Outer synchronization of complex networks with nondelayed and time-varying delayed couplings via pinning control or impulsive control. (English) Zbl 1342.93017

Summary: The outer synchronization problem between two complex networks with nondelayed and time-varying delayed couplings via two different control schemes, namely, pinning control and impulsive control, is considered. Firstly, by applying pinning control to a fraction of the network nodes and using a suitable Lyapunov function, we obtain some new and useful synchronization criteria, which guarantee the outer synchronization between two complex networks. Secondly, impulsive control is added to the nodes of corresponding response network. Based on the generalized inequality about time-varying delayed different equation, sufficient conditions for outer synchronization are derived. Finally, some examples are presented to demonstrate the effectiveness and feasibility of the results obtained in this paper.


93A15 Large-scale systems
93A14 Decentralized systems
93D30 Lyapunov and storage functions
93C15 Control/observation systems governed by ordinary differential equations
Full Text: DOI


[1] Arenas, A.; Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C., Synchronization in complex networks, Physics Reports, 469, 3, 93-153 (2008)
[2] Yu, D. C.; Righero, M.; Kocarev, L., Estimating topology of networks, Physical Review Letters, 97, 18 (2006)
[3] Strogatz, S. H.; Stewart, I., Coupled oscillators and biological synchronization, Scientific American, 269, 6, 102-109 (1993)
[4] Gray, C. M., Synchronous oscillations in neuronal systems: mechanisms and functions, Journal of Computational Neuroscience, 1, 1-2, 11-38 (1994)
[5] Vieira, M. D., Chaos and synchronized chaos in an earthquake model, Physical Review Letters, 82, 1, 201-204 (1999)
[6] Kuhnert, L.; Agladze, K. I.; Krinsky, V. I., Image processing using light-sensitive chemical waves, Nature, 337, 6204, 244-247 (1989)
[7] Wang, S. H.; Kuang, J. Y.; Li, J. H.; Luo, Y. L.; Lu, H. P.; Hu, G., Chaos-based secure communications in a large community, Physical Review E, 66, 6 (2002)
[8] Wu, Y., Adaptive impulsive outer synchronization between drive-response dynamical networks, Communications in Theoretical Physics, 61, 50, 590-594 (2014) · Zbl 1290.93073
[9] Wang, X. F.; Chen, G., Pinning control of scale-free dynamical networks, Physica A, 310, 3-4, 521-531 (2002) · Zbl 0995.90008
[10] Chen, T.; Liu, X.; Lu, W., Pinning complex networks by a single controller, IEEE Transactions on Circuits and Systems I: Regular Papers, 54, 6, 1317-1326 (2007) · Zbl 1374.93297
[11] Xiang, L.; Zhu, J. J. H., On pinning synchronization of general coupled networks, Nonlinear Dynamics, 64, 4, 339-348 (2011)
[12] Xia, W.; Cao, J., Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19, 1 (2009) · Zbl 1311.93061
[13] Zhou, J.; Wu, Q.; Xiang, L., Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization, Nonlinear Dynamics, 69, 3, 1393-1403 (2012) · Zbl 1253.93105
[14] Montbrió, E.; Kurths, J.; Blasius, B., Synchronization of two interacting populations of oscillators, Physical Review E, 70 (2004)
[15] Lu, J.; Ho, D. W. C.; Cao, J.; Kurths, J., Single impulsive controller for globally exponential synchronization of dynamical networks, Nonlinear Analysis: Real World Applications, 14, 1, 581-593 (2013) · Zbl 1254.93133
[16] Cao, J. D.; Ho, D. W. C.; Yang, Y. Q., Projective synchronization of a class of delayed chaotic systems via impulsive control, Physics Letters A, 373, 35, 3128-3133 (2009) · Zbl 1233.34017
[17] Liu, B.; Liu, X.; Chen, G.; Wang, H., Robust impulsive synchronization of uncertain dynamical networks, IEEE Transactions on Circuits and Systems. I: Regular Papers, 52, 7, 1431-1441 (2005) · Zbl 1374.82016
[18] Li, C.; Sun, W.; Kurths, J., Synchronization between two coupled complex networks, Physical Review E, 76, 4 (2007)
[19] Li, Z.; Xue, X., Outer synchronization of coupled networks using arbitrary coupling strength, Chaos, 20, 2 (2010) · Zbl 1311.34116
[20] Sun, W.; Yan, Z. Z.; Chen, S. H.; Lü, J. H., Outer synchronization of complex networks by impulse, Communications in Theoretical Physics, 56, 5, 885-890 (2011) · Zbl 1247.05230
[21] Wang, J. W.; Ma, Q. H.; Li, Z.; Mohammed, S. A. E., Mixed outer synchronization of coupled complex networks with time-varying coupling delay, Chaos, 21, 1-8 (2011) · Zbl 1345.34102
[22] Zheng, S., Inner-outer synchronization analysis of two complex networks with delayed and non-delayed coupling, Journal of Information and Computing Science, 7, 1, 11-18 (2012)
[23] He, P.; Jing, C. G.; Fan, T.; Chen, C. Z., Outer synchronization of complex networks with multiple coupling time-varying delays, International Journal of Control and Automation, 6, 4, 197-216 (2013)
[24] Zheng, S.; Shao, W., Mixed outer synchronization of dynamical networks with nonidentical nodes and output coupling, Nonlinear Dynamics, 73, 4, 2343-2352 (2013) · Zbl 1281.05125
[25] Liu, X.; Chen, T., Synchronization analysis for nonlinearly-coupled complex networks with an asymmetrical coupling matrix, Physica A, 387, 16-17, 4429-4439 (2008)
[26] Xia, W.; Cao, J., Pinning synchronization of delayed dynamical networks via periodically intermittent control, Chaos, 19 (2009) · Zbl 1311.93061
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