×

Synchronization of complex delayed dynamical networks with nonlinearly coupled nodes. (English) Zbl 1197.34092

Summary: We study the global synchronization of nonlinearly coupled complex delayed dynamical networks with both directed and undirected graphs. Via Lyapunov-Krasovskii stability theory and the network topology, we investigate the global synchronization of such networks. Under the assumption that coupling coefficients are known, a family of delay-independent decentralized nonlinear feedback controllers are designed to globally synchronize the networks. When coupling coefficients are unavailable, an adaptive mechanism is introduced to synthesize a family of delay-independent decentralized adaptive controllers which guarantee the global synchronization of the uncertain networks. Two numerical examples of directed and undirected delayed dynamical network are given, respectively, using the Lorenz system as the nodes of the networks, which demonstrate the effectiveness of proposed results.
Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34K20 Stability theory of functional-differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37N35 Dynamical systems in control
93D15 Stabilization of systems by feedback
93D21 Adaptive or robust stabilization
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albert, R.; Barabasi, A. L., Statistical mechanics of complex networks, Rev Mod Phys, 74, 47-91 (2002) · Zbl 1205.82086
[2] Wang, X. F.; Chen, G., Complex networks: small-world, scale-free, and beyond, IEEE Trans Circuits Syst Mag, 3, 6-20 (2003)
[3] Strogatz, S. H., Exploring complex networks, Nature, 410, 268-276 (2001) · Zbl 1370.90052
[4] Newman, M., The structure and functions of complex networks, SIAM Rev, 45, 167-256 (2003) · Zbl 1029.68010
[5] Latora, V.; Marchiori, M., How the science of complex networks can help developing strategies against terrorism, Chaos, Solitons & Fractals, 20, 69-75 (2004) · Zbl 1122.91372
[6] Lu, W. L.; Chen, T. P., New approach to synchronization analysis of linearly coupled ordinary differential systems, Phys D, 213, 214-230 (2006) · Zbl 1105.34031
[7] Li, X., Sync in complex dynamical networks: stability, evolution, control, and application, Int J Comput Cognition, 3, 16-26 (2005)
[8] Gu, Y. Q.; Shao, C.; Pu, X. C., Complete synchronization and stability of star-shaped complex networks, Chaos, Solitons & Fractals, 28, 480-488 (2006) · Zbl 1083.37025
[9] Yu, Y. G.; Zhang, S. C., Global synchronization of three coupled chaotic systems with ring connection, Chaos, Solitons & Fractals, 24, 1233-1242 (2005) · Zbl 1081.37020
[10] Checco, P.; Biey, M.; Kocarev, L., Synchronization in random networks with given expected degree sequences, Chaos, Solitons & Fractals, 35, 562-577 (2008) · Zbl 1138.93050
[11] Wang, X. F.; Chen, G., Synchronization in small-word dynamical networks, Int J Bifurc. Chaos, 12, 187-192 (2002)
[12] Gade, P. M.; Hu, C. K., Synchronous chaos in coupled map with small-world interactions, Phys Rev E, 62, 6409-6413 (2000)
[13] Hong, H.; Choi, M. Y.; Kim, B. J., Synchronization on small-world networks, Phys Rev E, 65, 026139 (2002)
[14] Barahona, M.; Pecora, L. M., Synchronization in small-world systems, Phys Rev Lett, 89, 054101 (2002)
[15] Rangarajan, G.; Ding, M., Stability of synchronized chaos in coupled dynamical systems, Phys Lett A, 296, 204-209 (2002) · Zbl 0994.37026
[16] Sorrentino, F.; Bernardo, M.; Cuellar, G. H.; Boccaletti, S., Synchronization in weighted scale-free networks with degree-degree correlation, Phys D, 224, 123-129 (2006) · Zbl 1117.34049
[17] Li, C.; Chen, G., Synchronization in general complex dynamical networks with coupling delays, Phys A, 343, 263-278 (2004)
[18] Gao, H.; Lam, J.; Chen, G., New criteria for synchronization stability of general complex dynamical networks with coupling delays, Phys Lett A, 360, 263-273 (2006) · Zbl 1236.34069
[19] Zhou, J.; Chen, T., Synchronization in general complex delayed dynamical networks, IEEE Trans Circuits Syst-I, 53, 733-744 (2006) · Zbl 1374.37056
[20] Li, P.; Zhang, Y.; Zhang, L., Global synchronization of a class of delayed complex networks, Chaos, Solitons & Fractals, 30, 903-908 (2006) · Zbl 1142.34374
[21] Lu, J.; Ho, D. W.C., Local and global synchronization in generalcomplex dynamical networks with delay decoupling, Chaos, Solitons & Fractals, 37, 1497-1510 (2008) · Zbl 1142.93426
[22] Wang, W. W.; Cao, J. D., Synchronization in an array of linearly coupled networks with time-varying delay, Phys A, 366, 197-211 (2006)
[23] Lü, J.; Chen, G., A time-varying complex dynamical network model and its controlled synchronization criteria, IEEE Trans Automatic Control, 50, 841-846 (2005) · Zbl 1365.93406
[24] Chavez, M.; Hwang, D. U.; Amann, A.; Hentschel, H.; Boccaletti, S., Synchronization is enhanced in weighted complex networks, Phys Rev Lett, 94, 218701 (2005)
[25] Pecora, L. M.; Carroll, T. L., Master stability functions for synchronized coupled systems, Phys Rev Lett, 80, 2109-2112 (1998)
[26] Nishikawa, T.; Motter, A. E., Maximum performance at minimum cost in network synchronization, Phys D, 224, 77-89 (2006) · Zbl 1117.34048
[27] Belykh, I.; Hasler, M., Synchronization and graph topology, Int J Bifurcation Chaos, 15, 3423-3433 (2005) · Zbl 1107.34047
[28] Zhou, J.; Lu, J.; Lü, J., Adaptive synchronization of an uncertain complex dynamical network, IEEE Trans Automatic Control, 51, 652-656 (2006) · Zbl 1366.93544
[29] Li, Z.; Chen, G., Robust adaptive synchronization of uncertain dynamical networks, Phys Lett A, 324, 166-178 (2004) · Zbl 1123.93316
[30] Liao, T.; Tsai, S., Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, Solitons & Fractals, 11, 1387-1396 (2000) · Zbl 0967.93059
[31] Watts, D. J.; Strogatz, S. H., Collective dynamical of ‘small-world’ networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139
[32] Barabasi, A. L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512 (1999) · Zbl 1226.05223
[33] Barabasi, A. L.; Albert, R.; Jeong, H., Mean-field theory for scale-free random networks, Phys A, 272, 173-187 (1999)
[34] Ozean, N.; Arik, S., Gobal robust stability analysis of neural networks with multiple time delays, IEEE Trans Circuits Syst-I, 53, 166-176 (2006)
[35] Kolmanovskii, B.; Myshkis, A. D., Introduction to the theory and applications of functional differential equations (1999), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0917.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.