American Institute of Mathematical Sciences

Advanced
  • Previous Article
    (Almost) Everything you always wanted to know about deterministic control problems in stratified domains
  • NHM Home
  • This Issue
  • Next Article
    Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks
December  2015, 10(4): 787-807. doi: 10.3934/nhm.2015.10.787

Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics

Seung-Yeal Ha 1, , Se Eun Noh 2, and  Jinyeong Park 3,

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.
Keywords: practical synchronization., external force, intrinsic dynamics, Kuramoto model.
Mathematics Subject Classification: Primary: 70F99; Secondary: 92B2.
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
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

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

[1]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2021, 13 (1) : 1-23. doi: 10.3934/jgm.2020032
[2]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225
[3]

Juan Manuel Pastor, Javier García-Algarra, Javier Galeano, José María Iriondo, José J. Ramasco. A simple and bounded model of population dynamics for mutualistic networks. Networks & Heterogeneous Media, 2015, 10 (1) : 53-70. doi: 10.3934/nhm.2015.10.53
[4]

Yu Jin, Xiao-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1991-2010. doi: 10.3934/dcdsb.2020362
[5]

Daniel Amin, Mikael Vejdemo-Johansson. Intrinsic disease maps using persistent cohomology. Foundations of Data Science, 2021  doi: 10.3934/fods.2021008
[6]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[7]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[8]

Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109
[9]

Wei Wang, Wanbiao Ma, Xiulan Lai. Sufficient conditions for global dynamics of a viral infection model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3989-4011. doi: 10.3934/dcdsb.2020271
[10]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[11]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[12]

Rafael López, Óscar Perdomo. Constant-speed ramps for a central force field. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3447-3464. doi: 10.3934/dcds.2021003
[13]

Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269
[14]

Zhikun She, Xin Jiang. Threshold dynamics of a general delayed within-host viral infection model with humoral immunity and two modes of virus transmission. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3835-3861. doi: 10.3934/dcdsb.2020259
[15]

Shuting Chen, Zengji Du, Jiang Liu, Ke Wang. The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021098
[16]

Jicheng Liu, Meiling Zhao. Normal deviation of synchronization of stochastic coupled systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021079
[17]

Prabir Panja, Soovoojeet Jana, Shyamal kumar Mondal. Dynamics of a stage structure prey-predator model with ratio-dependent functional response and anti-predator behavior of adult prey. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 391-405. doi: 10.3934/naco.2020033
[18]

Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021015
[19]

Xiaochun Gu, Fang Han, Zhijie Wang, Kaleem Kashif, Wenlian Lu. Enhancement of gamma oscillations in E/I neural networks by increase of difference between external inputs. Electronic Research Archive, , () : -. doi: 10.3934/era.2021035
[20]

Gelasio Salaza, Edgardo Ugalde, Jesús Urías. Master--slave synchronization of affine cellular automaton pairs. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 491-502. doi: 10.3934/dcds.2005.13.491

2019 Impact Factor: 1.053

Tools

Metrics

  • PDF downloads (126)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences