Adaptive synchronization in tree-like dynamical networks. (English) Zbl 1125.34031

The authors investigate the synchronization in three-like dynamical networks, which can be described by the following system of coupled ordinary differential equations
\[ \dot x_i = f(x_i) + c \sum_{j=1}^{N} a_{ij} \Gamma x_j, \quad i=1,\dots, N, \]
where \(f(x_i)=(f_1(x_i),\dots,f_n(x_i))^T\): \(\mathbb R^n\to \mathbb R^n\), \(x_i=(x_{i1},\dots,x_{in})\in R^n\) are the state variables of the nodes, \(c>0\) is the coupling strength, \(\Gamma\) is a diagonal matrix. The structure of the network is described by the coupling matrix \(A=(a_{ij})\).
The main result reports the possibility of finding a coupling \(c(x)\) such that the system will be completely synchronized, i.e. \(\| x_i(t)-x_j(t)\| \to 0\) for \(t\to \infty\), any \(i,j\) and all initial conditions.


34D05 Asymptotic properties of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34H05 Control problems involving ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI


[1] Barabási, A. L.; Albert, R., Emergence of scaling in random networks, Science, 286, 509 (1999) · Zbl 1226.05223
[2] Barahona, M.; Pecora, L. M., Synchronization in small-world systems, Phys. Rev. Lett., 89, 054101 (2002)
[3] Cao, J.; Li, P.; Wang, W., Global synchronization in arrays of delayed neural networks with constant and delayed coupling, Phys. Lett. A, 353, 318-325 (2006)
[4] Cao, J.; Lu, J., Adaptive synchronization of neural networks with or without time-varying delays, Chaos, 16, 013133 (2006) · Zbl 1144.37331
[5] Cao, J.; Song, Q., Stability in Cohen-Grossberg type BAM neural networks with time-varying delays, Nonlinearity, 19, 7, 1601-1617 (2006) · Zbl 1118.37038
[6] Cao, J.; Yu, W.; Qu, Y., A new complex network model and convergence dynamics in reputation computation for virtual organizations, Phys. Lett. A, 356, 6, 414-425 (2006) · Zbl 1160.91403
[7] Chen, G.; Zhou, J.; Liu, Z. R., Global synchronization of coupled delayed neural networks and applications to chaotic CNN models, Int. J. Bifurcation Chaos, 14, 2229-2240 (2004) · Zbl 1077.37506
[8] Deo, N., Graph Theory with Applications to Engineering and Computer Science (1980), Prentice-Hall: Prentice-Hall New Delhi
[9] Hong, H.; Choi, M. Y.; Kim, B. J., Synchronization on small-world networks, Phys. Rev. E, 52, 026139 (2002)
[10] LaSalle, J.; Lefschtg, S., Stability by Lyapunov’s Direct Method with Application (1961), Academic Press: Academic Press New York
[11] Li, C. G.; Chen, G., Synchronization in general complex dynamical networks with coupling delays, Physica A, 343, 263-278 (2004)
[12] Li, X.; Chen, G., Synchronization and desynchronization of complex dynamical networks: an engineering viewpoint, IEEE Trans. Circuits Syst. I, 50, 1381-1390 (2003) · Zbl 1368.37087
[13] Lorenz, E. N., Deterministic non-periodic flows, J. Atmos. Sci, 20, 130 (1963)
[14] Lu, W.; Chen, T., Synchronization of coupled connected neural networks with delays, IEEE Trans. Circuits Syst. I, 51, 2491-2503 (2004) · Zbl 1371.34118
[15] Lü, J. H.; Chen, G., A time-varying complex dynamical network model and its controlled synchronization criteria, IEEE Trans. Autom. Control, 50, 841-846 (2005) · Zbl 1365.93406
[16] Lü, J. H.; Yu, X. H.; Chen, G.; Cheng, D. Z., Characterizing the synchronizability of small-world dynamical networks, IEEE Trans. Circuits Syst. I, 51, 787-796 (2004) · Zbl 1374.34220
[17] Lü, J. H.; Yu, X. H.; Chen, G. R., Chaos synchronization of general complex dynamical networks, Physica A, 334, 281-302 (2004)
[18] Stilwell, D. J.; Bollt, E. M.; Roberson, D. G., Sufficient conditions for fast switching synchronization in time-varying network topologies, SIAM J. Appl. Dynamical Syst., 5, 1, 140-156 (2006) · Zbl 1145.37345
[19] Strogatz, S. H., Exploring complex networks, Nature, 410, 268-276 (2001) · Zbl 1370.90052
[20] Wang, W.; Cao, J., Synchronization in an array of linearly coupled networks with time-varying delay, Physica A, 366, 197-211 (2006)
[21] Wang, X. F.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. I, 49, 54-62 (2002) · Zbl 1368.93576
[22] Wang, X. F.; Chen, G., Synchronization in small-world dynamical networks, Int. J. Bifurcation Chaos, 12, 187-192 (2002)
[23] Wang, Z. D.; Ho, D. W.C., State estimation for delayed neural networks, IEEE Trans. Neural Networks, 16, 1, 279-284 (2005)
[24] Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442 (1998) · Zbl 1368.05139
[25] Wu, C. W., Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling, IEEE Trans. Circuits Syst. II, 52, 282-286 (2005)
[26] Wu, C. W., Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18, 1057-1064 (2005) · Zbl 1089.37024
[27] Wu, C. W.; Chua, L. O., Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst. I, 42, 430-447 (1995) · Zbl 0867.93042
[28] Yu, W.; Cao, J., Adaptive Q-S (lag, anticipated, and complete) time-varying synchronization and parameters identification of uncertain delayed neural networks, Chaos, 16, 023119 (2006) · Zbl 1146.93371
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