# American Institute of Mathematical Sciences

July 2019, 39(7): 3897-3921. doi: 10.3934/dcds.2019157

## The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

 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

Received  April 2018 Published  April 2019

In our previous work [3], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model.

In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Rényi, small-world, as well as certain weighted graphs on a circle.

Citation: Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
##### References:
 [1] 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. [2] ____, 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. [3] H. Chiba and G. S. Medvedev, The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and the transition point formulas, Discrete and Continuous Dynamical Systems - A, 39 (2019), 131-155. doi: 10.3934/dcds.2019006. [4] H. Chiba, G. S. Medvedev and M. Mizhura, Bifurcations in the Kuramoto model on graphs, Chaos, 28 (2018), 073109, 10pp. doi: 10.1063/1.5039609. [5] 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. [6] H. Dietert, Stability of partially locked states in the Kuramoto model through Landau damping with Sobolev regularity, arXiv e-prints, 2017. [7] ____, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl., (9) 105 (2016), 451–489. doi: 10.1016/j.matpur.2015.11.001. [8] R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347. [9] B. Fernandez, D. 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. [10] F. D. Gakhov, Boundary Value Problems, Dover Publications, Inc., New York, 1990, Translated from the Russian, Reprint of the 1966 translation. [11] I. M. Gel$'$fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4: Applications of harmonic analysis. Translated by Amiel Feinstein Academic Press, New York - London, 1964. [12] H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1967), 366-383. doi: 10.2969/jmsj/01930366. [13] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975), 420–422. Lecture Notes in Phys., 39. Springer, Berlin, 1975. [14] L. Lovász, Large Networks and Graph Limits, AMS, Providence, RI, 2012. [15] 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. [16] 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. [17] ____, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803. doi: 10.1007/s00205-013-0706-9. [18] ____, Small-world networks of Kuramoto oscillators, Phys. D, 266 (2014), 13-22. doi: 10.1016/j.physd.2013.09.008. [19] G.S. Medvedev and X. Tang, Stability of twisted states in the Kuramoto model on Cayley and random graphs, Journal of Nonlinear Science, 25 (2015), 1169-1208. doi: 10.1007/s00332-015-9252-y. [20] C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. [21] S. H. Strogatz, 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)00094-4. [22] 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. [23] S. H. Strogatz, R. 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. [24] D. A. Wiley, S. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103, 8pp. doi: 10.1063/1.2165594.

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##### References:
 [1] 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. [2] ____, 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. [3] H. Chiba and G. S. Medvedev, The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and the transition point formulas, Discrete and Continuous Dynamical Systems - A, 39 (2019), 131-155. doi: 10.3934/dcds.2019006. [4] H. Chiba, G. S. Medvedev and M. Mizhura, Bifurcations in the Kuramoto model on graphs, Chaos, 28 (2018), 073109, 10pp. doi: 10.1063/1.5039609. [5] 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. [6] H. Dietert, Stability of partially locked states in the Kuramoto model through Landau damping with Sobolev regularity, arXiv e-prints, 2017. [7] ____, Stability and bifurcation for the Kuramoto model, J. Math. Pures Appl., (9) 105 (2016), 451–489. doi: 10.1016/j.matpur.2015.11.001. [8] R. M. Dudley, Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, vol. 74, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511755347. [9] B. Fernandez, D. 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. [10] F. D. Gakhov, Boundary Value Problems, Dover Publications, Inc., New York, 1990, Translated from the Russian, Reprint of the 1966 translation. [11] I. M. Gel$'$fand and N. Ya. Vilenkin, Generalized Functions, Vol. 4: Applications of harmonic analysis. Translated by Amiel Feinstein Academic Press, New York - London, 1964. [12] H. Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan, 19 (1967), 366-383. doi: 10.2969/jmsj/01930366. [13] Y. Kuramoto, Self-entrainment of a population of coupled non-linear oscillators, International Symposium on Mathematical Problems in Theoretical Physics (Kyoto Univ., Kyoto, 1975), 420–422. Lecture Notes in Phys., 39. Springer, Berlin, 1975. [14] L. Lovász, Large Networks and Graph Limits, AMS, Providence, RI, 2012. [15] 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. [16] 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. [17] ____, The nonlinear heat equation on W-random graphs, Arch. Ration. Mech. Anal., 212 (2014), 781-803. doi: 10.1007/s00205-013-0706-9. [18] ____, Small-world networks of Kuramoto oscillators, Phys. D, 266 (2014), 13-22. doi: 10.1016/j.physd.2013.09.008. [19] G.S. Medvedev and X. Tang, Stability of twisted states in the Kuramoto model on Cayley and random graphs, Journal of Nonlinear Science, 25 (2015), 1169-1208. doi: 10.1007/s00332-015-9252-y. [20] C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201. doi: 10.1007/s11511-011-0068-9. [21] S. H. Strogatz, 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)00094-4. [22] 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. [23] S. H. Strogatz, R. 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. [24] D. A. Wiley, S. H. Strogatz and M. Girvan, The size of the sync basin, Chaos, 16 (2006), 015103, 8pp. doi: 10.1063/1.2165594.
Deformation of the integral path for the Laplace inversion formula
Formation of partially phase-locked solutions near a bifurcation with two-dimensional null space. The KM with intrinsic frequencies from the standard normal distribution, graphon (6.21), and random initial condition was for suffiently large time to reach a stationary regime. The values of $K$ are a) $3.5$, b) $4$, and c) $5$. The asymptotic state in (a) combines oscillators grouped around a $1$-twisted state with those distributed randomly around $\mathbb{S}$. For increasing values of $K$, the noisy twisted states become more distinct (b, c)
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