Models of coupled oscillator networks play an important role in describing collective synchronization dynamics in biological and technological systems. The Kuramoto model describes oscillator’s phase evolution and explains the transition from incoherent to coherent oscillations under simplifying assumptions, including all-to-all coupling with uniform strength. Real world networks, however, often display heterogeneous connectivity and coupling weights that influence the critical threshold for this transition. We formulate a general mean-field theory (Vlasov–Focker Planck equation) for stochastic Kuramoto-type phase oscillator models, valid for coupling graphs/networks with heterogeneous connectivity and coupling strengths, using graphop theory in the mean-field limit. Considering symmetric odd-valued coupling functions, we mathematically prove an exact formula for the critical threshold for the incoherence–coherence transition. We numerically test the predicted threshold using large finite-size representations of the network model. For a large class of graph models, we find that the numerical tests agree very well with the predicted threshold obtained from mean-field theory. However, the prediction is more difficult in practice for graph structures that are sufficiently sparse. Our findings open future research avenues toward a deeper understanding of mean-field theories for heterogeneous systems.
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November 2022
Research Article|
November 04 2022
Graphop mean-field limits and synchronization for the stochastic Kuramoto model
Special Collection:
Dynamics of Oscillator Populations
Marios Antonios Gkogkas
;
Marios Antonios Gkogkas
(Conceptualization, Formal analysis, Writing – original draft, Writing – review & editing)
1
Department of Mathematics, Technical University of Munich
, 85748 Garching b. München, Germany
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Benjamin Jüttner
;
Benjamin Jüttner
2
Department of Applied Mathematics and Computer Science, Technical University of Denmark
, 2800 Kgs. Lyngby, Denmark
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Christian Kuehn
;
Christian Kuehn
(Conceptualization, Formal analysis, Supervision, Writing – original draft, Writing – review & editing)
1
Department of Mathematics, Technical University of Munich
, 85748 Garching b. München, Germany
3
Complexity Science Hub Vienna
, 1070 Vienna, Austria
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Erik Andreas Martens
Erik Andreas Martens
a)
(Conceptualization, Formal analysis, Supervision, Writing – original draft, Writing – review & editing)
4
Centre for Mathematical Sciences, Lund University
, Sölvegatan 18, 221 00 Lund, Sweden
a)Author to whom correspondence should be addressed: [email protected]
Search for other works by this author on:
Marios Antonios Gkogkas
1
Benjamin Jüttner
2
Christian Kuehn
1,3
Erik Andreas Martens
4,a)
1
Department of Mathematics, Technical University of Munich
, 85748 Garching b. München, Germany
2
Department of Applied Mathematics and Computer Science, Technical University of Denmark
, 2800 Kgs. Lyngby, Denmark
3
Complexity Science Hub Vienna
, 1070 Vienna, Austria
4
Centre for Mathematical Sciences, Lund University
, Sölvegatan 18, 221 00 Lund, Sweden
a)Author to whom correspondence should be addressed: [email protected]
Chaos 32, 113120 (2022)
Article history
Received:
March 31 2022
Accepted:
August 22 2022
Citation
Marios Antonios Gkogkas, Benjamin Jüttner, Christian Kuehn, Erik Andreas Martens; Graphop mean-field limits and synchronization for the stochastic Kuramoto model. Chaos 1 November 2022; 32 (11): 113120. https://doi.org/10.1063/5.0094009
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