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Adaptive synchronization of uncertain dynamical networks with delayed coupling. (English) Zbl 1182.92007
Summary: We propose a simple scheme for the synchronization of an uncertain complex dynamical network with delayed coupling. Based on the Lyapunov stability theory of functional differential equations, certain controllers can be designed for ensuring the states of uncertain dynamical network with coupling delays to globally asymptotically synchronize by combining the adaptive method and linear feedback with the updated feedback strength. Different update gains ηi will lead to different rates toward synchrony, the choice of which depends on the concrete systems and network models. This strategy can be applied to any complex dynamical network (regular, small-world, scale-free or random). Numerical examples with respectively nearest-neighbor coupling and scale-free structure are given to demonstrate the effectiveness of our presented scheme.
92B20General theory of neural networks (mathematical biology)
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
[1]Watts, D.J., Strogatz, S.H.: Collective dynamics of ’small-world’ networks. Nature 393(6684), 440–442 (1998) · doi:10.1038/30918
[2]Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999) · Zbl 1226.05223 · doi:10.1126/science.286.5439.509
[3]Strogatz, S.H.: Exploring complex networks. Nature 410, 268–276 (2001) · doi:10.1038/35065725
[4]Lu, J.Q., He, J., Cao, J.D., Gao, Z.Q.: Topology influences performance in the associative memory neural networks. Phys. Lett. A 354(5–6), 335–343 (2006) · doi:10.1016/j.physleta.2006.01.085
[5]Liu, Y., Takiguchi, Y., Davis, P., Aida, T., Saito, S., Liu, J.M.: Injection locking and synchronization of chaos in semiconductor lasers. Appl. Phys. Lett. 80, 4306–4308 (2002) · doi:10.1063/1.1485127
[6]Kim, K.T., Kim, M.S., Chong, Y., Niemeyer, J.: Simulations of collective synchronization in Josephson junction arrays. Appl. Phys. Lett. 88, 062501 (2006) · doi:10.1063/1.2171796
[7]Lu, J.Q., Cao, J.D.: Adaptive synchronization in tree-like dynamical networks. Nonlinear Anal. Real World Appl. 8(4), 1252–1260 (2007) · Zbl 1125.34031 · doi:10.1016/j.nonrwa.2006.07.010
[8]Lu, J.Q., Ho, D.W.C.: Local and global synchronization in general complex dynamical networks with delay coupling. Chaos Solitons Fractals (2006). doi: 10.1016/j.chaos.2006.10.030
[9]Baek, S.J., Ott, E.: Onset of synchronization in systems of globally coupled chaotic maps. Phys. Rev. E 69(6), 66210 (2004) · doi:10.1103/PhysRevE.69.066210
[10]Donetti, L., Hurtado, P.I., Munoz, M.A.: Entangled networks, synchronization, and optimal network topology. Phys. Rev. Lett. 95, 188701 (2005) · doi:10.1103/PhysRevLett.95.188701
[11]Belykh, I., de Lange, E., Hasler, M.: Synchronization of bursting neurons: what matters in the network topology. Phys. Rev. Lett. 94(18), 188101 (2005) · doi:10.1103/PhysRevLett.94.188101
[12]Zhou, J., Lu, J., Lü, J.: Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Autom. Control 51(4), 652–656 (2006) · doi:10.1109/TAC.2006.872760
[13]Wang, W., Cao, J.: Synchronization in an array of linearly coupled networks with time-varying delay. Physica A: Stat. Mech. Appl. 366, 197–211 (2006) · doi:10.1016/j.physa.2005.10.047
[14]Wang, X.F.: Complex networks: topology, dynamics and synchronization. Int. J. Bifurc. Chaos 12(5), 885–916 (2002) · Zbl 1044.37561 · doi:10.1142/S0218127402004802
[15]Lu, J.Q., Ho, D.W.C., Liu, M.: Globally exponential synchronization in an array of asymmetric coupled neural networks. Phys. Lett. A 369, 444–451 (2007) · Zbl 1209.37108 · doi:10.1016/j.physleta.2007.05.036
[16]Pikovsky, A., Rosenblum, M., Kurths, J., Hilborn, R.C.: Synchronization: a universal concept in nonlinear science. Am. J. Phys. 70, 655 (2002) · doi:10.1119/1.1475332
[17]Li, C., Chen, L., Aihara, K.: Synchronization of coupled nonidentical genetic oscillators. Phys. Biol. 3, 37–44 (2006) · doi:10.1088/1478-3975/3/1/004
[18]Li, C., Chen, L., Aihara, K.: Stochastic synchronization of genetic oscillator networks. BMC Syst. Biol. 1, 6 (2007) · doi:10.1186/1752-0509-1-6
[19]Pecora, L.M., Carroll, T.L.: Synchronization in chaotic systems. Phys. Rev. Lett. 64(8), 821–824 (1990) · doi:10.1103/PhysRevLett.64.821
[20]Li, C., Chen, G.: Synchronization in general complex dynamical networks with coupling delays. Physica A: Stat. Mech. Appl. 343, 263–278 (2004) · doi:10.1016/j.physa.2004.05.058
[21]Cao, J.D., Lu, J.Q.: Adaptive synchronization of neural networks with or without time-varying delay. Chaos 16(1), 013133 (2006) · Zbl 1144.37331 · doi:10.1063/1.2178448
[22]Jiang, Y.: Globally coupled maps with time delay interactions. Phys. Lett. A 267(5–6), 342–349 (2000) · doi:10.1016/S0375-9601(00)00135-3
[23]Masoller, C., Martı, A.C., Zanette, D.H.: Synchronization in an array of globally coupled maps with delayed interactions. Physica A: Stat. Mech. Appl. 325(1–2), 186–191 (2003) · Zbl 1026.37023 · doi:10.1016/S0378-4371(03)00197-3
[24]Choi, M.Y., Kim, H.J., Kim, D., Hong, H.: Synchronization in a system of globally coupled oscillators with time delay. Phys. Rev. E 61(1), 371–381 (2000) · doi:10.1103/PhysRevE.61.371
[25]Heil, T., Fischer, I., Elsässer, W., Mulet, J., Mirasso, C.R.: Chaos synchronization and spontaneous symmetry-breaking in symmetrically delay-coupled semiconductor lasers. Phys. Rev. Lett. 86(5), 795–798 (2001) · doi:10.1103/PhysRevLett.86.795
[26]Earl, M.G., Strogatz, S.H.: Synchronization in oscillator networks with delayed coupling: a stability criterion. Phys. Rev. E 67(3), 36204 (2003) · doi:10.1103/PhysRevE.67.036204
[27]Atay, F.M., Jost, J., Wende, A.: Delays, connection topology, and synchronization of coupled chaotic maps. Phys. Rev. Lett. 92(14), 144101 (2004) · doi:10.1103/PhysRevLett.92.144101
[28]Wünsche, H.J., Bauer, S., Kreissl, J., Ushakov, O., Korneyev, N., Henneberger, F., Wille, E., Erzgräber, H., Peil, M., Elsäßer, W., et al.: Synchronization of delay-coupled oscillators: a study of semiconductor lasers. Phys. Rev. Lett. 94(16), 163901 (2005) · doi:10.1103/PhysRevLett.94.163901
[29]Cao, J., Li, P., Wang, W.: Global synchronization in arrays of delayed neural networks with constant and delayed coupling. Phys. Lett. A 353(4), 318–325 (2006) · doi:10.1016/j.physleta.2005.12.092
[30]Chen, M., Zhou, D.: Synchronization in uncertain complex networks. Chaos 16(1), 013101 (2006) · Zbl 1144.37338 · doi:10.1063/1.2126581
[31]Li, Z., Chen, G.: Robust adaptive synchronization of uncertain dynamical networks. Phys. Lett. A 324(2-3), 166–178 (2004) · Zbl 1123.93316 · doi:10.1016/j.physleta.2004.02.058
[32]Hale, J.K.: Diffusive coupling, dissipation, and synchronization. J. Dyn. Differ. Equ. 9(1), 1–52 (1997) · Zbl 1091.34532 · doi:10.1007/BF02219051
[33]Wu, C.W.: Synchronization in Coupled Chaotic Circuits and Systems. World Scientific, Singapore (2002)
[34]Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963) · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[35]Leonov, G., Bunin, A., Koksch, N.: Attractor localization of the Lorenz system. Z. Angew. Math. Mech. 67(2), 649–656 (1987) · Zbl 0653.34040 · doi:10.1002/zamm.19870671215