 1071–1082 (2011)
Huang, T., Li, C., Yu, W., Chen, G.: Synchronization of delayed chaotic systems with parameter mismatches by using intermittent linear state feedback. Nonlinearity 22, 569–584 (2009)
Long, S., Xu, D.: Delay-dependent stability analysis for impulsive neural networks with time varying delays. Neurocomputing 71, 1705–1713 (2008)
Zhou, J., **ang, L., Liu, Z.: Synchronization in complex delayed dynamical networks with impulsive effects. Physica A 384, 684–692 (2007)
Li, K., Lai, C.H.: Adaptive-impulsive synchronization of uncertain complex dynamical networks. Phys. Lett. A 372, 1601–1606 (2008)
Cai, S., Zhou, J., **ang, L., Liu, Z.: Robust impulsive synchronization of complex delayed dynamical networks. Phys. Lett. A 372, 4990–4995 (2008)
Lu, W., Chen, T.: New approach to synchronization analysis of linearly coupled ordinary differential systems. Physica D 213, 214–230 (2006)
Guo, W., Austin, F., Chen, S.: Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling. Commun. Nonlinear Sci. Numer. Simul. 15, 1631–1639 (2010)
Lorenz, E.N.: Deterministic non-periodic flow. J. Atmos. Sci. 20, 130–141 (1963)
Chen, G., Ueta, T.: Yet another chaotic attractor. Int. J. Bifurc. Chaos 9, 1465–1466 (1999)
Lü, J., Chen, G.: A new chaotic attractor coined. Int. J. Bifurc. Chaos 12, 659–661 (2002)
Berman, A., Plemmons, R.J.: Nonnegative Matrices in Mathematical Sciences. Academic Press, New York (1979)
Chavez, M., Hwang, D.U., Amann, A., Hentschel, H.G.E., Boccaletti, S.: Synchronization is enhanced in weighted complex networks. Phys. Rev. Lett. 94, 218701 (2005)
Hramov, A.E., Khramova, A.E., Koronovskii, A.A., Boccaletti, S.: Synchronization in networks of slightly nonidentical elements. Int. J. Bifurc. Chaos 18, 845–850 (2008)
Acknowledgements
The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
This work was supported in part by the National Basic Research Program of China (973 Program) (2012CB720003), in part by the National Natural Science Foundation of China under Grants 61074057, 61121003, 61004023, and 91016004, in part by the Innovation Foundation of BUAA for Ph.D. Graduates.
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Wang, JL., Wu, HN. Synchronization criteria for impulsive complex dynamical networks with time-varying delay. Nonlinear Dyn 70, 13–24 (2012). https://doi.org/10.1007/s11071-012-0427-x
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DOI: https://doi.org/10.1007/s11071-012-0427-x