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March  2020, 25(3): 1059-1081. doi: 10.3934/dcdsb.2019208

Emergent collective behaviors of stochastic kuramoto oscillators

Seung-Yeal Ha 1,2, , Dongnam Ko 3,, , Chanho Min 4, and  Xiongtao Zhang 5,

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 00826, Korea (Republic of)
2. 

Korea Institute for Advanced Study, Hoegiro 85, Seoul 02455, Korea (Republic of)
3. 

DeustoTech, University of Deusto, and Facultad de Ingeniería, Universidad de Deusto, Avenida de las Universidades 24, Bilbao 48007, Basque Country, Spain
4. 

Department of Mathematical Sciences, Seoul National University, Seoul 00826, Korea (Republic of)
5. 

Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei Province, China

* Corresponding author: Dongnam Ko

Received  September 2018 Revised  May 2019 Published  March 2020 Early access  September 2019

Fund Project: The work of S.-Y. Ha is supported by the NRF grant (2017R1A2B2001864). The work of D. Ko is supported by the European Research Council under the European Union's Horizon 2020 research and innovation programme (ERC-2015-AdG-694126-DyCon) and partially supported by CNCS-UEFISCDI Grant No. PN-III-P4-ID-PCE-2016-0035. The work of X. Zhang is supported by the National Natural Science Foundation of China (Grant No. 11801194).

We study the collective dynamics of Kuramoto ensemble under uncertain coupling strength. For a finite ensemble, we can model the dynamics of the Kuramoto ensemble by the stochastic Kuramoto system with multiplicative noise. In contrast, for an infinite ensemble, the dynamics is effectively described by the Kuramoto-Sakaguchi-Fokker-Planck(KS-FP) equation with state dependent degenerate diffusion. We present emergent synchronization estimates for the stochastic and kinetic models, which yield the stability of the phase-locked state for identical Kuramoto ensemble with the same natural frequencies. We also provide a brief explanation on the mean-field limit between two models.

Keywords: Kuramoto model, mean-field limit, multiplicative noise, synchronization, random environment.
Mathematics Subject Classification: Primary: 92B25, 93E15; Secondary: 70K20.
Citation: Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete and Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. Prez 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. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.

[3]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[4]

D. Benedetto, E. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[5]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer-Verlag, London, 2006. doi: 10.1007/1-84628-186-5.

[6]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343.  doi: 10.1016/j.aml.2011.09.011.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[8]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model, Ergodic Theory Dynam. Systems, 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.

[9]

Y. Choi, S.-Y. Ha, S. 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.

[10]

L. DeVille, Transitions amongst synchronous solutions in the stochastic Kuramoto model, Nonlinearity, 25 (2012), 1473-1494.  doi: 10.1088/0951-7715/25/5/1473.

[11]

X. Ding and R. Wu, A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales, Stoch. Proc. Appl., 78 (1998), 155-171.  doi: 10.1016/S0304-4149(98)00051-9.

[12]

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.

[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]

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.

[15]

S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differential Equations, 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017.

[16]

S.-Y. Ha, Y. H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, Physica D: Nonlinear Phenomena, 2019, 132154, arXiv: 1610.01703. doi: 10.1016/j.physd.2019.132154.

[17]

S.-Y. Ha, H. W. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[18]

A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296–4301. doi: 10.23919/ACC.2004.1383983.

[19]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[20]

Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes Theor. Phys., 39 (1975), 420–422. doi: 10.1007/BFb0013365.

[21]

C. Lancellotti, On the Vlasov Limit for systems of nonlinearly coupled oscillators without noise, Transport. Theor. Stat., 34 (2005), 523-535.  doi: 10.1080/00411450508951152.

[22]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[23]

R. 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.

[24]

B. Øksendal, Stochastic Differential Equations - An Introduction with Applications, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.

[25]

S. H. Park and S. Kim, Noise-induced phase transitions in globally coupled active rotators, Phys. Rev. E, 53 (1996), 3425. doi: 10.1103/PhysRevE.53.3425.

[26]

P. Reimann, C. Van den Broeck and R. Kawai, Nonequilibrium noise in coupled phase oscillators, Phys. Rev. E, 60 (1999), 6402. doi: 10.1103/PhysRevE.60.6402.

[27]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[28]

A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Ete de Probabilites de Saint-Flour XIX - 1989, Springer-Verlag, Berlin, Heidelberg, 1464 (1991), 165–251. doi: 10.1007/BFb0085166.

[29]

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. 

[30]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

show all references

References:
[1]

J. A. Acebron, L. L. Bonilla, C. J. Prez 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. Rogge, Stability of phase locking and existence of frequency in networks of globally coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.

[3]

S. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Physics, 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895.

[4]

D. Benedetto, E. Caglioti and U. Montemagno, On the complete phase synchronization for the Kuramoto model in the mean-field limit, Commun. Math. Sci., 13 (2015), 1775-1786.  doi: 10.4310/CMS.2015.v13.n7.a6.

[5]

N. Berglund and B. Gentz, Noise-Induced Phenomena in Slow-Fast Dynamical Systems, Springer-Verlag, London, 2006. doi: 10.1007/1-84628-186-5.

[6]

F. Bolley, J. A. Cañizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339-343.  doi: 10.1016/j.aml.2011.09.011.

[7]

J. Buck and E. Buck, Biology of synchronous flashing of fireflies, Nature, 211 (1966), 562-564.  doi: 10.1038/211562a0.

[8]

H. Chiba, A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model, Ergodic Theory Dynam. Systems, 35 (2015), 762-834.  doi: 10.1017/etds.2013.68.

[9]

Y. Choi, S.-Y. Ha, S. 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.

[10]

L. DeVille, Transitions amongst synchronous solutions in the stochastic Kuramoto model, Nonlinearity, 25 (2012), 1473-1494.  doi: 10.1088/0951-7715/25/5/1473.

[11]

X. Ding and R. Wu, A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales, Stoch. Proc. Appl., 78 (1998), 155-171.  doi: 10.1016/S0304-4149(98)00051-9.

[12]

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.

[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]

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.

[15]

S.-Y. Ha, J. Jeong, S. E. Noh, Q. Xiao and X. Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differential Equations, 262 (2017), 2554-2591.  doi: 10.1016/j.jde.2016.11.017.

[16]

S.-Y. Ha, Y. H. Kim, J. Morales and J. Park, Emergence of phase concentration for the Kuramoto-Sakaguchi equation, Physica D: Nonlinear Phenomena, 2019, 132154, arXiv: 1610.01703. doi: 10.1016/j.physd.2019.132154.

[17]

S.-Y. Ha, H. W. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci., 14 (2016), 1073-1091.  doi: 10.4310/CMS.2016.v14.n4.a10.

[18]

A. Jadbabaie, N. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2004), 4296–4301. doi: 10.23919/ACC.2004.1383983.

[19]

Y. Kuramoto, Chemical Oscillations, Waves and Turbulence, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69689-3.

[20]

Y. Kuramoto, Self-entrainment of a population of coupled nonlinear oscillators, in International Symposium on Mathematical Problems in Theoretical Physics, Lecture Notes Theor. Phys., 39 (1975), 420–422. doi: 10.1007/BFb0013365.

[21]

C. Lancellotti, On the Vlasov Limit for systems of nonlinearly coupled oscillators without noise, Transport. Theor. Stat., 34 (2005), 523-535.  doi: 10.1080/00411450508951152.

[22]

X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, New York, 1994.

[23]

R. 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.

[24]

B. Øksendal, Stochastic Differential Equations - An Introduction with Applications, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.

[25]

S. H. Park and S. Kim, Noise-induced phase transitions in globally coupled active rotators, Phys. Rev. E, 53 (1996), 3425. doi: 10.1103/PhysRevE.53.3425.

[26]

P. Reimann, C. Van den Broeck and R. Kawai, Nonequilibrium noise in coupled phase oscillators, Phys. Rev. E, 60 (1999), 6402. doi: 10.1103/PhysRevE.60.6402.

[27]

H. Sakaguchi, Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79 (1988), 39-46.  doi: 10.1143/PTP.79.39.

[28]

A.-S. Sznitman, Topics in propagation of chaos, in Ecole d'Ete de Probabilites de Saint-Flour XIX - 1989, Springer-Verlag, Berlin, Heidelberg, 1464 (1991), 165–251. doi: 10.1007/BFb0085166.

[29]

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. 

[30]

M. Verwoerd and O. Mason, Global phase-locking in finite populations of phase-coupled oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 134-160.  doi: 10.1137/070686858.

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