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

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.
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
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

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

[1]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[2]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[3]

Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021001

[4]

Guillaume Cantin, M. A. Aziz-Alaoui. Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020283

[5]

Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China. Electronic Research Archive, 2021, 29 (1) : 1661-1679. doi: 10.3934/era.2020085

[6]

Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165

[7]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[8]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346

[9]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344

[10]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006

[11]

Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021001

[12]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[13]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[14]

Shigui Ruan. Nonlinear dynamics in tumor-immune system interaction models with delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 541-602. doi: 10.3934/dcdsb.2020282

[15]

Yicheng Liu, Yipeng Chen, Jun Wu, Xiao Wang. Periodic consensus in network systems with general distributed processing delays. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021002

[16]

Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020

[17]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[18]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[19]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[20]

Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (25)

[Back to Top]