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Exponential synchronization of weighted general delay coupled and non-delay coupled dynamical networks. (English) Zbl 1205.93124

Summary: Time delays commonly exist in the real world. In the present work, we investigate the exponential synchronization of weighted general delay coupled and non-delay coupled complex dynamical networks with different topological structures. Based on the Lyapunov stability theory, the suitable controllers are designed to make the controlled dynamical network exponentially synchronize an isolated node with any pre-specified exponential convergence rate, and proved theoretically. The synchronization scheme is applicable to the undirected networks as well as the directed ones. The derived controllers are simple and can be readily used in practical applications. Furthermore, the coupling matrix is not necessary to be irreducible and the network node dynamics need not satisfy the very strong and conservative uniform Lipschitz condition. Numerical simulations further validate the effectiveness and feasibility of our synchronization method.

MSC:

93D15 Stabilization of systems by feedback
37N35 Dynamical systems in control
93C23 Control/observation systems governed by functional-differential equations
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[1] Watts, D. J.; Strogatz, S. H., Collective dynamics of small-world networks, Nature, 393, 6684, 409-410 (1998) · Zbl 1368.05139
[2] Barabási, A. L.; Albert, R., Emergence of scaling in random networks, Science, 286, 5439, 509-512 (1999) · Zbl 1226.05223
[3] Strogatz, S. H., Exploring complex networks, Nature, 410, 6825, 268-276 (2001) · Zbl 1370.90052
[4] Albert, R.; Barabási, A. L., Statistical mechanics of complex networks, Reviews of Modern Physics, 74, 1, 47-97 (2002) · Zbl 1205.82086
[5] Lü, J.; Yu, X. H.; Chen, G.; Cheng, D. Z., Characterizing the synchronizability of small-world dynamical networks, IEEE Transactions on Circuits and Systems I, 51, 4, 787-796 (2004) · Zbl 1374.34220
[6] Boccaleti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D. U., Complex networks: structure and dynamics, Physics Reports, 424, 4-5, 175-308 (2006) · Zbl 1371.82002
[7] Kaneko, K., Theory and Applications of Coupled Map Lattices (1993), Wiley: Wiley New York · Zbl 0777.00014
[8] Gelover-Santiago, A. L.; Lima, R.; Matinez-Mekler, G., Synchronization and cluster periodic solutions in globally coupled maps, Physica A, 283, 1-2, 131-135 (2000)
[9] Gu, Y. Q.; Shao, C.; Fu, X. C., Complete synchronization and stability of star-shaped complex networks, Chaos, Solitons & Fractals, 28, 2, 480-488 (2006) · Zbl 1083.37025
[10] Wang, X. F.; Chen, G., Synchronization in small-world dynamical networks, International Journal of Bifurcation and Chaos, 12, 1, 187-192 (2002)
[11] Wang, X. F.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Transactions on Circuits and Systems I, 49, 1, 54-62 (2002) · Zbl 1368.93576
[12] Li, X.; Wang, X.; Chen, G., Pinning a complex dynamical network to its equilibrium, IEEE Transactions on Circuits and Systems I, 51, 3, 2074-2087 (2004) · Zbl 1374.94915
[13] Lü, J.; Chen, G., A time-varying complex dynamical network models and its controlled synchronization criteria, IEEE Transactions on Automatic Control, 50, 6, 841-846 (2005) · Zbl 1365.93406
[14] Wu, C. W., Synchronization and convergence of linear dynamics in random directed networks, IEEE Transactions on Automatic Control, 51, 7, 1207-1210 (2006) · Zbl 1366.93537
[15] Yu, W.; Cao, J.; Lü, J., Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM Journal on Applied Dynamical Systems, 7, 1, 108-133 (2008) · Zbl 1161.94011
[16] Cao, J.; Li, P.; Wang, W., Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Physics Letters A, 353, 4, 318-325 (2006)
[17] He, G.; Yang, J., Adaptive synchronization in nonlinearly coupled dynamical networks, Chaos, Solitons & Fractals, 38, 5, 1254-1259 (2008) · Zbl 1154.93424
[18] Arenas, A.; Diaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C., Synchronization in complex networks, Physics Reports, 469, 1, 93-153 (2008)
[19] Yu, W.; Chen, G.; Lü, J., On pinning synchronization of complex dynamical networks, Automatica, 45, 2, 429-435 (2009) · Zbl 1158.93308
[20] Song, Q.; Cao, J.; Liu, F., Synchronization of complex dynamical networks with nonidentical nodes, Physics Letters A, 374, 4, 544-551 (2010) · Zbl 1234.05218
[21] Wu, X.; Zheng, W.; Zhou, J., Generalized outer synchronization between complex dynamical networks, Chaos, 19, 1, 013109 (2009) · Zbl 1311.34119
[22] Dhamala, M.; Jirsa, V. K.; Ding, M., Enhancement of neural synchrony by time delay, Physical Review Letters, 92, 7, 074104 (2004)
[23] Thomas, M.; Morari, M., (Delay Effects on Stability: A Robust Control Approach. Delay Effects on Stability: A Robust Control Approach, Lecture Notes in Control and Information Science, vol. 269 (2001), Springer: Springer Berlin, Heidelberg, New York) · Zbl 0997.93001
[24] Xu, X.; Chen, Z.; Si, G.; Hu, X.; Luo, P., A novel definition of generalized synchronization on networks and a numerical simulation example, Computers & Mathematics with Applications, 56, 11, 2789-2794 (2008) · Zbl 1165.34379
[25] Liu, Z. X.; Chen, Z. Q.; Yuan, Z. Z., Pinning control of weighted general complex dynamical networks with time delay, Physica A, 375, 1, 345-354 (2007)
[26] Wen, S.; Chen, S.; Guo, W., Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling, Physics Letters A, 372, 42, 6340-6346 (2008) · Zbl 1225.05223
[27] Wang, L.; Dai, H. P.; Dong, H.; Shen, Y. H.; Sun, Y. X., Adaptive synchronization of weighted complex dynamical networks with coupling time-varying delays, Physics Letters A, 372, 20, 3632-3639 (2008) · Zbl 1220.90041
[28] Wu, C. W., Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18, 3, 1057-1064 (2005) · Zbl 1089.37024
[29] Li, P.; Yi, Z., Synchronization analysis of delayed complex networks with time-varying couplings, Physica A, 387, 14, 3729-3737 (2008)
[30] Lou, X.; Cui, B., Synchronization of neural networks based on parameter identification and via output or state coupling, Journal of Computational and Applied Mathematics, 222, 2, 440-457 (2008) · Zbl 1168.65041
[31] Lu, J.; Cao, J., Adaptive synchronization of uncertain dynamical networks with delayed coupling, Nonlinear Dynamics, 53, 1-2, 107-115 (2008) · Zbl 1182.92007
[32] Xiang, L. Y.; Liu, Z. X.; Chen, Z. Q.; Chen, F.; Yuan, Z. Z., Pinning control of complex dynamical networks with general topology, Physica A, 379, 1, 298-306 (2007)
[33] Chen, G.; Ueta, T., Yet another chaotic attractor, International Journal of Bifurcation and Chaos, 9, 7, 1465-1466 (1999) · Zbl 0962.37013
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