American Institute of Mathematical Sciences

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January  2011, 7(1): 87-101. doi: 10.3934/jimo.2011.7.87

Cluster synchronization for linearly coupled complex networks

Xiwei Liu 1, , Tianping Chen 2, and  Wenlian Lu 3,

1. 

The Key Laboratory of Embedded System and Service Computing, Ministry of Education, Department of Computer Science and Technology, Tongji University, Shanghai, 200092, China
2. 

Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China
3. 

Center for Computational Systems Biology, Laboratory of Mathematics for Nonlinear Sciences, School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

Received  December 2009 Revised  October 2010 Published  January 2011

In this paper, the cluster synchronization for an array of linearly coupled identical chaotic systems is investigated. New coupling schemes (or coupling matrices) are proposed, by which global cluster synchronization of linearly coupled chaotic systems can be realized. Here, the number and the size of clusters (or groups) can be arbitrary. Some sufficient criteria to ensure global cluster synchronization are derived. Moreover, for any given coupling matrix, new coupled complex networks with adaptive coupling strengths are proposed, which can synchronize coupled chaotic systems by clusters. Numerical simulations are finally given to show the validity of the theoretical results.
Keywords: adaptive coupling strengths., complex networks, Cluster synchronization, left eigenvector.
Mathematics Subject Classification: Primary: 34C15, 90B10; Secondary: 68M12, 68T0.
Citation: Xiwei Liu, Tianping Chen, Wenlian Lu. Cluster synchronization for linearly coupled complex networks. Journal of Industrial & Management Optimization, 2011, 7 (1) : 87-101. doi: 10.3934/jimo.2011.7.87
References:
[1]

R. Albert and A. Barabsi, Statistical mechanics of complex networks, Rev. Modern Phys., 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.  Google Scholar

[2]

I. Belykh, V. Belykh and E. Mosekilde, Cluster synchronization modes in an ensemble of coupled chaotic oscillators, Phys. Rev. E, 63 (2001), 036216. doi: 10.1103/PhysRevE.63.036216.  Google Scholar

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I. Belykh, V. Belykh, K. Nevidin and M. Hasler, Persistent clusters in lattices of coupled nonidentical chaotic systems, Chaos, 13 (2003), 165-178. doi: 10.1063/1.1514202.  Google Scholar

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V. Belykh, I. Belykh and M. Hasler, Connected graph stability method for synchronized coupled chaotic systems, Physica D, 195 (2004), 159-187. doi: 10.1016/j.physd.2004.03.012.  Google Scholar

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S. Boccaletti, A. Farini and F. Arecchi, Adaptive synchronization of chaos for secure communication, Phys. Rev. E, 55 (1997), 4979-4981. doi: 10.1103/PhysRevE.55.4979.  Google Scholar

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K. Kaneko, Relevance of dynamic clustering to biological networks, Physica D, 75 (1994), 55. doi: 10.1016/0167-2789(94)90274-7.  Google Scholar

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Y. Kuang, "Delay Differential Equations in Populaiton Dynamics," Academic Press, New York, 1993. Google Scholar

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Y. Li and S. Chen, Optimal traffic signal control for an $M\times N$ traffic network, Journal of Industrial and Management Optimization, 4 (2008), 661-672. Google Scholar

[9]

T. Liao and S. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, Solitons and Fractals, 11 (2000), 1387-1396. doi: 10.1016/S0960-0779(99)00051-X.  Google Scholar

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X. Liu and T. Chen, Exponential synchronization of the linearly coupled dynamical networks with delays, Chin. Ann. Math. Ser. B, 28 (2007), 737-746. doi: 10.1007/s11401-006-0194-4.  Google Scholar

[11]

W. Lu and T. Chen, Synchronization of coupled connected neural networks with delays, IEEE Trans. Circuits Syst.-I, 51 (2004), 2491-2503.  Google Scholar

[12]

W. Lu and T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213 (2006), 214-230. doi: 10.1016/j.physd.2005.11.009.  Google Scholar

[13]

Z. Ma, Z. Liu and G. Zhang, A new method to realize cluster synchronization in connected chaotic networks, Chao, 16 (2006), 023103. Google Scholar

[14]

R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098.  Google Scholar

[15]

M. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

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L. Pecora and T. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

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A. Pogromsky, G. Santoboni and H. Nijmeijer, An ultimate bound on the trajectories of the Lorenz system and its applications, Nonlinearity, 16 (2003), 1597-1605. doi: 10.1088/0951-7715/16/5/303.  Google Scholar

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W. Qin and G. Chen, Coupling schemes for cluster synchronization in coupled Josephson equations, Physica D, 197 (2004), 375-391. doi: 10.1016/j.physd.2004.07.011.  Google Scholar

[19]

N. Rulkov, Images of synchronized chaos: experiments with circuits, Chaos, 6 (1996), 262-279. doi: 10.1063/1.166174.  Google Scholar

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

[21]

X. Wang and G. Chen, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. -I, 49 (2002), 54-62.  Google Scholar

[22]

X. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. Bifur. Chaos, 12 (2002), 187-192. doi: 10.1142/S0218127402004292.  Google Scholar

[23]

D. Watts and S. Strogatz, Collective dynamics of small-world, Nature, 393 (1998), 440-442. doi: 10.1038/30918.  Google Scholar

[24]

G. Wei and Y. Q. Jia, Synchronization-based image edge detection, Europhys. Lett., 59 (2002), 814-819. doi: 10.1209/epl/i2002-00115-8.  Google Scholar

[25]

C. Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18 (2005), 1057-1064. doi: 10.1088/0951-7715/18/3/007.  Google Scholar

[26]

C. Wu and L. Chua, Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst.-I, 42 (1995), 430-447.  Google Scholar

[27]

Q. Xie, G. Chen and E. Bollt, Hybrid chaos synchronization and its application in information processing, Math. Comput. Model., 35 (2002), 145-163. doi: 10.1016/S0895-7177(01)00157-1.  Google Scholar

[28]

T. Yang and L. O. Chua, Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication, Int. J. Bifur. Chaos, 7 (1997), 645-664. doi: 10.1142/S0218127497000443.  Google Scholar

show all references

References:
[1]

R. Albert and A. Barabsi, Statistical mechanics of complex networks, Rev. Modern Phys., 74 (2002), 47-97. doi: 10.1103/RevModPhys.74.47.  Google Scholar

[2]

I. Belykh, V. Belykh and E. Mosekilde, Cluster synchronization modes in an ensemble of coupled chaotic oscillators, Phys. Rev. E, 63 (2001), 036216. doi: 10.1103/PhysRevE.63.036216.  Google Scholar

[3]

I. Belykh, V. Belykh, K. Nevidin and M. Hasler, Persistent clusters in lattices of coupled nonidentical chaotic systems, Chaos, 13 (2003), 165-178. doi: 10.1063/1.1514202.  Google Scholar

[4]

V. Belykh, I. Belykh and M. Hasler, Connected graph stability method for synchronized coupled chaotic systems, Physica D, 195 (2004), 159-187. doi: 10.1016/j.physd.2004.03.012.  Google Scholar

[5]

S. Boccaletti, A. Farini and F. Arecchi, Adaptive synchronization of chaos for secure communication, Phys. Rev. E, 55 (1997), 4979-4981. doi: 10.1103/PhysRevE.55.4979.  Google Scholar

[6]

K. Kaneko, Relevance of dynamic clustering to biological networks, Physica D, 75 (1994), 55. doi: 10.1016/0167-2789(94)90274-7.  Google Scholar

[7]

Y. Kuang, "Delay Differential Equations in Populaiton Dynamics," Academic Press, New York, 1993. Google Scholar

[8]

Y. Li and S. Chen, Optimal traffic signal control for an $M\times N$ traffic network, Journal of Industrial and Management Optimization, 4 (2008), 661-672. Google Scholar

[9]

T. Liao and S. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications, Chaos, Solitons and Fractals, 11 (2000), 1387-1396. doi: 10.1016/S0960-0779(99)00051-X.  Google Scholar

[10]

X. Liu and T. Chen, Exponential synchronization of the linearly coupled dynamical networks with delays, Chin. Ann. Math. Ser. B, 28 (2007), 737-746. doi: 10.1007/s11401-006-0194-4.  Google Scholar

[11]

W. Lu and T. Chen, Synchronization of coupled connected neural networks with delays, IEEE Trans. Circuits Syst.-I, 51 (2004), 2491-2503.  Google Scholar

[12]

W. Lu and T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems, Physica D, 213 (2006), 214-230. doi: 10.1016/j.physd.2005.11.009.  Google Scholar

[13]

Z. Ma, Z. Liu and G. Zhang, A new method to realize cluster synchronization in connected chaotic networks, Chao, 16 (2006), 023103. Google Scholar

[14]

R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators, SIAM J. Appl. Math., 50 (1990), 1645-1662. doi: 10.1137/0150098.  Google Scholar

[15]

M. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

[16]

L. Pecora and T. Carroll, Master stability functions for synchronized coupled systems, Phys. Rev. Lett., 80 (1998), 2109-2112. doi: 10.1103/PhysRevLett.80.2109.  Google Scholar

[17]

A. Pogromsky, G. Santoboni and H. Nijmeijer, An ultimate bound on the trajectories of the Lorenz system and its applications, Nonlinearity, 16 (2003), 1597-1605. doi: 10.1088/0951-7715/16/5/303.  Google Scholar

[18]

W. Qin and G. Chen, Coupling schemes for cluster synchronization in coupled Josephson equations, Physica D, 197 (2004), 375-391. doi: 10.1016/j.physd.2004.07.011.  Google Scholar

[19]

N. Rulkov, Images of synchronized chaos: experiments with circuits, Chaos, 6 (1996), 262-279. doi: 10.1063/1.166174.  Google Scholar

[20]

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

[21]

X. Wang and G. Chen, Synchronization in scale-free dynamical networks: robustness and fragility, IEEE Trans. Circuits Syst. -I, 49 (2002), 54-62.  Google Scholar

[22]

X. Wang and G. Chen, Synchronization in small-world dynamical networks, Int. J. Bifur. Chaos, 12 (2002), 187-192. doi: 10.1142/S0218127402004292.  Google Scholar

[23]

D. Watts and S. Strogatz, Collective dynamics of small-world, Nature, 393 (1998), 440-442. doi: 10.1038/30918.  Google Scholar

[24]

G. Wei and Y. Q. Jia, Synchronization-based image edge detection, Europhys. Lett., 59 (2002), 814-819. doi: 10.1209/epl/i2002-00115-8.  Google Scholar

[25]

C. Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity, 18 (2005), 1057-1064. doi: 10.1088/0951-7715/18/3/007.  Google Scholar

[26]

C. Wu and L. Chua, Synchronization in an array of linearly coupled dynamical systems, IEEE Trans. Circuits Syst.-I, 42 (1995), 430-447.  Google Scholar

[27]

Q. Xie, G. Chen and E. Bollt, Hybrid chaos synchronization and its application in information processing, Math. Comput. Model., 35 (2002), 145-163. doi: 10.1016/S0895-7177(01)00157-1.  Google Scholar

[28]

T. Yang and L. O. Chua, Impulsive control and synchronization of nonlinear dynamical systems and application to secure communication, Int. J. Bifur. Chaos, 7 (1997), 645-664. doi: 10.1142/S0218127497000443.  Google Scholar

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