June 2018, 13(2): 297-322. doi: 10.3934/nhm.2018013

Uniform stability and mean-field limit for the augmented Kuramoto model

1. 

Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul 08826, Korea

2. 

Korea Institute for Advanced Study, Hoegiro 87, Seoul 02455, Korea

3. 

Department of Mathematical Sciences, Seoul National University, Seoul 08826, Korea

4. 

Department of Mathematics and Research Institute of Natural Sciences, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, Korea

5. 

Center for Mathematical sciences, Huazhong University of Science and Technology, Wuhan, China

* Corresponding author: Jinyeong Park

Received  July 2017 Revised  March 2018 Published  May 2018

Fund Project: The works of S.-Y. Ha and X. Zhang are supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1401-03). The work of J. Kim is supported by the German Research Foundation (DFG), project number IRTG2235

We present two uniform estimates on stability and mean-field limit for the "augmented Kuramoto model (AKM)" arising from the second-order lifting of the first-order Kuramoto model (KM) for synchronization. In particular, we address three issues such as synchronization estimate, uniform stability and mean-field limit which are valid uniformly in time for the AKM. The derived mean-field equation for the AKM corresponds to the dissipative Vlasov-McKean type equation. The kinetic Kuramoto equation for distributed natural frequencies is not compatible with the frequency variance functional approach for the complete synchronization. In contrast, the kinetic equation for the AKM has a similar structural similarity with the kinetic Cucker-Smale equation which admits the Lyapunov functional approach for the variance. We present sufficient frameworks leading to the uniform stability and mean-field limit for the AKM.

Citation: Seung-Yeal Ha, Jeongho Kim, Jinyeong Park, Xiongtao Zhang. Uniform stability and mean-field limit for the augmented Kuramoto model. Networks & Heterogeneous Media, 2018, 13 (2) : 297-322. doi: 10.3934/nhm.2018013
References:
[1]

J. A. AcebronL. 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.

[2]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.

[3]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823. doi: 10.1007/s10955-015-1426-3.

[4]

D. BenedettoE. 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]

J. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197.

[6]

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

[7]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415. doi: 10.1007/s10955-014-1005-z.

[8]

L. CasettiM. Pettini and E. G. D. Cohen, Phase transitions and topology changes in configuration space, J. Statist. Phys., 111 (2003), 1091-1123. doi: 10.1023/A:1023044014341.

[9]

Y.-P. ChoiS.-Y. HaS. 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]

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.

[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 networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012.

[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]

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.

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[16]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[17]

S.-Y. HaT. Y. 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. HaH. K. Kim and J.-Y. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462. doi: 10.1088/0951-7715/28/5/1441.

[19]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci, 4 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10.

[20]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267. doi: 10.4171/EMSS/17.

[21]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear to Kinet. and Relat. Model.

[22]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103. doi: 10.1090/S0033-569X-2010-01200-7.

[23]

S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations, 251 (2011), 2685-2695. doi: 10.1016/j.jde.2011.04.004.

[24]

S.-Y. HaZ. 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.

[25]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2014), 4296-4301.

[26]

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

[27]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.

[28]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535. doi: 10.1080/00411450508951152.

[29]

D. Mehta, N. S. Daleo, F. Dörfler and J. D. Hauenstein, Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis, Chaos, 25 (2015), 053103, 7pp. doi: 10.1063/1.4919696.

[30]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347. doi: 10.1007/s00332-006-0806-x.

[31]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266. doi: 10.1016/j.physd.2005.01.017.

[32]

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.

[33]

H. Neunzert, An introduction to the nonlinear boltzmann-vlasov equation, In Kinetic Theories and the Boltzmann Equation(Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.

[34]

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

[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]

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.

[37]

M. Verwoerd and O. Mason, A convergence result for the Kurmoto model with all-to-all couplings, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920. doi: 10.1137/090771946.

[38]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453. doi: 10.1137/080725726.

[39]

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.

[40]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[41]

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.

show all references

References:
[1]

J. A. AcebronL. 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.

[2]

D. Aeyels and J. Rogge, Existence of partial entrainment and stability of phase-locking behavior of coupled oscillators, Prog. Theor. Phys., 112 (2004), 921-941.

[3]

D. BenedettoE. Caglioti and U. Montemagno, Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162 (2016), 813-823. doi: 10.1007/s10955-015-1426-3.

[4]

D. BenedettoE. 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]

J. Bronski, L. Deville and M. J. Park, Fully synchronous solutions and the synchronization phase transition for the finite-N Kuramoto model, Chaos, 22 (2012), 033133, 17pp. doi: 10.1063/1.4745197.

[6]

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

[7]

J. A. CarrilloY.-P. ChoiS.-Y. HaM.-J. Kang and Y. Kim, Contractivity of transport distances for the kinetic Kuramoto equation, J. Stat. Phys., 156 (2014), 395-415. doi: 10.1007/s10955-014-1005-z.

[8]

L. CasettiM. Pettini and E. G. D. Cohen, Phase transitions and topology changes in configuration space, J. Statist. Phys., 111 (2003), 1091-1123. doi: 10.1023/A:1023044014341.

[9]

Y.-P. ChoiS.-Y. HaS. 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]

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.

[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 networks of phase oscillators: A survey, Automatica, 50 (2014), 1539-1564. doi: 10.1016/j.automatica.2014.04.012.

[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]

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.

[15]

S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci., 7 (2009), 297-325. doi: 10.4310/CMS.2009.v7.n2.a2.

[16]

S.-Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic description of flocking, Kinet. and Relat. Model., 1 (2008), 415-435. doi: 10.3934/krm.2008.1.415.

[17]

S.-Y. HaT. Y. 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. HaH. K. Kim and J.-Y. Park, Remarks on the complete synchronization of Kuramoto oscillators, Nonlinearity, 28 (2015), 1441-1462. doi: 10.1088/0951-7715/28/5/1441.

[19]

S.-Y. HaH. K. Kim and S. W. Ryoo, Emergence of phase-locked states for the Kuramoto model in a large coupling regime, Commun. Math. Sci, 4 (2016), 1073-1091. doi: 10.4310/CMS.2016.v14.n4.a10.

[20]

S.-Y. HaD. KoJ. Park and X. Zhang, Collective synchronization of classical and quantum oscillators, EMS Surveys in Mathematical Sciences, 3 (2016), 209-267. doi: 10.4171/EMSS/17.

[21]

S.-Y. Ha, J. Kim and X. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, To appear to Kinet. and Relat. Model.

[22]

S.-Y. HaC. LattanzioB. Rubino and M. Slemrod, Flocking and synchronization of particle models, Quart. Appl. Math., 69 (2011), 91-103. doi: 10.1090/S0033-569X-2010-01200-7.

[23]

S.-Y. Ha and M. Slemrod, A fast-slow dynamical systems theory for the Kuramoto type phase model, J. Differential Equations, 251 (2011), 2685-2695. doi: 10.1016/j.jde.2011.04.004.

[24]

S.-Y. HaZ. 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.

[25]

A. JadbabaieN. Motee and M. Barahona, On the stability of the Kuramoto model of coupled nonlinear oscillators, Proceedings of the American Control Conference, (2014), 4296-4301.

[26]

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

[27]

Y. Kuramoto, International symposium on mathematical problems in mathematical physics, Lecture Notes in Theoretical Physics, 30 (1975), 420.

[28]

C. Lancellotti, On the vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory and Statistical Physics, 34 (2005), 523-535. doi: 10.1080/00411450508951152.

[29]

D. Mehta, N. S. Daleo, F. Dörfler and J. D. Hauenstein, Algebraic geometrization of the Kuramoto model: Equilibria and stability analysis, Chaos, 25 (2015), 053103, 7pp. doi: 10.1063/1.4919696.

[30]

R. Mirollo and S. H. Strogatz, The spectrum of the partially locked state for the Kuramoto model, J. Nonlinear Science, 17 (2007), 309-347. doi: 10.1007/s00332-006-0806-x.

[31]

R. Mirollo and S. H. Strogatz, The spectrum of the locked state for the Kuramoto model of coupled oscillators, Physica D, 205 (2005), 249-266. doi: 10.1016/j.physd.2005.01.017.

[32]

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.

[33]

H. Neunzert, An introduction to the nonlinear boltzmann-vlasov equation, In Kinetic Theories and the Boltzmann Equation(Montecatini, 1981), 60-110, Lecture Notes in Math., 1048, Springer, Berlin, 1984. doi: 10.1007/BFb0071878.

[34]

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

[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]

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.

[37]

M. Verwoerd and O. Mason, A convergence result for the Kurmoto model with all-to-all couplings, SIAM J. Appl. Dyn. Syst., 10 (2011), 906-920. doi: 10.1137/090771946.

[38]

M. Verwoerd and O. Mason, On computing the critical coupling coefficient for the Kuramoto model on a complete bipartite graph, SIAM J. Appl. Dyn. Syst., 8 (2009), 417-453. doi: 10.1137/080725726.

[39]

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.

[40]

C. Villani, Optimal Transport, Old and New, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.

[41]

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.

[1]

Seung-Yeal Ha, Jeongho Kim, Xiongtao Zhang. Uniform stability of the Cucker-Smale model and its application to the Mean-Field limit. Kinetic & Related Models, 2018, 11 (5) : 1157-1181. doi: 10.3934/krm.2018045

[2]

Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299

[3]

Gerasimenko Viktor. Heisenberg picture of quantum kinetic evolution in mean-field limit. Kinetic & Related Models, 2011, 4 (1) : 385-399. doi: 10.3934/krm.2011.4.385

[4]

Michael Herty, Mattia Zanella. Performance bounds for the mean-field limit of constrained dynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 2023-2043. doi: 10.3934/dcds.2017086

[5]

Rong Yang, Li Chen. Mean-field limit for a collision-avoiding flocking system and the time-asymptotic flocking dynamics for the kinetic equation. Kinetic & Related Models, 2014, 7 (2) : 381-400. doi: 10.3934/krm.2014.7.381

[6]

Franco Flandoli, Matti Leimbach. Mean field limit with proliferation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3029-3052. doi: 10.3934/dcdsb.2016086

[7]

Yufeng Shi, Tianxiao Wang, Jiongmin Yong. Mean-field backward stochastic Volterra integral equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1929-1967. doi: 10.3934/dcdsb.2013.18.1929

[8]

Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303

[9]

Jianhui Huang, Xun Li, Jiongmin Yong. A linear-quadratic optimal control problem for mean-field stochastic differential equations in infinite horizon. Mathematical Control & Related Fields, 2015, 5 (1) : 97-139. doi: 10.3934/mcrf.2015.5.97

[10]

Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287

[11]

Diogo Gomes, Marc Sedjro. One-dimensional, forward-forward mean-field games with congestion. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 901-914. doi: 10.3934/dcdss.2018054

[12]

Juan Li, Wenqiang Li. Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs. Mathematical Control & Related Fields, 2015, 5 (3) : 501-516. doi: 10.3934/mcrf.2015.5.501

[13]

Hancheng Guo, Jie Xiong. A second-order stochastic maximum principle for generalized mean-field singular control problem. Mathematical Control & Related Fields, 2018, 8 (2) : 451-473. doi: 10.3934/mcrf.2018018

[14]

Xianping Wu, Xun Li, Zhongfei Li. A mean-field formulation for multi-period asset-liability mean-variance portfolio selection with probability constraints. Journal of Industrial & Management Optimization, 2018, 14 (1) : 249-265. doi: 10.3934/jimo.2017045

[15]

Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89

[16]

Josselin Garnier, George Papanicolaou, Tzu-Wei Yang. Mean field model for collective motion bistability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-29. doi: 10.3934/dcdsb.2018210

[17]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173

[18]

Levon Nurbekyan. One-dimensional, non-local, first-order stationary mean-field games with congestion: A Fourier approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 963-990. doi: 10.3934/dcdss.2018057

[19]

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

[20]

Seung-Yeal Ha, Jaeseung Lee, Zhuchun Li. Emergence of local synchronization in an ensemble of heterogeneous Kuramoto oscillators. Networks & Heterogeneous Media, 2017, 12 (1) : 1-24. doi: 10.3934/nhm.2017001

2017 Impact Factor: 1.187

Metrics

  • PDF downloads (71)
  • HTML views (150)
  • Cited by (0)

[Back to Top]