## Adaptive lag synchronization for uncertain complex dynamical network with delayed coupling.(English)Zbl 1238.93053

Summary: This paper proposes an adaptive control method to achieve the lag synchronization between uncertain complex dynamical network having delayed coupling and a non-identical reference node. Unknown parameters of both the network and reference node are estimated by adaptive laws obtained by Lyapunov’s stability theory. With the estimated parameters, the proposed method guarantees the globally asymptotical synchronization of the network in spite of unknown bounded disturbances. The effectiveness of our work is verified through a numerical example and simulation.

### MSC:

 93C40 Adaptive control/observation systems 93C73 Perturbations in control/observation systems 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory 93A15 Large-scale systems
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### References:

 [1] Wang, X.F., Complex networks: topology, dynamics and synchronization, International journal of bifurcation and chaos, 12, 885-916, (2002) · Zbl 1044.37561 [2] Chen, T.; Liu, X.; Lu, W., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE transactions on circuits and systems I: fundamental theory and applications, 49, 1, 54-62, (2002) · Zbl 1368.93576 [3] Boccaletti, S.; Latora, V.; Moreno, Y.; Chavez, M.; Hwang, D., Complex networks: structure and dynamics, Physics reports, 424, 175-308, (2006) · Zbl 1371.82002 [4] Li, Z.; Feng, G.; Hill, D., Controlling complex dynamical networks with coupling delays to a desired orbit, Physics letters A, 359, 42-46, (2006) · Zbl 1209.93136 [5] Wang, X.F.; Chen, G., Pinning control of scale-free dynamical networks, Physica A: statistical mechanics and its applications, 310, 521-531, (2002) · Zbl 0995.90008 [6] Lu, W., Adaptive dynamical networks via neighborhood information: synchronization and pinning control, Chaos, 17, 023122, (2007) · Zbl 1159.37366 [7] Jiang, G.-P.; Tang, W.K.-S.; Chen, G., A state-observer-based approach for synchronization in complex dynamical networks, IEEE transactions on circuits and systems I: regular papers, 53, 2739-2745, (2006) · Zbl 1374.37128 [8] Zheng, S.; Dong, G.; Bi, Q., Impulsive synchronization of complex networks with non-delayed and delayed coupling, Physics letters A, 373, 4255-4259, (2009) · Zbl 1234.05220 [9] Song, Q.; Cao, J.; Liu, F., Synchronization of complex dynamical networks with nonidentical nodes, Physics letters A, 374, 544-551, (2010) · Zbl 1234.05218 [10] Zhou, J.; Lu, J.; Lü, J., Adaptive synchronization of an uncertain complex dynamical network, IEEE transactions on automatic control, 51, 652-656, (2006) · Zbl 1366.93544 [11] He, G.; Yang, J., Adaptive synchronization in nonlinearly coupled dynamical networks, Chaos, solitons and fractals, 38, 1254-1259, (2008) · Zbl 1154.93424 [12] Liu, H.; Lu, J.-A.; Lu, J.; Hill, D.J., Structure identification of uncertain general complex dynamical networks with time delay, Automatica, 45, 1799-1807, (2009) · Zbl 1185.93031 [13] Zhu, Q.; Cao, J., Adaptive synchronization of chaotic cohen – crossberg neural networks with mixed time delays, Nonlinear dynamics, 61, 517-534, (2010) · Zbl 1204.93064 [14] Zhu, Q.; Cao, J., Adaptive synchronization under almost every initial data for stochastic neural networks with time-varying delays and distributed delays, Communications in nonlinear science and numerical simulation, 16, 2139-2159, (2011) · Zbl 1221.93247 [15] Rulkov, N.F.; Sushchik, M.M.; Tsimring, L.S., Generalized synchronization of chaos in directionally coupled chaotic systems, Physical review E, 51, 980-994, (1995) [16] Zhang, Y.; Xu, S.; Chu, Y.; Lu, J., Robust global synchronization of complex networks with neutral-type delayed nodes, Applied mathematics and computation, 216, 768-778, (2010) · Zbl 1364.34110 [17] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., From phase to lag synchronization in coupled chaotic oscillators, Physics review letters, 78, 4193-4196, (1997) [18] Rosenblum, M.G.; Pikovsky, A.S.; Kurths, J., Phase synchronization of chaotic oscillators, Physics review letters, 76, 1804-1807, (1996) [19] Mainieri, R.; Rehacek, J., Projective synchronization in three-dimensional chaotic systems, Physics review letters, 82, 3042-3045, (1999) [20] Voss, H.U., Anticipating chaotic synchronization, Physics review E, 61, 5115-5119, (2000) [21] Shahverdiev, E.M.; Sivaprakasam, S.; Shore, K.A., Lag synchronization in time-delayed systems, Physics letters A, 292, 320-324, (2002) · Zbl 0979.37022 [22] Miao, Q.; Tang, Y.; Lu, S.; Fang, J., Lag synchronization of a class of chaotic systems with unknown parameters, Nonlinear dynamics, 57, 107-112, (2009) · Zbl 1176.34097 [23] Wang, L.; Yuan, Z.; Chen, X.; Zhou, Z., Lag synchronization of chaotic systems with parameter mismatches, Communications in nonlinear science and numerical simulation, 16, 987-992, (2011) · Zbl 1221.37226 [24] Guo, W., Lag synchronization of the complex networks via pinning control, Nonlinear analysis: real world applications, 12, 2579-2585, (2011) · Zbl 1223.93057 [25] Li, H., New criteria for synchronization stability of continuous complex dynamical networks with non-delayed and delayed coupling, Communications in nonlinear science and numerical simulation, 16, 1027-1043, (2011) · Zbl 1221.34198 [26] Zhou, J.; Wu, Q.; Xiang, L.; Cai, S.; Liu, Z., Impulsive synchronization seeking in general complex delayed dynamical networks, Nonlinear analysis: hybrid systems, 5, 513-524, (2011) · Zbl 1238.93050 [27] Roopaei, M.; Sahraei, B.R.; Lin, T.-C., Adaptive sliding mode control in a novel class of chaotic systems, Communications in nonlinear science and numerical simulation, 15, 4158-4170, (2010) · Zbl 1222.93124 [28] Hu, C.; Yu, J.; Jiang, H.; Teng, Z., Synchronization of complex community networks with nonidentical nodes and adaptive coupling strength, Physics letters A, 375, 873-879, (2011) · Zbl 1242.05255 [29] Cai, S.; He, Q.; Hao, J.; Liu, Z., Exponential synchronization of complex networks with nonidentical time-delayed dynamical nodes, Physics letters A, 374, 2539-2550, (2010) · Zbl 1236.05185 [30] Khalil, H.K., Nonlinear systems, (1996), Prentice-Hall Upper Saddle River, NJ · Zbl 0626.34052
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