×
Gottwald, Georg A.

Finite-size effects in a stochastic Kuramoto model. (English) Zbl 1390.34094

Chaos 27, No. 10, 101103, 6 p. (2017).
Summary: We present a collective coordinate approach to study the collective behaviour of a finite ensemble of \(N\) stochastic Kuramoto oscillators using two degrees of freedom: one describing the shape dynamics of the oscillators and one describing their mean phase. Contrary to the thermodynamic limit \(N\to\infty\) in which the mean phase of the cluster of globally synchronized oscillators is constant in time, the mean phase of a finite-size cluster experiences Brownian diffusion with a variance proportional to \(1/N\). This finite-size effect is quantitatively well captured by our collective coordinate approach.{
©2017 American Institute of Physics}

Cited in 9 Documents

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
34F05 Ordinary differential equations and systems with randomness
34D06 Synchronization of solutions to ordinary differential equations
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Acebrón, J. A.; Bonilla, L. L.; Pérez Vicente, C. J.; Ritort, F.; Spigler, R., The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys., 77, 137-185 (2005) · doi:10.1103/RevModPhys.77.137
[2] Arenas, A.; Diaz-Guilera, A.; Kurths, J.; Moreno, Y.; Zhou, C., Synchronization in complex networks, Phys. Rep., 469, 93-153 (2008) · doi:10.1016/j.physrep.2008.09.002
[3] Bertini, L.; Giacomin, G.; Pakdaman, K., Dynamical aspects of mean field plane rotators and the Kuramoto model, J. Stat. Phys., 138, 270-290 (2010) · Zbl 1187.82067 · doi:10.1007/s10955-009-9908-9
[4] Bertini, L.; Giacomin, G.; Poquet, C., Synchronization and random long time dynamics for mean-field plane rotators, Probab. Theory Relat. Fields, 160, 593-653 (2014) · Zbl 1309.60093 · doi:10.1007/s00440-013-0536-6
[5] Bhowmik, D.; Shanahan, M., How well do oscillator models capture the behaviour of biological neurons?, 1-8 (2012)
[6] Breakspear, M.; Heitmann, S.; Daffertshofer, A., Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model, Front. Human Neurosci., 4, 190 (2010) · doi:10.3389/fnhum.2010.00190
[7] Brede, M.; Kalloniatis, A. C., Frustration tuning and perfect phase synchronization in the Kuramoto-Sakaguchi model, Phys. Rev. E, 93, 062315 (2016) · doi:10.1103/PhysRevE.93.062315
[8] Buice, M. A.; Chow, C. C., Correlations, fluctuations, and stability of a finite-size network of coupled oscillators, Phys. Rev. E, 76, 031118 (2007) · doi:10.1103/PhysRevE.76.031118
[9] Cabral, J.; Hugues, E.; Sporns, O.; Deco, G., Role of local network oscillations in resting-state functional connectivity, NeuroImage, 57, 130-139 (2011) · doi:10.1016/j.neuroimage.2011.04.010
[10] Cox, S. M.; Gottwald, G. A., A bistable reaction-diffusion system in a stretching flow, Physica D, 216, 307-318 (2006) · Zbl 1102.37052 · doi:10.1016/j.physd.2006.03.007
[11] Dörfler, F.; Bullo, F., Synchronization in complex networks of phase oscillators: A survey, Automatica, 50, 1539-1564 (2014) · Zbl 1296.93005 · doi:10.1016/j.automatica.2014.04.012
[12] Dorogovtsev, S. N.; Goltsev, A. V.; Mendes, J. F. F., Critical phenomena in complex networks, Rev. Mod. Phys., 80, 1275-1335 (2008) · doi:10.1103/RevModPhys.80.1275
[13] Faisal, A. A.; Selen, L. P. J.; Wolpert, D. M., Noise in the nervous system, Nat. Rev. Neurosci., 9, 292-303 (2008) · doi:10.1038/nrn2258
[14] Filatrella, G.; Nielsen, A. H.; Pedersen, N. F., Analysis of a power grid using a Kuramoto-like model, Eur. Phys. J. B, 61, 485-491 (2008) · doi:10.1140/epjb/e2008-00098-8
[15] Giacomin, G.; Poquet, C., Noise, interaction, nonlinear dynamics and the origin of rhythmic behaviors, Braz. J. Probab. Stat., 29, 460-493 (2015) · Zbl 1330.92017 · doi:10.1214/14-BJPS258
[16] Goldwyn, J. H.; Shea-Brown, E., The what and where of adding channel noise to the Hodgkin-Huxley equations, PLoS Comput. Biol., 7, e1002247-e1002249 (2011) · doi:10.1371/journal.pcbi.1002247
[17] Gottwald, G. A., Model reduction for networks of coupled oscillators, Chaos, 25, 053111-053112 (2015) · Zbl 1374.34106 · doi:10.1063/1.4921295
[18] Gottwald, G. A.; Kramer, L., On propagation failure in one- and two-dimensional excitable media, Chaos, 14, 855-863 (2004) · Zbl 1080.92502 · doi:10.1063/1.1772552
[19] Hildebrand, E. J.; Buice, M. A.; Chow, C. C., Kinetic theory of coupled oscillators, Phys. Rev. Lett., 98, 054101 (2007) · doi:10.1103/PhysRevLett.98.054101
[20] Hong, H.; Ha, M.; Park, H., Finite-size scaling in complex networks, Phys. Rev. Lett., 98, 258701 (2007) · doi:10.1103/PhysRevLett.98.258701
[21] Kuramoto, Y., Chemical Oscillations, Waves, and Turbulence, 19, viii+156 (1984) · Zbl 0558.76051
[22] Luçon, E., Large population asymptotics for interacting diffusions in a quenched random environment, From Particle Systems to Partial Differential Equations II, 129, 231-251 (2015) · Zbl 1342.60176
[23] Luçon, E.; Poquet, C., Long time dynamics and disorder-induced traveling waves in the stochastic Kuramoto model, Ann. Inst. Henri Poincaré, 53, 1196-1240 (2017) · Zbl 1386.60323
[24] Luçon, E.; Stannat, W., Transition from Gaussian to non-Gaussian fluctuations for mean-field diffusions in spatial interaction, Ann. Appl. Probab., 26, 3840-3909 (2016) · Zbl 1358.60104 · doi:10.1214/16-AAP1194
[25] Marvel, S. A.; Mirollo, R. E.; Strogatz, S. H., Identical phase oscillators with global sinusoidal coupling evolve by Möbius group action, Chaos, 19, 043104-043111 (2009) · Zbl 1311.34082 · doi:10.1063/1.3247089
[26] Menon, S. N.; Gottwald, G. A., Bifurcations in reaction-diffusion systems in chaotic flows, Phys. Rev. E, 71, 066201 (2005) · doi:10.1103/PhysRevE.71.066201
[27] Menon, S. N.; Gottwald, G. A., Bifurcations of flame filaments in chaotically mixed combustion reactions, Phys. Rev. E, 75, 016209 (2007) · doi:10.1103/PhysRevE.75.016209
[28] Menon, S. N.; Gottwald, G. A., On bifurcations in a chaotically stirred excitable medium, Physica D, 238, 461-475 (2009) · Zbl 1163.37013 · doi:10.1016/j.physd.2008.11.012
[29] Neda, Z.; Ravasz, E.; Brechet, Y.; Vicsek, T.; Barabasi, A. L., The sound of many hands clapping, Nature, 403, 849-850 (2000) · doi:10.1038/35002660
[30] Osipov, G. V.; Kurths, J.; Zhou, C., Synchronization in Oscillatory Networks, 37c (2007) · Zbl 1137.37018
[31] Ott, E.; Antonsen, T. M., Low dimensional behavior of large systems of globally coupled oscillators, Chaos, 18, 037113-037116 (2008) · Zbl 1309.34058 · doi:10.1063/1.2930766
[32] Pantaleone, J., Stability of incoherence in an isotropic gas of oscillating neutrinos, Phys. Rev. D, 58, 073002 (1998) · doi:10.1103/PhysRevD.58.073002
[33] Pikovsky, A.; Rosenblum, M., Dynamics of heterogeneous oscillator ensembles in terms of collective variables, Physica D, 240, 872-881 (2011) · Zbl 1233.37014 · doi:10.1016/j.physd.2011.01.002
[34] Pikovsky, A.; Rosenblum, M., Dynamics of globally coupled oscillators: Progress and perspectives, Chaos, 25, 097616 (2015) · Zbl 1374.34001 · doi:10.1063/1.4922971
[35] Pikovsky, A.; Rosenblum, M.; Kurths, J., Synchronization: A Universal Concept in Nonlinear Sciences (2001) · Zbl 0993.37002
[36] Pinto, R. S.; Saa, A., Optimal synchronization of Kuramoto oscillators: A dimensional reduction approach, Phys. Rev. E, 92, 6, 062801 (2015) · doi:10.1103/PhysRevE.92.062801
[37] Rodrigues, F. A.; Peron, T. K. D.; Ji, P.; Kurths, J., The Kuramoto model in complex networks, Phys. Rep., 610, 1-98 (2016) · Zbl 1357.34089 · doi:10.1016/j.physrep.2015.10.008
[38] Sakaguchi, H., Cooperative phenomena in coupled oscillator systems under external fields, Prog. Theor. Phys., 79, 39-46 (1988) · doi:10.1143/PTP.79.39
[39] Scott, A., Nonlinear Science: Emergence and Dynamics of Coherent Structures (Oxford Texts in Applied and Engineering Mathematics), 8, xxiv+480 (2003) · Zbl 1072.35003
[40] Sheeba, J. H.; Stefanovska, A.; McClintock, P. V. E., Neuronal synchrony during anesthesia: A thalamocortical model, Biophys. J., 95, 2722-2727 (2008) · doi:10.1529/biophysj.108.134635
[41] Sonnenschein, B.; Schimansky-Geier, L., Onset of synchronization in complex networks of noisy oscillators, Phys. Rev. E, 85, 051116 (2012) · doi:10.1103/PhysRevE.85.051116
[42] Sonnenschein, B.; Schimansky-Geier, L., Approximate solution to the stochastic Kuramoto model, Phys. Rev. E, 88, 052111 (2013) · doi:10.1103/PhysRevE.88.052111
[43] Strogatz, S. H., From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators, Physica D, 143, 1-20 (2000) · Zbl 0956.00057 · doi:10.1016/S0167-2789(00)00094-4
[44] Tang, L.-H., To synchronize or not to synchronize, that is the question: Finite-size scaling and fluctuation effects in the Kuramoto model, J. Stat. Mech.: Theor. Exp., 2011, P01034 · doi:10.1088/1742-5468/2011/01/P01034
[45] Watanabe, S.; Strogatz, S. H., Integrability of a globally coupled oscillator array, Phys. Rev. Lett., 70, 2391-2394 (1993) · Zbl 1063.34505 · doi:10.1103/PhysRevLett.70.2391
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.