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Graphop mean-field limits and synchronization for the stochastic Kuramoto model

Networks of coupled oscillators appear in an impressive range of systems in nature and technology where they display collective dynamics, such as synchronization.1–3 The Kuramoto model describes the phase evolution of oscillators4,5 and explains the transition from incoherent to coherent synchronized oscillations for a critical threshold of the coupling strength under simplifying assumptions, such as all-to-all coupling with uniform strength;6,7 however, real world networks often display strong heterogeneity in connectivity and coupling strength, which affect the critical threshold.8

We derive a mean-field theory for stochastic Kuramoto-type models and extend it to a large class of heterogeneous graph/network structures via graphop descriptions valid for the mean-field limit.

We prove a mathematically exact formula for the critical threshold, which we test numerically for large finite-size representations of the network model.

M. A. Gkogkas, B. Jüttner, C. Kuehn, E. A. Martens, Graphop mean-field limits and synchronization for the stochastic Kuramoto model, Chaos 32 (2022) 113120

 

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