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Cluster synchronization for linearly coupled complex networks
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 |
References:
[1] |
R. Albert and A. Barabsi, Statistical mechanics of complex networks,, Rev. Modern Phys., 74 (2002), 47.
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).
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.
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.
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.
doi: 10.1103/PhysRevE.55.4979. |
[6] |
K. Kaneko, Relevance of dynamic clustering to biological networks,, Physica D, 75 (1994).
doi: 10.1016/0167-2789(94)90274-7. |
[7] |
Y. Kuang, "Delay Differential Equations in Populaiton Dynamics,", Academic Press, (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. |
[9] |
T. Liao and S. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications,, Chaos, 11 (2000), 1387.
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.
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.
|
[12] |
W. Lu and T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems,, Physica D, 213 (2006), 214.
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). |
[14] |
R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators,, SIAM J. Appl. Math., 50 (1990), 1645.
doi: 10.1137/0150098. |
[15] |
M. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.
doi: 10.1137/S003614450342480. |
[16] |
L. Pecora and T. Carroll, Master stability functions for synchronized coupled systems,, Phys. Rev. Lett., 80 (1998), 2109.
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.
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.
doi: 10.1016/j.physd.2004.07.011. |
[19] |
N. Rulkov, Images of synchronized chaos: experiments with circuits,, Chaos, 6 (1996), 262.
doi: 10.1063/1.166174. |
[20] |
S. Strogatz, Exploring complex networks,, Nature, 410 (2001), 268.
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.
|
[22] |
X. Wang and G. Chen, Synchronization in small-world dynamical networks,, Int. J. Bifur. Chaos, 12 (2002), 187.
doi: 10.1142/S0218127402004292. |
[23] |
D. Watts and S. Strogatz, Collective dynamics of small-world,, Nature, 393 (1998), 440.
doi: 10.1038/30918. |
[24] |
G. Wei and Y. Q. Jia, Synchronization-based image edge detection,, Europhys. Lett., 59 (2002), 814.
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.
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.
|
[27] |
Q. Xie, G. Chen and E. Bollt, Hybrid chaos synchronization and its application in information processing,, Math. Comput. Model., 35 (2002), 145.
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.
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.
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).
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.
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.
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.
doi: 10.1103/PhysRevE.55.4979. |
[6] |
K. Kaneko, Relevance of dynamic clustering to biological networks,, Physica D, 75 (1994).
doi: 10.1016/0167-2789(94)90274-7. |
[7] |
Y. Kuang, "Delay Differential Equations in Populaiton Dynamics,", Academic Press, (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. |
[9] |
T. Liao and S. Tsai, Adaptive synchronization of chaotic systems and its application to secure communications,, Chaos, 11 (2000), 1387.
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.
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.
|
[12] |
W. Lu and T. Chen, New approach to synchronization analysis of linearly coupled ordinary differential systems,, Physica D, 213 (2006), 214.
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). |
[14] |
R. Mirollo and S. Strogatz, Synchronization of pulse-coupled biological oscillators,, SIAM J. Appl. Math., 50 (1990), 1645.
doi: 10.1137/0150098. |
[15] |
M. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167.
doi: 10.1137/S003614450342480. |
[16] |
L. Pecora and T. Carroll, Master stability functions for synchronized coupled systems,, Phys. Rev. Lett., 80 (1998), 2109.
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.
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.
doi: 10.1016/j.physd.2004.07.011. |
[19] |
N. Rulkov, Images of synchronized chaos: experiments with circuits,, Chaos, 6 (1996), 262.
doi: 10.1063/1.166174. |
[20] |
S. Strogatz, Exploring complex networks,, Nature, 410 (2001), 268.
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.
|
[22] |
X. Wang and G. Chen, Synchronization in small-world dynamical networks,, Int. J. Bifur. Chaos, 12 (2002), 187.
doi: 10.1142/S0218127402004292. |
[23] |
D. Watts and S. Strogatz, Collective dynamics of small-world,, Nature, 393 (1998), 440.
doi: 10.1038/30918. |
[24] |
G. Wei and Y. Q. Jia, Synchronization-based image edge detection,, Europhys. Lett., 59 (2002), 814.
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.
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.
|
[27] |
Q. Xie, G. Chen and E. Bollt, Hybrid chaos synchronization and its application in information processing,, Math. Comput. Model., 35 (2002), 145.
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.
doi: 10.1142/S0218127497000443. |
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