American Institute of Mathematical Sciences

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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 and Continuous Dynamical Systems - B, 2011, 16 (1) : 393-408. doi: 10.3934/dcdsb.2011.16.393
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S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276. doi: 10.1038/35065725.

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J. Wu and L. Jiao, Synchronization in complex delayed dynamical networks with nonsymmetric coupling, Physica A, 386 (2007), 513-530. doi: 10.1016/j.physa.2007.07.052.

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T. Liu, G. M. Dimirovskib and J. Zhao, Exponential synchronization of complex delayed dynamical networks with general topology, Physica A, 387 (2008), 643-652. doi: 10.1016/j.physa.2007.09.019.

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P. Li and Z. Yi, Synchronization analysis of delayed complex networks with time-varying couplings, Physica A, 387 (2008), 3729-3737. doi: 10.1016/j.physa.2008.02.008.

[6]

C. Li and G. Chen, Synchronization in general complex dynamical networks with coupling delays, Physica A, 343 (2004), 263-278. doi: 10.1016/j.physa.2004.05.058.

[7]

J. Lu and D. W. C. Ho, Local and global synchronization in general complex dynamical networks with delay coupling, Chaos, Solitons & Fractals, 37 (2008), 1497-1510. doi: 10.1016/j.chaos.2006.10.030.

[8]

C. P. Li, W. G. Sun and J. Kurths, Synchronization of complex dynamical networks with time delays, Physica A, 361 (2006), 24-34. doi: 10.1016/j.physa.2005.07.007.

[9]

X. Q. Wu, Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay, Physica A, 387 (2008), 997-1008. doi: 10.1016/j.physa.2007.10.030.

[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), 108-133. doi: 10.1137/070679090.

[11]

J. Lu and J. Cao, Synchronization-based approach for parameters identification in delayed chaotic neural networks, Physica A, 382 (2007), 672-682. doi: 10.1016/j.physa.2007.04.021.

[12]

J. Lu and J. D. Cao, Adaptive synchronization of uncertain dynamical networks with delayed coupling, Nonlinear Dynamics, 53 (2008), 107-115. doi: 10.1007/s11071-007-9299-x.

[13]

H. Huang, G. Feng and J. Cao, Exponential synchronization of chaotic Lur'e systems with delayed feedback control, Nonlinear Dynamics, 57 (2009), 441-453. doi: 10.1007/s11071-008-9454-z.

[14]

S. Cai, J. Zhou, L. Xiang and Z. Liu, Robust impulsive synchronization of complex delayed dynamical networks, Physics Letters A, 372 (2008), 4990-4995. doi: 10.1016/j.physleta.2008.05.077.

[15]

J. Lü, X. Yu and G. Chen, Chaos synchronization of general complex dynamical networks, Physica A, 334 (2004), 281-302. doi: 10.1016/j.physa.2003.10.052.

[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), 1369-1380. doi: 10.1016/j.physa.2007.10.063.

[17]

S. Albeverio and Christof Cebulla, Synchronizability of stochastic network ensembles in a model of interacting dynamical units, Physica A, 386 (2007), 503-512. doi: 10.1016/j.physa.2007.07.036.

[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), 6340-6346. doi: 10.1016/j.physleta.2008.08.059.

[19]

D. Goldstein and K. Kobayashi, On the complexity of network synchronization, SIAM Journal on Computing, 35 (2005), 567-589. doi: 10.1137/S0097539705447086.

[20]

J. Zhou, L. Xiang and Z. Liu, Global synchronization in general complex delayed dynamical networks and its applications, Physica A, 385 (2007), 729-742. doi: 10.1016/j.physa.2007.07.006.

[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), 345-354. doi: 10.1016/j.physa.2006.09.009.

[22]

W. W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays, Nonlinear Dynamics, 48 (2007), 165-174. doi: 10.1007/s11071-006-9080-6.

[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), 1041-1047. doi: 10.1142/S0218127406015295.

[24]

X. Liao, S. Guo and C. Li, Stability and bifurcation analysis in tri-neuron model with time delay, Nonlinear Dynamics, 49 (2007), 319-345. doi: 10.1007/s11071-006-9137-6.

[25]

J. Wu and L. Jiao, Synchronization in complex dynamical networks with nonsymmetric coupling, Physica D, 237 (2008), 2487-2498. doi: 10.1016/j.physd.2008.03.002.

show all references

References:
[1]

S. H. Strogatz, Exploring complex networks, Nature, 410 (2001), 268-276. doi: 10.1038/35065725.

[2]

R. Albert and A. L. BarabIasi, Statistical mechanics of complex networks, Rev. Mod. Phys., 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.

[3]

J. Wu and L. Jiao, Synchronization in complex delayed dynamical networks with nonsymmetric coupling, Physica A, 386 (2007), 513-530. doi: 10.1016/j.physa.2007.07.052.

[4]

T. Liu, G. M. Dimirovskib and J. Zhao, Exponential synchronization of complex delayed dynamical networks with general topology, Physica A, 387 (2008), 643-652. doi: 10.1016/j.physa.2007.09.019.

[5]

P. Li and Z. Yi, Synchronization analysis of delayed complex networks with time-varying couplings, Physica A, 387 (2008), 3729-3737. doi: 10.1016/j.physa.2008.02.008.

[6]

C. Li and G. Chen, Synchronization in general complex dynamical networks with coupling delays, Physica A, 343 (2004), 263-278. doi: 10.1016/j.physa.2004.05.058.

[7]

J. Lu and D. W. C. Ho, Local and global synchronization in general complex dynamical networks with delay coupling, Chaos, Solitons & Fractals, 37 (2008), 1497-1510. doi: 10.1016/j.chaos.2006.10.030.

[8]

C. P. Li, W. G. Sun and J. Kurths, Synchronization of complex dynamical networks with time delays, Physica A, 361 (2006), 24-34. doi: 10.1016/j.physa.2005.07.007.

[9]

X. Q. Wu, Synchronization-based topology identification of weighted general complex dynamical networks with time-varying coupling delay, Physica A, 387 (2008), 997-1008. doi: 10.1016/j.physa.2007.10.030.

[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), 108-133. doi: 10.1137/070679090.

[11]

J. Lu and J. Cao, Synchronization-based approach for parameters identification in delayed chaotic neural networks, Physica A, 382 (2007), 672-682. doi: 10.1016/j.physa.2007.04.021.

[12]

J. Lu and J. D. Cao, Adaptive synchronization of uncertain dynamical networks with delayed coupling, Nonlinear Dynamics, 53 (2008), 107-115. doi: 10.1007/s11071-007-9299-x.

[13]

H. Huang, G. Feng and J. Cao, Exponential synchronization of chaotic Lur'e systems with delayed feedback control, Nonlinear Dynamics, 57 (2009), 441-453. doi: 10.1007/s11071-008-9454-z.

[14]

S. Cai, J. Zhou, L. Xiang and Z. Liu, Robust impulsive synchronization of complex delayed dynamical networks, Physics Letters A, 372 (2008), 4990-4995. doi: 10.1016/j.physleta.2008.05.077.

[15]

J. Lü, X. Yu and G. Chen, Chaos synchronization of general complex dynamical networks, Physica A, 334 (2004), 281-302. doi: 10.1016/j.physa.2003.10.052.

[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), 1369-1380. doi: 10.1016/j.physa.2007.10.063.

[17]

S. Albeverio and Christof Cebulla, Synchronizability of stochastic network ensembles in a model of interacting dynamical units, Physica A, 386 (2007), 503-512. doi: 10.1016/j.physa.2007.07.036.

[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), 6340-6346. doi: 10.1016/j.physleta.2008.08.059.

[19]

D. Goldstein and K. Kobayashi, On the complexity of network synchronization, SIAM Journal on Computing, 35 (2005), 567-589. doi: 10.1137/S0097539705447086.

[20]

J. Zhou, L. Xiang and Z. Liu, Global synchronization in general complex delayed dynamical networks and its applications, Physica A, 385 (2007), 729-742. doi: 10.1016/j.physa.2007.07.006.

[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), 345-354. doi: 10.1016/j.physa.2006.09.009.

[22]

W. W. Yu, A LMI-based approach to global asymptotic stability of neural networks with time varying delays, Nonlinear Dynamics, 48 (2007), 165-174. doi: 10.1007/s11071-006-9080-6.

[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), 1041-1047. doi: 10.1142/S0218127406015295.

[24]

X. Liao, S. Guo and C. Li, Stability and bifurcation analysis in tri-neuron model with time delay, Nonlinear Dynamics, 49 (2007), 319-345. doi: 10.1007/s11071-006-9137-6.

[25]

J. Wu and L. Jiao, Synchronization in complex dynamical networks with nonsymmetric coupling, Physica D, 237 (2008), 2487-2498. doi: 10.1016/j.physd.2008.03.002.

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