American Institute of Mathematical Sciences

Advanced
July  2011, 16(1): 393-408. doi: 10.3934/dcdsb.2011.16.393

Local and global exponential synchronization of complex delayed dynamical networks with general topology

Jin-Liang Wang 1, , Zhi-Chun Yang 2, , Tingwen Huang 3, and  Mingqing Xiao 4,

1. 

National Key Laboratory of Science and Technology on Holistic Control, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China
2. 

College of Mathematics Science, Chongqing Normal University, Chongqing 400047, China
3. 

Department of Mathematics and Science, Texas A&M University at Qatar, PO Box 23874, Doha, Qatar
4. 

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901, United States

Received  June 2010 Revised  January 2011 Published  April 2011

In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
Keywords: Complex networks, time-varying delays, exponential synchronization..
Mathematics Subject Classification: Primary: 37N99, 93D05; Secondary: 93A3.
Citation: Jin-Liang Wang, Zhi-Chun Yang, Tingwen Huang, Mingqing Xiao. Local and global exponential synchronization of complex delayed dynamical networks with general topology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393
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J. Lu and D. W. C. Ho, Local and global synchronization in general complex dynamical networks with delay coupling,, Chaos, 37 (2008), 1497. doi: 10.1016/j.chaos.2006.10.030. Google Scholar

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C. P. Li, W. G. Sun and J. Kurths, Synchronization of complex dynamical networks with time delays,, Physica A, 361 (2006), 24. doi: 10.1016/j.physa.2005.07.007. Google Scholar

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X. Q. Wu, Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,, Physica A, 387 (2008), 997. doi: 10.1016/j.physa.2007.10.030. Google Scholar

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W. Yu, J. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 108. doi: 10.1137/070679090. Google Scholar

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J. Lu and J. Cao, Synchronization-based approach for parameters identification in delayed chaotic neural networks,, Physica A, 382 (2007), 672. doi: 10.1016/j.physa.2007.04.021. Google Scholar

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J. Lu and J. D. Cao, Adaptive synchronization of uncertain dynamical networks with delayed coupling,, Nonlinear Dynamics, 53 (2008), 107. doi: 10.1007/s11071-007-9299-x. Google Scholar

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H. Huang, G. Feng and J. Cao, Exponential synchronization of chaotic Lur'e systems with delayed feedback control,, Nonlinear Dynamics, 57 (2009), 441. doi: 10.1007/s11071-008-9454-z. Google Scholar

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S. Cai, J. Zhou, L. Xiang and Z. Liu, Robust impulsive synchronization of complex delayed dynamical networks,, Physics Letters A, 372 (2008), 4990. doi: 10.1016/j.physleta.2008.05.077. Google Scholar

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J. Lü, X. Yu and G. Chen, Chaos synchronization of general complex dynamical networks,, Physica A, 334 (2004), 281. doi: 10.1016/j.physa.2003.10.052. Google Scholar

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Z. Li, L. Jiao and J. J. Lee, Robust adaptive global synchronization of complex dynamical networks by adjusting time-varying coupling strength,, Physica A, 387 (2008), 1369. doi: 10.1016/j.physa.2007.10.063. Google Scholar

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S. Albeverio and Christof Cebulla, Synchronizability of stochastic network ensembles in a model of interacting dynamical units,, Physica A, 386 (2007), 503. doi: 10.1016/j.physa.2007.07.036. Google Scholar

[18]

S. Wen, S. Chen and W. Guo, Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling,, Physics Letters A, 372 (2008), 6340. doi: 10.1016/j.physleta.2008.08.059. Google Scholar

[19]

D. Goldstein and K. Kobayashi, On the complexity of network synchronization,, SIAM Journal on Computing, 35 (2005), 567. doi: 10.1137/S0097539705447086. Google Scholar

[20]

J. Zhou, L. Xiang and Z. Liu, Global synchronization in general complex delayed dynamical networks and its applications,, Physica A, 385 (2007), 729. doi: 10.1016/j.physa.2007.07.006. Google Scholar

[21]

Z. X. Liu, Z. Q. Chen and Z. Z. Yuan, Pinning control of weighted general complex dynamical networks with time delay,, Physica A, 375 (2007), 345. doi: 10.1016/j.physa.2006.09.009. Google Scholar

[22]

W. W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays,, Nonlinear Dynamics, 48 (2007), 165. doi: 10.1007/s11071-006-9080-6. Google Scholar

[23]

C. Li and X. Liao, Anti-synchronization of a class of coupled chaotic systems via linear feedback control,, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 16 (2006), 1041. doi: 10.1142/S0218127406015295. Google Scholar

[24]

X. Liao, S. Guo and C. Li, Stability and bifurcation analysis in tri-neuron model with time delay,, Nonlinear Dynamics, 49 (2007), 319. doi: 10.1007/s11071-006-9137-6. Google Scholar

[25]

J. Wu and L. Jiao, Synchronization in complex dynamical networks with nonsymmetric coupling,, Physica D, 237 (2008), 2487. doi: 10.1016/j.physd.2008.03.002. Google Scholar

show all references

References:
[1]

S. H. Strogatz, Exploring complex networks,, Nature, 410 (2001), 268. doi: 10.1038/35065725. Google Scholar

[2]

R. Albert and A. L. BarabIasi, Statistical mechanics of complex networks,, Rev. Mod. Phys., 74 (2002), 47. doi: 10.1103/RevModPhys.74.47. Google Scholar

[3]

J. Wu and L. Jiao, Synchronization in complex delayed dynamical networks with nonsymmetric coupling,, Physica A, 386 (2007), 513. doi: 10.1016/j.physa.2007.07.052. Google Scholar

[4]

T. Liu, G. M. Dimirovskib and J. Zhao, Exponential synchronization of complex delayed dynamical networks with general topology,, Physica A, 387 (2008), 643. doi: 10.1016/j.physa.2007.09.019. Google Scholar

[5]

P. Li and Z. Yi, Synchronization analysis of delayed complex networks with time-varying couplings,, Physica A, 387 (2008), 3729. doi: 10.1016/j.physa.2008.02.008. Google Scholar

[6]

C. Li and G. Chen, Synchronization in general complex dynamical networks with coupling delays,, Physica A, 343 (2004), 263. doi: 10.1016/j.physa.2004.05.058. Google Scholar

[7]

J. Lu and D. W. C. Ho, Local and global synchronization in general complex dynamical networks with delay coupling,, Chaos, 37 (2008), 1497. doi: 10.1016/j.chaos.2006.10.030. Google Scholar

[8]

C. P. Li, W. G. Sun and J. Kurths, Synchronization of complex dynamical networks with time delays,, Physica A, 361 (2006), 24. doi: 10.1016/j.physa.2005.07.007. Google Scholar

[9]

X. Q. Wu, Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay,, Physica A, 387 (2008), 997. doi: 10.1016/j.physa.2007.10.030. Google Scholar

[10]

W. Yu, J. Cao and J. Lü, Global synchronization of linearly hybrid coupled networks with time-varying delay,, SIAM Journal on Applied Dynamical Systems, 7 (2008), 108. doi: 10.1137/070679090. Google Scholar

[11]

J. Lu and J. Cao, Synchronization-based approach for parameters identification in delayed chaotic neural networks,, Physica A, 382 (2007), 672. doi: 10.1016/j.physa.2007.04.021. Google Scholar

[12]

J. Lu and J. D. Cao, Adaptive synchronization of uncertain dynamical networks with delayed coupling,, Nonlinear Dynamics, 53 (2008), 107. doi: 10.1007/s11071-007-9299-x. Google Scholar

[13]

H. Huang, G. Feng and J. Cao, Exponential synchronization of chaotic Lur'e systems with delayed feedback control,, Nonlinear Dynamics, 57 (2009), 441. doi: 10.1007/s11071-008-9454-z. Google Scholar

[14]

S. Cai, J. Zhou, L. Xiang and Z. Liu, Robust impulsive synchronization of complex delayed dynamical networks,, Physics Letters A, 372 (2008), 4990. doi: 10.1016/j.physleta.2008.05.077. Google Scholar

[15]

J. Lü, X. Yu and G. Chen, Chaos synchronization of general complex dynamical networks,, Physica A, 334 (2004), 281. doi: 10.1016/j.physa.2003.10.052. Google Scholar

[16]

Z. Li, L. Jiao and J. J. Lee, Robust adaptive global synchronization of complex dynamical networks by adjusting time-varying coupling strength,, Physica A, 387 (2008), 1369. doi: 10.1016/j.physa.2007.10.063. Google Scholar

[17]

S. Albeverio and Christof Cebulla, Synchronizability of stochastic network ensembles in a model of interacting dynamical units,, Physica A, 386 (2007), 503. doi: 10.1016/j.physa.2007.07.036. Google Scholar

[18]

S. Wen, S. Chen and W. Guo, Adaptive global synchronization of a general complex dynamical network with non-delayed and delayed coupling,, Physics Letters A, 372 (2008), 6340. doi: 10.1016/j.physleta.2008.08.059. Google Scholar

[19]

D. Goldstein and K. Kobayashi, On the complexity of network synchronization,, SIAM Journal on Computing, 35 (2005), 567. doi: 10.1137/S0097539705447086. Google Scholar

[20]

J. Zhou, L. Xiang and Z. Liu, Global synchronization in general complex delayed dynamical networks and its applications,, Physica A, 385 (2007), 729. doi: 10.1016/j.physa.2007.07.006. Google Scholar

[21]

Z. X. Liu, Z. Q. Chen and Z. Z. Yuan, Pinning control of weighted general complex dynamical networks with time delay,, Physica A, 375 (2007), 345. doi: 10.1016/j.physa.2006.09.009. Google Scholar

[22]

W. W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays,, Nonlinear Dynamics, 48 (2007), 165. doi: 10.1007/s11071-006-9080-6. Google Scholar

[23]

C. Li and X. Liao, Anti-synchronization of a class of coupled chaotic systems via linear feedback control,, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 16 (2006), 1041. doi: 10.1142/S0218127406015295. Google Scholar

[24]

X. Liao, S. Guo and C. Li, Stability and bifurcation analysis in tri-neuron model with time delay,, Nonlinear Dynamics, 49 (2007), 319. doi: 10.1007/s11071-006-9137-6. Google Scholar

[25]

J. Wu and L. Jiao, Synchronization in complex dynamical networks with nonsymmetric coupling,, Physica D, 237 (2008), 2487. doi: 10.1016/j.physd.2008.03.002. Google Scholar

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