Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling. (English) Zbl 1239.93060

Summary: This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.


93C40 Adaptive control/observation systems
93C10 Nonlinear systems in control theory
34H10 Chaos control for problems involving ordinary differential equations
Full Text: DOI


[1] Erdos, P.; Renyi, A., On the evolution of random graphs, Publ math inst hung acad sci, 5, 17-60, (1959)
[2] Watts, D.J.; Strogatz, S.H., Collective dynamics of small-world networks, Nature, 393, 440-442, (1998) · Zbl 1368.05139
[3] Barbaasi, A.L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509-512, (1999) · Zbl 1226.05223
[4] Wu, C.W.; Chua, L.O., Synchronization in an array of linearly coupled dynamical systems, IEEE T circuits I, 42, 430-447, (1995) · Zbl 0867.93042
[5] Li, X.; Chen, G., Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint, IEEE T circuits I, 50, 1381-1390, (2003) · Zbl 1368.37087
[6] Lü, J.; Yu, X.; Chen, G., Chaos synchronization of general complex dynamical networks, Physica A, 334, 281-302, (2004)
[7] Ma, Z.; Liu, Z.; Zhang, G., A new method to realize cluster synchronization in connected chaotic networks, Chaos, 16, 023103, (2006) · Zbl 1146.37330
[8] Wang, L.; Dai, H.; Sun, Y., Synchronization criteria for a generalized complex delayed dynamical network model, Physica A, 383, 703-713, (2007)
[9] Zhou, J.; Xiang, L.; Liu, Z., Global synchronization in general complex delayed dynamical networks and its applications, Physica A, 385, 729-742, (2007)
[10] Wang, X.F.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE T circuits I, 49, 54-62, (2002) · Zbl 1368.93576
[11] Wang, X.F.; Chen, G., Synchronization in scale-free dynamical networks, Int J bifurcat chaos, 12, 187-192, (2002)
[12] Cao, J.; Li, P.; Wang, W., Global synchronization in arrays of delayed neural networks with constant and coupling, Phys lett A, 353, 865-872, (2006)
[13] Lu, W.; Chen, T., New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213, 214-230, (2006) · Zbl 1105.34031
[14] Lu, W.; Chen, T.; Chen, G., Synchronization analysis of linearly coupled systems described by differential equations with a coupling delay, Physica D, 221, 118-134, (2006) · Zbl 1111.34056
[15] Zhou, J.; Xiang, L.; Liu, Z., Synchronization in complex delayed dynamical networks with impulsive effects, Physica A, 384, 684-692, (2007)
[16] He, G.; Yang, J., Adaptive synchronization in nonlinearly coupled dynamical networks, Chaos solitons fract, 38, 1254-1259, (2008) · Zbl 1154.93424
[17] Wu, W.; Chen, T., Global synchronization criteria of linearly coupled neural network systems with time-varying coupling, IEEE T neural networ, 19, 319-332, (2008)
[18] Arenas, A.; Diaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C., Synchronization in complex networks, Phys rep, 469, 93-153, (2008)
[19] Liu, T.; Zhao, J.; Hill, David J., Synchronization of complex delayed dynamical networks with nonlinearly coupled nodes, Chaos solitons fract, 40, 1506-1519, (2009) · Zbl 1197.34092
[20] Guo, W.; Austin, F.; Chen, S., Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling, Commun nonlinear sci numer simulat, 15, 1631-1639, (2010) · Zbl 1221.34213
[21] Li, C.P.; Sun, W.G.; Kurths, J., Synchronization between two coupled complex networks, Phys rev E, 76, 046204, (2007)
[22] Tang, H.W.; Chen, L.; Lu, J.A.; Tse, Chi K., Adaptive synchronization between two complex networks with nonidentical topological structures, Physica A, 387, 5623-5630, (2008)
[23] Li, Y.; Liu, Z.; Zhang, J.B., Synchronization between different networks, Chinese phys lett, 25, 874-877, (2008)
[24] Zheng, S.; Dong, G.; Bi, Q., Impulsive synchronization of complex networks with non-delayed and delayed coupling, Phys lett A, 373, 4255-4259, (2009) · Zbl 1234.05220
[25] Li, C.P.; Xu, C.X.; Sun, W.G.; Kurths, J., Outer synchronization of coupled discrete-time networks, Chaos, 19, 013106, (2009) · Zbl 1311.34115
[26] Wu, X.; Zheng, W.; Zhou, J., Generalized outer synchronization between complex dynamical networks, Chaos, 19, 013109, (2009) · Zbl 1311.34119
[27] Zhou, J.; Chen, T., Synchronization in general complex delayed dynamical networks, IEEE T circuits I, 53, 733-744, (2006) · Zbl 1374.37056
[28] Lasalle, J.P., The extent of asymptotic stability, Proc natl acad sci USA, 46, 3, 363-365, (1960) · Zbl 0094.28602
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