June 2018, 13(2): 323-337. doi: 10.3934/nhm.2018014

Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring

1. 

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

2. 

Department of Mathematics and Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China

1Corresponding Author: Zhuchun Li

Received  September 2017 Published  May 2018

Fund Project: This work was supported by National Natural Science Foundation of China Grants 11401135, 11671109 and 11731010. Z. Li was also supported by the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201501 and HIT.PIRS.201610)

We consider the dynamics of bidirectionally coupled identical Kuramoto oscillators in a ring, where each oscillator is influenced sinusoidally by two neighboring oscillator. Our purpose is to understand its dynamics in the following aspects: 1. identify all the phase-locked states (or equilibria) with stability or instability; 2. estimate the basins for stable phase-locked states; 3. identify the convergence rate towards phase-locked states. The crucial tool in this work is the celebrated theory of Łojasiewicz inequality.

Citation: Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014
References:
[1]

P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems, Systems Control Letters, 55 (2006), 573-577. doi: 10.1016/j.sysconle.2006.01.002.

[2]

J. A. AcebrónL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. doi: 10.1103/RevModPhys.77.137.

[3]

J. BolteA. Daniilidis and A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optimiz., 17 (2007), 1205-1233. doi: 10.1137/050644641.

[4]

Y. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phaselocked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011.

[5]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.

[6]

R. Delabays, T. Coletta and P. Jacquod, Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks, J. Math. Phys., 57 (2016), 032701, 21pp. doi: 10.1063/1.4943296.

[7]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X.

[8]

F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012.

[9]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542.

[10]

S.-Y. HaS.-E. Noh and J. Park, practical synchronization of generalized kuramoto systems with an intrinsic dynamics, Netw. Heterog. Media, 10 (2015), 787-807. doi: 10.3934/nhm.2015.10.787.

[11]

S.-Y. Ha and M. Kang, On the Basin of Attractors for the Unidirectionally Coupled Kuramoto Model in a Ring, SIAM. J. Appl. Math., 72 (2012), 1549-1574. doi: 10.1137/110829416.

[12]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics. Lecture Notes Phys., 39 (1975), 420.

[13]

Z. Li and X. Xue, Convergence for gradient-type systems with periodicity and analyticity, Submitted, 2017.

[14]

Z. Li, X. Xue and D. Yu, On the Łojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 0227041, 20pp. doi: 10.1063/1.4908104.

[15]

Z. LiX. Xue and D. Yu, Synchronization and tansient stability in power grids based on Łojasiewicz inequalities, SIAM J. Control Optim., 52 (2014), 2482-2511. doi: 10.1137/130950604.

[16]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. in Les Équations aux Dérivées Partielles, Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87-89.

[17]

P. Monzón and F. Paganini, Global considerations on the Kuramoto model of sinusoidally coupled oscillators, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, (2005), 3923-3928. doi: 10.1109/CDC.2005.1582774.

[18]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchrnization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[19]

J. A. Rogge and D. Aeyels, Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A: Math. Gen., 37 (2004), 11135-11148. doi: 10.1088/0305-4470/37/46/004.

[20]

D. A. Wiley, S. H. Strogatz and L. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103, 8pp. doi: 10.1063/1.2165594.

show all references

References:
[1]

P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems, Systems Control Letters, 55 (2006), 573-577. doi: 10.1016/j.sysconle.2006.01.002.

[2]

J. A. AcebrónL. L. BonillaC. J. P. Pérez VicenteF. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77 (2005), 137-185. doi: 10.1103/RevModPhys.77.137.

[3]

J. BolteA. Daniilidis and A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optimiz., 17 (2007), 1205-1233. doi: 10.1137/050644641.

[4]

Y. ChoiS.-Y. HaS. Jung and Y. Kim, Asymptotic formation and orbital stability of phaselocked states for the Kuramoto model, Physica D, 241 (2012), 735-754. doi: 10.1016/j.physd.2011.11.011.

[5]

N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Automatic Control, 54 (2009), 353-357. doi: 10.1109/TAC.2008.2007884.

[6]

R. Delabays, T. Coletta and P. Jacquod, Multistability of phase-locking and topological winding numbers in locally coupled Kuramoto models on single-loop networks, J. Math. Phys., 57 (2016), 032701, 21pp. doi: 10.1063/1.4943296.

[7]

F. Dörfler and F. Bullo, On the critical coupling for Kuramoto oscillators, SIAM. J. Appl. Dyn. Syst., 10 (2011), 1070-1099. doi: 10.1137/10081530X.

[8]

F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012.

[9]

G. B. Ermentrout, Synchronization in a pool of mutually coupled oscillators with random frequencies, J. Math. Biol., 22 (1985), 1-9. doi: 10.1007/BF00276542.

[10]

S.-Y. HaS.-E. Noh and J. Park, practical synchronization of generalized kuramoto systems with an intrinsic dynamics, Netw. Heterog. Media, 10 (2015), 787-807. doi: 10.3934/nhm.2015.10.787.

[11]

S.-Y. Ha and M. Kang, On the Basin of Attractors for the Unidirectionally Coupled Kuramoto Model in a Ring, SIAM. J. Appl. Math., 72 (2012), 1549-1574. doi: 10.1137/110829416.

[12]

Y. Kuramoto, International Symposium on Mathematical Problems in Mathematical Physics. Lecture Notes Phys., 39 (1975), 420.

[13]

Z. Li and X. Xue, Convergence for gradient-type systems with periodicity and analyticity, Submitted, 2017.

[14]

Z. Li, X. Xue and D. Yu, On the Łojasiewicz exponent of Kuramoto model, J. Math. Phys., 56 (2015), 0227041, 20pp. doi: 10.1063/1.4908104.

[15]

Z. LiX. Xue and D. Yu, Synchronization and tansient stability in power grids based on Łojasiewicz inequalities, SIAM J. Control Optim., 52 (2014), 2482-2511. doi: 10.1137/130950604.

[16]

S. Łojasiewicz, Une propriété topologique des sous-ensembles analytiques réels. in Les Équations aux Dérivées Partielles, Éditions du Centre National de la Recherche Scientifique, Paris, 1963, 87-89.

[17]

P. Monzón and F. Paganini, Global considerations on the Kuramoto model of sinusoidally coupled oscillators, Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, (2005), 3923-3928. doi: 10.1109/CDC.2005.1582774.

[18]

A. Pikovsky, M. Rosenblum and J. Kurths, Synchrnization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511755743.

[19]

J. A. Rogge and D. Aeyels, Stability of phase locking in a ring of unidirectionally coupled oscillators, J. Phys. A: Math. Gen., 37 (2004), 11135-11148. doi: 10.1088/0305-4470/37/46/004.

[20]

D. A. Wiley, S. H. Strogatz and L. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103, 8pp. doi: 10.1063/1.2165594.

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