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December  2015, 10(4): 787-807. doi: 10.3934/nhm.2015.10.787

Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747

2. 

Department of Mathematics, Myongji University, Yong-In, 449-728, South Korea

3. 

Department of Mathematical Sciences, Seoul National University, Seoul, 151-747

Received  September 2014 Revised  June 2015 Published  October 2015

We study the practical synchronization of the Kuramoto dynamics of units distributed over networks. The unit dynamics on the nodes of the network are governed by the interplay between their own intrinsic dynamics and Kuramoto coupling dynamics. We present two sufficient conditions for practical synchronization under homogeneous and heterogeneous forcing. For practical synchronization estimates, we employ the configuration diameter as a Lyapunov functional, and derive a Gronwall-type differential inequality for this value.
Citation: Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks & Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787
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show all references

References:
[1]

Rev. Mod. Phys., 77 (2005), 137-185. doi: 10.1103/RevModPhys.77.137.  Google Scholar

[2]

Chaos, 18 (2008), 037112, 10pp. doi: 10.1063/1.2952447.  Google Scholar

[3]

Graduate Text in Mathematics, 169. Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[4]

Nonlinear Dynam., 56 (2009), 57-68. doi: 10.1007/s11071-008-9379-6.  Google Scholar

[5]

Nature, 211 (1966), 562-564. doi: 10.1038/211562a0.  Google Scholar

[6]

Chaos, 18 (2008), 043128, 9pp. doi: 10.1063/1.3049136.  Google Scholar

[7]

Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011.  Google Scholar

[8]

Physica D, 240 (2011), 32-44. doi: 10.1016/j.physd.2010.08.004.  Google Scholar

[9]

IEEE Trans. Automatic Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.  Google Scholar

[10]

in Proceedings of the 30th Chinese Control Conference, Yantai, China 2011. Google Scholar

[11]

SIAM J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X.  Google Scholar

[12]

Physics Letters A, 262 (1999), 50-60. doi: 10.1016/S0375-9601(99)00667-2.  Google Scholar

[13]

Physica D, 239 (2010), 1692-1700. doi: 10.1016/j.physd.2010.05.003.  Google Scholar

[14]

Nonlinearity, 23 (2010), 3139-3156. doi: 10.1088/0951-7715/23/12/008.  Google Scholar

[15]

Communications in Mathematical Sciences, 12 (2014), 485-508. doi: 10.4310/CMS.2014.v12.n3.a5.  Google Scholar

[16]

in Proceedings of the American Control Conference. Boston Massachusetts 2004. Google Scholar

[17]

in Proceedings of 12th International Conference on Control, Automation and Systems, Jeju Island, Korea 2012. Google Scholar

[18]

Springer-Verlag Berlin 1984. doi: 10.1007/978-3-642-69689-3.  Google Scholar

[19]

Lecture notes in theoretical physics, 39 (1975), 420-422.  Google Scholar

[20]

Abstr. Appl. Anal., 2013 (2013), Art. ID 483269, 7 pp.  Google Scholar

[21]

Nonlinear Dynam., 69 (2012), 1285-1292. doi: 10.1007/s11071-012-0346-x.  Google Scholar

[22]

Int. J. Mod Phys C, 23 (2012), 1250073 14pp. Google Scholar

[23]

J. Nonlinear Sci., 17 (2007), 309-347. doi: 10.1007/s00332-006-0806-x.  Google Scholar

[24]

Physica D, 205 (2005), 249-266. doi: 10.1016/j.physd.2005.01.017.  Google Scholar

[25]

J. Stat. Phy., 63 (1991), 613-635. doi: 10.1007/BF01029202.  Google Scholar

[26]

Chaos, 18 (2008), 037113, 6pp. doi: 10.1063/1.2930766.  Google Scholar

[27]

Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.  Google Scholar

[28]

Prog. Theor. Phys., 79 (1988), 39-46. doi: 10.1143/PTP.79.39.  Google Scholar

[29]

in Dynamics and control of hybrid mechanical systems (eds. G. Leonov, H. Nijmeijer, A. Pogromsky and A. Fradkov), Singapore, World Scientific, (2010), 195-210. doi: 10.1142/9789814282321_0013.  Google Scholar

[30]

J. Math. Biol., 25 (1987), 327-347. doi: 10.1007/BF00276440.  Google Scholar

[31]

Springer New York 1980.  Google Scholar

[32]

J. Theor. Biol., 16 (1987), 15-42. doi: 10.1016/0022-5193(67)90051-3.  Google Scholar

[33]

Math. Ann., 71 (1912), 441-479. doi: 10.1007/BF01456804.  Google Scholar

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