January  2019, 39(1): 131-155. doi: 10.3934/dcds.2019006

The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas

1. 

Institute of Mathematics for Industry, Kyushu University/JST PRESTO, Fukuoka, 819-0395, Japan

2. 

Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

* Corresponding author: Georgi S. Medvedev

Received  September 2017 Revised  April 2018 Published  October 2018

Fund Project: The second author was partially supported by NSF DMS grant 1715161

In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdős-Rényi, small-world, and $ k$-nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas.

In the second part of this work [8], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.

Citation: Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 131-155. doi: 10.3934/dcds.2019006
References:
[1]

V. S. AfraimovichN. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative system, Izy. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060. Google Scholar

[2]

P. Billingsley, Probability and Measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995, A Wiley-Interscience Publication. Google Scholar

[3]

I. I. Blekhman, Sinkhronizatsiya V Prirode I Tekhnike, "Nauka", Moscow, 1981. Google Scholar

[4]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $ 1/N$ limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[5]

H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891. Google Scholar

[6]

____, 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. Google Scholar

[7]

____, A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions, Adv. Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001. Google Scholar

[8]

H. Chiba and G. S. Medvedev, The mean field analysis of the Kuramoto model on graphs Ⅱ. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations, submitted.Google Scholar

[9]

H. Chiba, G. S. Medvedev and M. S. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos, 28 (2018), 073109, 10pp. doi: 10.1063/1.5039609. Google Scholar

[10]

H. Chiba and I. Nishikawa, Center manifold reduction for large populations of globally coupled phase oscillators, Chaos, 21 (2011), 043103, 10pp. doi: 10.1063/1.3647317. Google Scholar

[11]

S. DelattreG. Giacomin and E. Luçon, A note on dynamical models on random graphs and Fokker-Planck equations, J. Stat. Phys., 165 (2016), 785-798. doi: 10.1007/s10955-016-1652-3. Google Scholar

[12]

H. Dietert, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl.(9), 105 (2016), 451-489. doi: 10.1016/j.matpur.2015.11.001. Google Scholar

[13]

R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen, 13 (1979), 48-58, 96. Google Scholar

[14]

F. Dorfler and F. Bullo, Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, SICON, 50 (2012), 1616-1642. doi: 10.1137/110851584. Google Scholar

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R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002, Revised reprint of the 1989 original. doi: 10.1017/CBO9780511755347. Google Scholar

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L. C. Evans, Partial Differential Equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

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B. FernandezD. Gérard-Varet and G. Giacomin, Landau damping in the Kuramoto model, Ann. Henri Poincaré, 17 (2016), 1793-1823. doi: 10.1007/s00023-015-0450-9. Google Scholar

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I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions. Vol. 4, Applications of harmonic analysis, Translated from the 1961 Russian original by Amiel Feinstein, Reprint of the 1964 English translation. Google Scholar

[19]

F. Golse, On the dynamics of large particle systems in the mean field limit, Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, Lect. Notes Appl. Math. Mech., Springer, [Cham], 3 (2016), 1-144. doi: 10.1007/978-3-319-26883-5_1. Google Scholar

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J. M. Hendrickx and A. Olshevsky, On symmetric continuum opinion dynamics, SIAM J. Control Optim., 54 (2016), 2893-2918. doi: 10.1137/130943923. Google Scholar

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F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, Applied Mathematical Sciences, vol. 126, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1828-9. Google Scholar

[22]

D. Kaliuzhnyi-Verbovetskyi and G. S. Medvedev, The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit, SIAM J. Math. Anal., 50 (2018), 2441-2465. doi: 10.1137/17M1134007. Google Scholar

[23]

____, The semilinear heat equation on sparse random graphs, SIAM J. Math. Anal., 49 (2017), 1333-1355. doi: 10.1137/16M1075831. Google Scholar

[24]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. Google Scholar

[25]

S. Yu. KourtchatovV. V. LikhanskiiA. P. NapartovichF. T. Arecchi and A. Lapucci, Theory of phase locking of globally coupled laser arrays, Phys. Rev. A, 52 (1995), 4089-4094. Google Scholar

[26]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenomena in Complex Systems, 5 (2002), 380-385. Google Scholar

[27]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975), Springer, Berlin, Lecture Notes in Phys., 39 (1975), 420-422. Google Scholar

[28]

C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys., 34 (2005), 523-535. doi: 10.1080/00411450508951152. Google Scholar

[29]

R. LevyW. D. HutchisonA. M. Lozano and J. O. Dostrovsky, High-frequency synchronization of neuronal activity in the subthalamic nucleus of parkinsonian patients with limb tremor, Journal of Neuroscience, 20 (2000), 7766-7775. Google Scholar

[30]

L. Lovász, Large Networks and Graph Limits, AMS, Providence, RI, 2012.Google Scholar

[31]

L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B, 96 (2006), 933-957. doi: 10.1016/j.jctb.2006.05.002. Google Scholar

[32]

E. Luçon and W. Stannat, Mean field limit for disordered diffusions with singular interactions, Ann. Appl. Probab., 24 (2014), 1946-1993. doi: 10.1214/13-AAP968. Google Scholar

[33]

I. G. Malkin, Metody Lyapunova i Puankare v Teorii Nelineĭnyh Kolebaniĭ, OGIZ, Moscow-Leningrad, 1949. Google Scholar

[34]

____, Some Problems of the Theory of Nonlinear Oscillations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. Google Scholar

[35]

G. S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Anal., 46 (2014), 2743-2766. doi: 10.1137/130943741. Google Scholar

[36]

____, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803. doi: 10.1007/s00205-013-0706-9. Google Scholar

[37]

____, Small-world networks of Kuramoto oscillators, Phys. D, 266 (2014), 13-22. doi: 10.1016/j.physd.2013.09.008. Google Scholar

[38]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[39]

H. Neunzert, Mathematical investigations on particle - in - cell methods, Fluid Dyn. Trans., vol. 9, 1978,229-254.Google Scholar

[40]

____, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Math., vol. 1048, Springer, Berlin, 1984, 60-110. doi: 10.1007/BFb0071878. Google Scholar

[41]

O. E. Omelchenko, Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Nonlinearity, 26 (2013), 2469-2498. doi: 10.1088/0951-7715/26/9/2469. Google Scholar

[42]

M. A. Porter and J. P. Gleeson, Dynamical Systems on Networks, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 4, Springer, Cham, 2016, A tutorial. doi: 10.1007/978-3-319-26641-1. Google Scholar

[43]

W. Ren, R. W. Beard and T. W. McLain, Coordination variables and consensus building in multiple vehicle systems, Cooperative control, Lecture Notes in Control and Inform. Sci., vol. 309, Springer, Berlin, 2005,171-188. doi: 10.1007/978-3-540-31595-7_10. Google Scholar

[44]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annual Review of Physiology, 55 (1993), 349-374. Google Scholar

[45]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32. Google Scholar

[46]

S. Strogatz, Sync. How order emerges from chaos in the universe, nature, and daily life. Hyperion Books, New York, 2003. Google Scholar

[47]

____, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00137-8. Google Scholar

[48]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635. doi: 10.1007/BF01029202. Google Scholar

[49]

S. H. StrogatzR. E. Mirollo and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett., 68 (1992), 2730-2733. doi: 10.1103/PhysRevLett.68.2730. Google Scholar

[50]

B. Szegedy, Limits of kernel operators and the spectral regularity lemma, European J. Combin., 32 (2011), 1156-1167. doi: 10.1016/j.ejc.2011.03.005. Google Scholar

[51]

R. D. TraubM. A. Whittington and M. O. Cunningham, Epileptic fast oscillations and synchrony in vitro, Epilepsia, 51 (2010), 28-28. Google Scholar

[52]

S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D: Nonlinear Phenomena, 74 (1994), 197-253. Google Scholar

[53]

D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. Google Scholar

[54]

N. Young, An Introduction to Hilbert Space, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9781139172011. Google Scholar

show all references

References:
[1]

V. S. AfraimovichN. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative system, Izy. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060. Google Scholar

[2]

P. Billingsley, Probability and Measure, third ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995, A Wiley-Interscience Publication. Google Scholar

[3]

I. I. Blekhman, Sinkhronizatsiya V Prirode I Tekhnike, "Nauka", Moscow, 1981. Google Scholar

[4]

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the $ 1/N$ limit of interacting classical particles, Comm. Math. Phys., 56 (1977), 101-113. doi: 10.1007/BF01611497. Google Scholar

[5]

H. Chiba, Continuous limit and the moments system for the globally coupled phase oscillators, Discrete Contin. Dyn. Syst., 33 (2013), 1891-1903. doi: 10.3934/dcds.2013.33.1891. Google Scholar

[6]

____, 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. Google Scholar

[7]

____, A spectral theory of linear operators on rigged Hilbert spaces under analyticity conditions, Adv. Math., 273 (2015), 324-379. doi: 10.1016/j.aim.2015.01.001. Google Scholar

[8]

H. Chiba and G. S. Medvedev, The mean field analysis of the Kuramoto model on graphs Ⅱ. Asymptotic stability of the incoherent state, center manifold reduction, and bifurcations, submitted.Google Scholar

[9]

H. Chiba, G. S. Medvedev and M. S. Mizuhara, Bifurcations in the Kuramoto model on graphs, Chaos, 28 (2018), 073109, 10pp. doi: 10.1063/1.5039609. Google Scholar

[10]

H. Chiba and I. Nishikawa, Center manifold reduction for large populations of globally coupled phase oscillators, Chaos, 21 (2011), 043103, 10pp. doi: 10.1063/1.3647317. Google Scholar

[11]

S. DelattreG. Giacomin and E. Luçon, A note on dynamical models on random graphs and Fokker-Planck equations, J. Stat. Phys., 165 (2016), 785-798. doi: 10.1007/s10955-016-1652-3. Google Scholar

[12]

H. Dietert, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl.(9), 105 (2016), 451-489. doi: 10.1016/j.matpur.2015.11.001. Google Scholar

[13]

R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen, 13 (1979), 48-58, 96. Google Scholar

[14]

F. Dorfler and F. Bullo, Synchronization and transient stability in power networks and non-uniform Kuramoto oscillators, SICON, 50 (2012), 1616-1642. doi: 10.1137/110851584. Google Scholar

[15]

R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002, Revised reprint of the 1989 original. doi: 10.1017/CBO9780511755347. Google Scholar

[16]

L. C. Evans, Partial Differential Equations, second ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019. Google Scholar

[17]

B. FernandezD. Gérard-Varet and G. Giacomin, Landau damping in the Kuramoto model, Ann. Henri Poincaré, 17 (2016), 1793-1823. doi: 10.1007/s00023-015-0450-9. Google Scholar

[18]

I. M. Gel'fand and N. Ya. Vilenkin, Generalized Functions. Vol. 4, Applications of harmonic analysis, Translated from the 1961 Russian original by Amiel Feinstein, Reprint of the 1964 English translation. Google Scholar

[19]

F. Golse, On the dynamics of large particle systems in the mean field limit, Macroscopic and large scale phenomena: coarse graining, mean field limits and ergodicity, Lect. Notes Appl. Math. Mech., Springer, [Cham], 3 (2016), 1-144. doi: 10.1007/978-3-319-26883-5_1. Google Scholar

[20]

J. M. Hendrickx and A. Olshevsky, On symmetric continuum opinion dynamics, SIAM J. Control Optim., 54 (2016), 2893-2918. doi: 10.1137/130943923. Google Scholar

[21]

F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks, Applied Mathematical Sciences, vol. 126, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-1828-9. Google Scholar

[22]

D. Kaliuzhnyi-Verbovetskyi and G. S. Medvedev, The mean field equation for the Kuramoto model on graph sequences with non-Lipschitz limit, SIAM J. Math. Anal., 50 (2018), 2441-2465. doi: 10.1137/17M1134007. Google Scholar

[23]

____, The semilinear heat equation on sparse random graphs, SIAM J. Math. Anal., 49 (2017), 1333-1355. doi: 10.1137/16M1075831. Google Scholar

[24]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Reprint of the 1980 edition. Google Scholar

[25]

S. Yu. KourtchatovV. V. LikhanskiiA. P. NapartovichF. T. Arecchi and A. Lapucci, Theory of phase locking of globally coupled laser arrays, Phys. Rev. A, 52 (1995), 4089-4094. Google Scholar

[26]

Y. Kuramoto and D. Battogtokh, Coexistence of coherence and incoherence in nonlocally coupled phase oscillators, Nonlinear Phenomena in Complex Systems, 5 (2002), 380-385. Google Scholar

[27]

Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975), Springer, Berlin, Lecture Notes in Phys., 39 (1975), 420-422. Google Scholar

[28]

C. Lancellotti, On the Vlasov limit for systems of nonlinearly coupled oscillators without noise, Transport Theory Statist. Phys., 34 (2005), 523-535. doi: 10.1080/00411450508951152. Google Scholar

[29]

R. LevyW. D. HutchisonA. M. Lozano and J. O. Dostrovsky, High-frequency synchronization of neuronal activity in the subthalamic nucleus of parkinsonian patients with limb tremor, Journal of Neuroscience, 20 (2000), 7766-7775. Google Scholar

[30]

L. Lovász, Large Networks and Graph Limits, AMS, Providence, RI, 2012.Google Scholar

[31]

L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B, 96 (2006), 933-957. doi: 10.1016/j.jctb.2006.05.002. Google Scholar

[32]

E. Luçon and W. Stannat, Mean field limit for disordered diffusions with singular interactions, Ann. Appl. Probab., 24 (2014), 1946-1993. doi: 10.1214/13-AAP968. Google Scholar

[33]

I. G. Malkin, Metody Lyapunova i Puankare v Teorii Nelineĭnyh Kolebaniĭ, OGIZ, Moscow-Leningrad, 1949. Google Scholar

[34]

____, Some Problems of the Theory of Nonlinear Oscillations, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow, 1956. Google Scholar

[35]

G. S. Medvedev, The nonlinear heat equation on dense graphs and graph limits, SIAM J. Math. Anal., 46 (2014), 2743-2766. doi: 10.1137/130943741. Google Scholar

[36]

____, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803. doi: 10.1007/s00205-013-0706-9. Google Scholar

[37]

____, Small-world networks of Kuramoto oscillators, Phys. D, 266 (2014), 13-22. doi: 10.1016/j.physd.2013.09.008. Google Scholar

[38]

S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar

[39]

H. Neunzert, Mathematical investigations on particle - in - cell methods, Fluid Dyn. Trans., vol. 9, 1978,229-254.Google Scholar

[40]

____, An introduction to the nonlinear Boltzmann-Vlasov equation, Kinetic Theories and the Boltzmann Equation (Montecatini, 1981), Lecture Notes in Math., vol. 1048, Springer, Berlin, 1984, 60-110. doi: 10.1007/BFb0071878. Google Scholar

[41]

O. E. Omelchenko, Coherence-incoherence patterns in a ring of non-locally coupled phase oscillators, Nonlinearity, 26 (2013), 2469-2498. doi: 10.1088/0951-7715/26/9/2469. Google Scholar

[42]

M. A. Porter and J. P. Gleeson, Dynamical Systems on Networks, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol. 4, Springer, Cham, 2016, A tutorial. doi: 10.1007/978-3-319-26641-1. Google Scholar

[43]

W. Ren, R. W. Beard and T. W. McLain, Coordination variables and consensus building in multiple vehicle systems, Cooperative control, Lecture Notes in Control and Inform. Sci., vol. 309, Springer, Berlin, 2005,171-188. doi: 10.1007/978-3-540-31595-7_10. Google Scholar

[44]

W. Singer, Synchronization of cortical activity and its putative role in information processing and learning, Annual Review of Physiology, 55 (1993), 349-374. Google Scholar

[45]

E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32. Google Scholar

[46]

S. Strogatz, Sync. How order emerges from chaos in the universe, nature, and daily life. Hyperion Books, New York, 2003. Google Scholar

[47]

____, From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Phys. D, 143 (2000), 1-20. doi: 10.1016/S0167-2789(00)00137-8. Google Scholar

[48]

S. H. Strogatz and R. E. Mirollo, Stability of incoherence in a population of coupled oscillators, J. Statist. Phys., 63 (1991), 613-635. doi: 10.1007/BF01029202. Google Scholar

[49]

S. H. StrogatzR. E. Mirollo and P. C. Matthews, Coupled nonlinear oscillators below the synchronization threshold: relaxation by generalized Landau damping, Phys. Rev. Lett., 68 (1992), 2730-2733. doi: 10.1103/PhysRevLett.68.2730. Google Scholar

[50]

B. Szegedy, Limits of kernel operators and the spectral regularity lemma, European J. Combin., 32 (2011), 1156-1167. doi: 10.1016/j.ejc.2011.03.005. Google Scholar

[51]

R. D. TraubM. A. Whittington and M. O. Cunningham, Epileptic fast oscillations and synchrony in vitro, Epilepsia, 51 (2010), 28-28. Google Scholar

[52]

S. Watanabe and S. H. Strogatz, Constants of motion for superconducting Josephson arrays, Physica D: Nonlinear Phenomena, 74 (1994), 197-253. Google Scholar

[53]

D. J. Watts and S. H. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. Google Scholar

[54]

N. Young, An Introduction to Hilbert Space, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1988. doi: 10.1017/CBO9781139172011. Google Scholar

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