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Sparse stabilization of dynamical systems driven by attraction and avoidance forces
Asymptotic synchronous behavior of Kuramoto type models with frustrations
1. | Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 151-747 |
2. | Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea |
3. | Department of Mathematics, Harbin Institute of Technology, Harbin 150001 |
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. 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. |
[2] |
D. Aeyels and J. A. Rogge, Stability of phase locking and existence of entrainment in networks of globally coupled oscillators, in Proc. 6th IFAC Symposium on Nonlinear Control Systems, 3 (2004), 1031-1036. |
[3] |
P. Ashwin and J. W. Swift, The dynamics of $n$ weakly coupled identical oscillators, J. Nonlinear Sci., 2 (1992), 69-108.
doi: 10.1007/BF02429852. |
[4] |
L. L. Bonilla, J. C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 67 (1992), 313-330.
doi: 10.1007/BF01049037. |
[5] |
H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, ().
|
[6] |
Y. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2010), 32-44.
doi: 10.1016/j.physd.2010.08.004. |
[7] |
Y. Choi, S.-Y. Ha, S.-E. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[8] |
N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357.
doi: 10.1109/TAC.2008.2007884. |
[9] |
H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076.
doi: 10.1103/PhysRevLett.68.1073. |
[10] |
F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89.
doi: 10.1016/j.physd.2007.06.025. |
[11] |
J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.
doi: 10.4310/CMS.2013.v11.n2.a7. |
[12] |
F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, submitted, (2013). |
[13] |
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. |
[14] |
F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.
doi: 10.1073/pnas.1212134110. |
[15] |
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. |
[16] |
S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators, Quart. Appl. Math., 71 (2013), 707-728.
doi: 10.1090/S0033-569X-2013-01302-0. |
[17] |
S.-Y. Ha, T. Ha and J. H. Kim, On the complete synchronization for the globally coupled Kuramoto model, Physica D, 239 (2010), 1692-1700.
doi: 10.1016/j.physd.2010.05.003. |
[18] |
S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103. |
[19] |
S.-Y. Ha, and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci., 12 (2014), 485-508.
doi: 10.4310/CMS.2014.v12.n3.a5. |
[20] |
S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.
doi: 10.1016/j.jde.2013.07.013. |
[21] |
S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto phase model, J. Differential Equations, 251 (2011), 2685-2695.
doi: 10.1016/j.jde.2011.04.004. |
[22] |
A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. |
[23] |
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69689-3. |
[24] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422. |
[25] |
C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113.
doi: 10.1063/1.3068353. |
[26] |
Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. |
[27] |
S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719. |
[28] |
R. E. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2007), 309-347.
doi: 10.1007/s00332-006-0806-x. |
[29] |
R. E. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillator, Physica D, 205 (2005), 249-266.
doi: 10.1016/j.physd.2005.01.017. |
[30] |
R. E. Mirollo and S. H. Strogatz, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys., 63 (1991), 613-635.
doi: 10.1007/BF01029202. |
[31] |
E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003.
doi: 10.1209/0295-5075/83/68003. |
[32] |
K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035.
doi: 10.1103/PhysRevE.57.5030. |
[33] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743. |
[34] |
H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys., 76 (1986), 576-581.
doi: 10.1143/PTP.76.576. |
[35] |
S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.
doi: 10.1016/S0167-2789(00)00094-4. |
[36] |
T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22. |
[37] |
J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.
doi: 10.1007/BF01048044. |
[38] |
A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
doi: 10.1016/0022-5193(67)90051-3. |
[39] |
Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707. |
show all references
References:
[1] |
J. A. Acebrón, L. L. Bonilla, C. J. P. Vicente, F. 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. |
[2] |
D. Aeyels and J. A. Rogge, Stability of phase locking and existence of entrainment in networks of globally coupled oscillators, in Proc. 6th IFAC Symposium on Nonlinear Control Systems, 3 (2004), 1031-1036. |
[3] |
P. Ashwin and J. W. Swift, The dynamics of $n$ weakly coupled identical oscillators, J. Nonlinear Sci., 2 (1992), 69-108.
doi: 10.1007/BF02429852. |
[4] |
L. L. Bonilla, J. C. Neu and R. Spigler, Nonlinear stability of incoherence and collective synchronization in a population of coupled oscillators, J. Stat. Phys., 67 (1992), 313-330.
doi: 10.1007/BF01049037. |
[5] |
H. Chiba, A proof of the Kuramoto's conjecture for a bifurcation structure of the infinite dimensional Kuramoto model,, preprint, ().
|
[6] |
Y. Choi, S.-Y. Ha and S.-B. Yun, Complete synchronization of Kuramoto oscillators with finite inertia, Physica D, 240 (2010), 32-44.
doi: 10.1016/j.physd.2010.08.004. |
[7] |
Y. Choi, S.-Y. Ha, S.-E. Jung and Y. Kim, Asymptotic formation and orbital stability of phase-locked states for the Kuramoto model, Physica D, 241 (2012), 735-754.
doi: 10.1016/j.physd.2011.11.011. |
[8] |
N. Chopra and M. W. Spong, On exponential synchronization of Kuramoto oscillators, IEEE Trans. Autom. Control, 54 (2009), 353-357.
doi: 10.1109/TAC.2008.2007884. |
[9] |
H. Daido, Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions, Phys. Rev. Lett., 68 (1992), 1073-1076.
doi: 10.1103/PhysRevLett.68.1073. |
[10] |
F. De Smet and D. Aeyels, Partial entrainment in the finite Kuramoto-Sakaguchi model, Physica D, 234 (2007), 81-89.
doi: 10.1016/j.physd.2007.06.025. |
[11] |
J.-G. Dong and X. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11 (2013), 465-480.
doi: 10.4310/CMS.2013.v11.n2.a7. |
[12] |
F. Dörfler and F. Bullo, Synchronization in complex oscillator networks: A survey, submitted, (2013). |
[13] |
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. |
[14] |
F. Dörfler, M. Chertkov and F. Bullo, Synchronization in complex oscillator networks and smart grids, Proceedings of the National Academy of Sciences, 110 (2013), 2005-2010.
doi: 10.1073/pnas.1212134110. |
[15] |
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. |
[16] |
S.-Y. Ha and M.-J. Kang, Fast and slow relaxations to bi-cluster configurations for the ensemble of Kuramoto oscillators, Quart. Appl. Math., 71 (2013), 707-728.
doi: 10.1090/S0033-569X-2013-01302-0. |
[17] |
S.-Y. Ha, T. Ha and J. H. Kim, On the complete synchronization for the globally coupled Kuramoto model, Physica D, 239 (2010), 1692-1700.
doi: 10.1016/j.physd.2010.05.003. |
[18] |
S.-Y. Ha, C. Lattanzio, B. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103. |
[19] |
S.-Y. Ha, and Z. Li, Complete synchronization of Kuramoto oscillators with hierarchical leadership, Commun. Math. Sci., 12 (2014), 485-508.
doi: 10.4310/CMS.2014.v12.n3.a5. |
[20] |
S.-Y. Ha, Z. Li and X. Xue, Formation of phase-locked states in a population of locally interacting Kuramoto oscillators, J. Differential Equations, 255 (2013), 3053-3070.
doi: 10.1016/j.jde.2013.07.013. |
[21] |
S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto phase model, J. Differential Equations, 251 (2011), 2685-2695.
doi: 10.1016/j.jde.2011.04.004. |
[22] |
A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proc. American Control Conf., 5 (2004), 4296-4301. |
[23] |
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984.
doi: 10.1007/978-3-642-69689-3. |
[24] |
Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics, 39 (1975), 420-422. |
[25] |
C. R. Laing, Chimera states in heterogeneous networks, Chaos, 19 (2009), 013113.
doi: 10.1063/1.3068353. |
[26] |
Z. Levnajić, Emergent multistability and frustration in phase-repulsive networks of oscillators, Phys. Rev. E, 84 (2011), 016231. |
[27] |
S. Lück and A. Pikovsky, Dynamics of multi-frequency oscillator ensembles with resonant coupling, Phys. Lett. A, 375 (2011), 2714-2719. |
[28] |
R. E. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Sci., 17 (2007), 309-347.
doi: 10.1007/s00332-006-0806-x. |
[29] |
R. E. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillator, Physica D, 205 (2005), 249-266.
doi: 10.1016/j.physd.2005.01.017. |
[30] |
R. E. Mirollo and S. H. Strogatz, Stability of incoherence in a population of coupled oscillators, J. Stat. Phys., 63 (1991), 613-635.
doi: 10.1007/BF01029202. |
[31] |
E. Oh, C. Choi, B. Kahng and D. Kim, Modular synchronization in complex networks with a gauge Kuramoto model, EPL, 83 (2008), 68003.
doi: 10.1209/0295-5075/83/68003. |
[32] |
K. Park, S. W. Rhee and M. Y. Choi, Glass synchronization in the network of oscillators with random phase shift, Phys. Rev. E, 57 (1998), 5030-5035.
doi: 10.1103/PhysRevE.57.5030. |
[33] |
A. Pikovsky, M. Rosenblum and J. Kurths, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511755743. |
[34] |
H. Sakaguchi and Y. Kuramoto, A soluble active rotator model showing phase transitions via mutual entrainment, Prog. Theor. Phys., 76 (1986), 576-581.
doi: 10.1143/PTP.76.576. |
[35] |
S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143 (2000), 1-20.
doi: 10.1016/S0167-2789(00)00094-4. |
[36] |
T. Tanaka, T. Aoki and T. Aoyagi, Dynamics in co-evolving networks of active elements, Forma, 24 (2009), 17-22. |
[37] |
J. L. van Hemmen and W. F. Wreszinski, Lyapunov function for the Kuramoto model of nonlinearly coupled oscillators, J. Stat. Phys., 72 (1993), 145-166.
doi: 10.1007/BF01048044. |
[38] |
A. T. Winfree, Biological rhythms and the behavior of populations of coupled oscillators, J. Theor. Biol., 16 (1967), 15-42.
doi: 10.1016/0022-5193(67)90051-3. |
[39] |
Z. G. Zheng, Frustration effect on synchronization and chaos in coupled oscillators, Chin. Phys. Soc., 10 (2011), 703-707. |
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