Synchronization analysis of linearly coupled networks of discrete time systems. (English) Zbl 1071.39011

The authors study coupled map networks of the form \[ x^i(t+1) = f(x^i(t)) + \sum_{j\neq i}^{m} b_{ij} [f(x^j(t))-f(x^i(t))], \quad i=1,\dots,m, \] where \(x^i(t)= (x_1^i(t),\dots,x_n^i(t))^T \in \mathbb{R}^n\) is the state variable of the \(i\)-th node, \(t\in \mathbb{N}\) is the discrete time, \(f:\mathbb{R}^n\to \mathbb{R}^n\) is continuous, \(B=(b_{ij})\) is the coupling matrix connecting the nodes, \(b_{ij}>0\) for all \(i\neq j\).
As the main result, the criteria are obtained for the synchronization of the subsystems, i.e. conditions when \(\| x^i(t)-x^j(t)\| \to 0\) as \(t\to\infty\) for some set of initial conditions. The global as well as local synchronization is considered. The obtained criteria reveal that two factors influence synchronization: dynamical behaviors at each node and coupling configuration.


39A11 Stability of difference equations (MSC2000)
93C55 Discrete-time control/observation systems
Full Text: DOI


[1] C.Hgenii, Horoloquim oscilatorium (Aqud F. Muget, Parisiis) (1673); C.Hgenii, Horoloquim oscilatorium (Aqud F. Muget, Parisiis) (1673)
[2] Strogatz, S. H.; Stewart, I., Coupled oscillators and biological synchronization, Sci. Am., 269, 6, 102-109 (1993)
[3] Gray, C. M., Synchronous oscillations in neural systems, J. Comput. Neurosci., 1, 11, 38 (1994)
[4] Glass, L., Synchronization and rhythmic processes in physiology, Nature, 410, 277-284 (2001)
[5] Millerioux, G.; Daafouz, J., An observer-based appraoch for input-independent global chaos synchronization of discrete-time switched systems, IEEE Trans. Circ. Syst. -1, 50, 10, 1270-1279 (2003) · Zbl 1368.93252
[6] de S. Vieira, M., Chaos and synchronization chaos in an earthquakes, Phys. Rev. Lett., 82, 1, 201-204 (1999)
[7] W. Lu, T. Chen, Synchronization of coupled connected neural networks with delays, IEEE Trans. CAS-1, in press.; W. Lu, T. Chen, Synchronization of coupled connected neural networks with delays, IEEE Trans. CAS-1, in press. · Zbl 1371.34118
[8] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys. Rev. Lett., 64, 8, 821-824 (1990) · Zbl 0938.37019
[9] Mirollo, R. E.; Strogatz, S. H., Synchronization properties of pulse-coupled biological oscillators, SIAM, J. Appl. Math., 50, 1645-1662 (1990) · Zbl 0712.92006
[10] Heagy, J. F.; Carroll, T. L.; Pecora, I. M., Synchronous chaos in coupled oscillator systems, Phys. Rev. E, 50, 1874-1885 (1994)
[11] Lakshmanan, M.; Murali, K., Chaos in Nonlinear Oscillators: Controlling and Synchronization (1996), World Scientific: World Scientific Singapore · Zbl 0868.58058
[12] Bohr, T.; Christensen, O. B., Size dependence, coherence, and scaling in turbulent coupled map lattices, Phys. Rev. Lett., 63, 2161-2164 (1989)
[13] Kaneko, K., Spatio-temporal intermittency in coupled map lattices, Prog. Theor. Phys., 74, 1033 (1985) · Zbl 0979.37505
[14] Kaneko, K., Theory and Applications of Coupled Map Lattices (1993), Wiley: Wiley New York · Zbl 0777.00014
[15] Jost, J.; Joy, M. P., Spectral properties and synchronization in coupled map lattices, Phys. Rev. E, 65 (2001 016201)
[16] Gade, P. M.; Hu, C.-K., Synchronous chaos in coupled map lattices, Phys. Rev. E, 65, 5, 6409-6413 (2000)
[17] Wang, X. F.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circ. Syst. -1, 49, 1, 54-62 (2002) · Zbl 1368.93576
[18] Wang, X. F.; Chen, G., Synchronization in small-world dynamical networks, Int. J. Birf. Chaos, 12, 1, 187-192 (2002)
[19] Wu, C. W.; Chua, L. O., Synchronziation in an array of linearly coupled dynamical systems, IEEE Trans. Circ. Syst. -1, 42, 8, 430-447 (1995) · Zbl 0867.93042
[20] Horn, P. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press New York · Zbl 0576.15001
[21] Boyd, S.; Ghaoui, L. E.I..; Feron, E.; Balakrishnana, U., Linear Matrix Inequalities in System and Control Theory (1994), SIAM: SIAM Philadelphia, PA · Zbl 0816.93004
[22] Watts, D. J.; Strogatz, S. H., Collective dynamics of ‘small-world’ networks, Nature, 399, 4, 440-442 (1998) · Zbl 1368.05139
[23] Newman, M. E.J.; Watts, D. J., Renormalization group analysis of the small-worlds network model, Phys. Lett. A, 263, 9, 341-346 (1999) · Zbl 0940.82029
[24] Barabasi, A.-L.; Albert, R., Emergence of scaling in random networks, Science, 286, 15, 509-512 (1999) · Zbl 1226.05223
[25] Wang, J.; Jing, Z., Topological structure of chaos in discrete-time neural networks with generalized input-output function, Int. J. Birf. Chaos, 11, 7, 1835-1851 (2001) · Zbl 1091.37525
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.