Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects. (English) Zbl 1204.34072

For a class of complex networks of interacting identical dynamical systems with impulsive effects, a problem of synchronization is studied. The coupling in the networks is considered to be with a time delay such that each element of the ensemble receives retarded signals from other oscillators. The coupling matrix belongs to a class of irreducible matrices with zero row sums and non-negative elements except for the main diagonal. Under further assumptions on the vector fields of the interacting oscillators and the coupling matrix, exponential stability of the synchronized manifold is demonstrated. The latter means that the synchronization errors decay to zero exponentially fast with time. Sufficient conditions for the exponential synchronization are derived by the geometrical decomposition of the network states over the eigenvectors of the coupling matrix and linear matrix inequality methods. Two numerical examples are also presented to illustrate the applicability of the obtained analytical results.


34D06 Synchronization of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
Full Text: DOI


[1] Strogatz, S.H., Exploring complex networks, Nature, 410, 268-276, (2001) · Zbl 1370.90052
[2] Watts, D.J.; Strogatz, S.H., Collective dynamics of ‘small-world’ networks, Nature, 393, 440-442, (1998) · Zbl 1368.05139
[3] Pecora, L.M., Synchronization conditions and desynchronization patterns in coupled limit-cycle and chaotic systems, Phys. rev. E, 58, 347-360, (1998)
[4] Pecora, L.M.; Carroll, T.; Johnson, G.; Mar, D.; Fink, K.S., Synchronization stability in coupled oscillator arrays: solution for arbitrary configuration, Internat. J. bifur. chaos, 10, 2, 273-290, (2000) · Zbl 1090.34542
[5] Rangarajan, G.; Ding, M.Z., Stability of synchronized chaos in coupled dynamical systems, Phys. lett. A, 296, 204-209, (2002) · Zbl 0994.37026
[6] Sorrentino, F.; Bernardo, M.; Garofalo, F., Synchronizability and synchronization dynamics of weighed and unweighed scale free networks with degree mixing, Internat. J. bifur. chaos, 17, 7, 2119-2143, (2007)
[7] Wang, X.; Chen, G., Synchronization in small-world dynamical networks, Internat. J. bifur. chaos, 12, 187-192, (2002)
[8] Wang, X.; Chen, G., Synchronization in scale-free dynamical networks: robustness and fragility, IEEE trans. circuit syst. I, 49, 54-62, (2002) · Zbl 1368.93576
[9] Lü, J.; Yu, X.; Chen, G., Chaos synchronization of general complex dynamical networks, Physica A, 334, 281-302, (2004)
[10] Lü, J.; Chen, G., A time-varying complex dynamical network model and its controlled synchronization criteria, IEEE trans. automat. control, 50, 6, 841-846, (2005) · Zbl 1365.93406
[11] Chen, M., Synchronization in time-varying networks: A matrix measure approach, Phys. rev. E, 76, 016104, (2007)
[12] Li, C.; Chen, G., Synchronization in complex dynamical networks with coupling delays, Physica A, 343, 263-278, (2004)
[13] Wu, J.; Jiao, L., Synchronization in complex delayed dynamical networks with nonsymmetric coupling, Physica A, 386, 513-530, (2007)
[14] Earl, M.G.; Strogate, S.H., Synchronization in oscillator networks with delayed coupling: A stability criterion, Phys. rev. E, 67, 036204, (2003)
[15] Buric, N.; Todorovie, K., Synchronization of noisy delayed feedback systems with delayed coupling, Phys. rev. E, 75, 026209, (2007)
[16] Liu, X.; Chen, T., Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling, Physica A, 381, 543-556, (2007)
[17] Wang, Q.; Chen, G.; Lu, Q.; Hao, F., Novel criteria of synchronization stability in complex networks with coupling delays, Physica A, 378, 527-536, (2007)
[18] Li, C.; Sun, W.; Kurths, J., Synchronization of complex dynamical networks with time delays, Physica A, 361, 24-34, (2006)
[19] Lu, J.; Ho, D.W.C., Local and global synchronization in general complex dynamical networks with delay coupling, Chaos solitons fractals, 37, 5, 1497-1510, (2008) · Zbl 1142.93426
[20] Liu, X.; Wang, J.; Huang, L., Global synchronization for a class of dynamical complex networks, Physica A, 386, 82-92, (2007)
[21] Zhou, J.; Chen, T., Synchronization in general complex delayed dynamical networks, IEEE trans. circuit syst. I, 53, 3, 733-744, (2006) · Zbl 1374.37056
[22] Wu, C.W.; Chua, L.O., Synchronization in an array of linearly coupled dynamical system, IEEE trans. cir. syst. I, 42, 8, 430-447, (1995) · Zbl 0867.93042
[23] 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)
[24] Wang, W.; Cao, J., Synchronization in an array of linearly coupled networks with time-varying, Physica A, 366, 197-211, (2006)
[25] Yu, W.; Cao, J.; Lü, J., Global synchronization of linearly hybrid coupled networks with time-varying delay, SIAM J. appl. dynam. syst., 7, 1, 108-133, (2008) · Zbl 1161.94011
[26] Lu, W.; Chen, T., New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213, 214-230, (2006) · Zbl 1105.34031
[27] 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
[28] Wu, W.; Chen, T., Global synchronization criteria of linearly coupled neural network systems with time-varying coupling, IEEE trans. neural netw., 19, 2, 319-332, (2008)
[29] Yang, Y.; Cao, J., Exponential lag synchronization of a class of chaotic delayed neural networks with impulsive effects, Physica A, 386, 1, 492-502, (2007)
[30] Zhou, J.; Xiang, L.; Liu, Z., Synchronization in complex delayed dynamical networks with impulsive effects, Physica A, 384, 2, 684-692, (2007)
[31] Zhu, W.; Xu, D.; Huang, Y., Global impulsive exponential synchronization of time-delayed coupled chaotic systems, Chaos solitons fractals, 35, 5, 904-912, (2008) · Zbl 1141.37019
[32] Li, P.; Cao, J.; Wang, Z., Robust impulsive synchronization of coupled delayed neural networks with uncertainties, Physica A, 373, 1, 261-272, (2007)
[33] Li, K.; Lai, C.H., Adaptive-impulsive synchronization of uncertain complex dynamical networks, Phys. lett. A, 372, 1601-1606, (2008) · Zbl 1217.05210
[34] Cai, S.; Zhou, J.; Xiang, L.; Liu, Z., Robust impulsive synchronization of complex delayed dynamical networks, Phys. lett. A, 372, 4990-4995, (2008) · Zbl 1221.34075
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.