We generalize the study of the noisy Kuramoto model, considered on a network of two interacting communities, to the case where the interaction strengths within and across communities are taken to be different in general. By developing a geometric interpretation of the self-consistency equations, we are able to separate the parameter space into ten regions in which we identify the maximum number of solutions in the steady state. Furthermore, we prove that in the steady state, only the angles and are possible between the average phases of the two communities and derive the solution boundary for the unsynchronized solution. Last, we identify the equivalence class relation in the parameter space corresponding to the symmetrically synchronized solution.
REFERENCES
1.
2.
D. M.
Abrams
, R.
Mirollo
, S. H.
Strogatz
, and D. A.
Wiley
, “Solvable model for chimera states of coupled oscillators
,” Phys. Rev. Lett.
101
, 084103
(2008
). 3.
A.
Almog
, M. R.
Buijink
, O.
Roethler
, S.
Michel
, J. H.
Meijer
, J. H. T.
Rohling
, and D.
Garlaschelli
, “Uncovering functional signature in neural systems via random matrix theory
,” PLoS Comput. Biol.
15
(5
), e1006934
(2019
). 4.
E.
Barreto
, B.
Hunt
, E.
Ott
, and P.
So
, “Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators
,” Phys. Rev. E
77
, 036107
(2008
). 5.
L.
Bertini
, G.
Giacomin
, and K.
Pakdaman
, “Dynamical aspects of mean field plane rotators and the Kuramoto model
,” J. Stat. Phys.
138
, 27
–290
(2010
). 6.
K.
Binmore
, The Breakdown of Social Contracts, in Social Dynamics, edited by S. N. Durlauf and H. Peyton Young (MIT Press, 2004).7.
R.
Buijink
, A.
Almog
, C.
Wit
, O.
Roethler
, A.
Olde Engberink
, J.
Meijer
, D.
Garlaschelli
, J.
Rohling
, and S.
Michel
, “Evidence for weakened intercellular coupling in the mammalian circadian clock under long photoperiod
,” PLoS One
11
(12
), e0168954
(2016
). 8.
D.
Chialvo
, “Emergent complex neural dynamics
,” Nat. Phys.
6
, 744
–750
(2010
). 9.
D.
Garlaschelli
, F.
den Hollander
, J.
Meylahn
, and B.
Zeegers
, “Synchronization of phase oscillators on the hierarchical lattice
,” J. Stat. Phys.
174
, 188
–218
(2019
). 10.
D. S.
Goldobin
, I. V.
Tyulkina
, L. S.
Klimenko
, and A.
Pikovsky
, “Collective mode reductions for populations of coupled noisy oscillators
,” Chaos
28
, 101101
(2018
). 11.
K. M.
Hannay
, D. B.
Forger
, and V.
Booth
, “Macroscopic models for networks of coupled biological oscillators
,” Sci. Adv.
4
, e1701047
(2018
). 12.
F.
den Hollander
, Large Deviations, Fields Institute Monographs Vol. 14 (American Mathematical Society, Providence, RI, 2000).13.
H.
Hong
and S. H.
Strogatz
, “Conformists and contrarians in a Kuramoto model with identical natural frequencies
,” Phys. Rev. E
84
, 046202
(2011
). 14.
H.
Hong
and S. H.
Strogatz
, “Kuramoto model of coupled oscillators with positive and negative coupling parameters: An example of conformist and contrarian oscillators
,” Phys. Rev. Lett.
106
, 054102
(2011
). 15.
H.
Hong
and S. H.
Strogatz
, “Mean-field behavior in coupled oscillators with attractive and repulsive interactions
,” Phys. Rev. E
85
, 056210
(2012
). 16.
A.
Jeffrey
and D.
Zwillinger
, Table of Integrals, Series, and Products
, 7th ed. (Academic Press
, 2007
).17.
Y.
Kawamura
, H.
Nakao
, K.
Arai
, H.
Kori
, and Y.
Kuramoto
, “Phase synchronization between collective rhythms of globally coupled oscillator groups: Noisy identical case
,” Chaos
20
(4
), 043109
(2010
). 18.
Y.
Kawamura
, H.
Nakao
, K.
Arai
, H.
Kori
, and Y.
Kuramoto
, “Phase synchronization between collective rhythms of globally coupled oscillator groups: Noiseless nonidentical case
,” Chaos
20
(4
), 043110
(2010
). 19.
T.
Kotwal
, X.
Jiang
, and D. M.
Abrams
, “Connecting the Kuramoto model and the chimera state
,” Phys. Rev. Lett.
119
, 264101
(2017
). 20.
S. G.
Krantz
and H. R.
Parks
, The Implicit Function Theorem. History, Theory and Applications
(Birkhäuser
, Boston
, 2002
).21.
J. M.
Meylahn
, “Two-community noisy Kuramoto model
,” Nonlinearity
33
(4
), 1847-
–1880
(2020
). 22.
E.
Montbrió
, J.
Kurths
, and B.
Blasius
, “Synchronization of two interacting populations of oscillators
,” Phys. Rev. E
70
, 056125
(2004
). 23.
K.
Okuda
and Y.
Kuramoto
, “Mutual entrainment between populations of coupled oscillators
,” Prog. Theor. Phys.
86
(6
), 1159
–1176
(1991
). 24.
A.
Pluchino
, V.
Latora
, and A.
Rapisarda
, “Changing opinions in a changing world: A new perspective in sociophysics
,” Int. J. Mod. Phys. C
16
(4
), 515-
–531
(2005
). 25.
J. H. T.
Rohling
and J. M.
Meylahn
, “Two-community noisy Kuramoto model suggests mechanism for splitting in the suprachiasmatic nucleus
,” J. Biol. Rhythms
35
(2
), 158-
–166
(2019
). 26.
J. H.
Sheeba
, V. K.
Chandrasekar
, A.
Stefanovska
, and P. V. E.
McClintock
, “Routes to synchrony between asymmetrically interacting oscillator ensembles
,” Phys. Rev. E
78
, 025201(R)
(2008
). 27.
J. H.
Sheeba
, V. K.
Chandrasekar
, A.
Stefanovska
, and P. V. E.
McClintock
, “Routes to synchrony between asymmetrically interacting oscillator ensembles
,” Phys. Rev. E
79
, 046210
(2009
). 28.
B.
Sonnenschein
, T. K. D.
M. Peron
, F. A.
Rodrigues
, J.
Kurths
, and L.
Schimansky-Geier
, “Collective dynamics in two populations of noisy oscillators with asymmetric interactions
,” Phys. Rev. E
91
, 062910
(2015
). 29.
B.
Sonnenschein
and L.
Schimansky-Geier
, “Approximate solution to the stochastic Kuramoto model
,” Phys. Rev. E
88
, 052111
(2013
). 30.
I. V.
Tyulkina
, D. S.
Goldobin
, L. S.
Klimenko
, and A.
Pikovsky
, “Dynamics of noisy oscillator populations beyond the Ott–Antonsen ansatz
,” Phys. Rev. Lett.
120
, 264101
(2018
). 31.
G. N.
Watson
, A Treatise on the Theory of Bessel Functions
, 2nd ed. (Cambridge University Press
, 1944
).32.
R.
Xiao
, J.
Li
, and T.
Chen
, “Modeling and intelligent optimization of social collective behavior with online public opinion synchronization
,” Int. J. Mach. Learn. Cybern.
10
, 1979
–1996
(2019
). © 2021 Author(s).
2021
Author(s)
You do not currently have access to this content.
