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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[26]

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

[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.

[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.

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