Approximate solution to the stochastic Kuramoto model

Abstract

We study Kuramoto phase oscillators with temporal fluctuations in the frequencies. The infinite-dimensional system can be reduced in a Gaussian approximation to two first-order differential equations. This yields a solution for the \emph{time-dependent} order parameter, which characterizes the synchronization between the oscillators. The known critical coupling strength is exactly recovered by the Gaussian theory. Extensive numerical experiments further show that the analytical results are very accurate below and sufficiently above the critical value. We obtain the asymptotic order parameter \emph{in closed form}, which suggests a tighter upper bound for the corresponding scaling. As a last point, we elaborate the Gaussian approximation in complex networks with distributed degrees.

6 pages, 3 figures

References in corpus (5)

• Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
• Exact Results for the Kuramoto Model with a Bimodal Frequency Distribution
• Partially integrable dynamics of hierarchical populations of coupled oscillators
• Heterogeneous connections induce oscillations in large scale networks
• Excitable elements controlled by noise and network structure

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• Coupled networks and networks with bimodal frequency distributions are equivalent
• Synchronization in coupled laser arrays with correlated and uncorrelated disorder
• Cooperative behavior between oscillatory and excitable units: the peculiar role of positive coupling-frequency correlations
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• Discordant synchronization patterns on directed networks of identical phase oscillators with attractive and repulsive couplings
• Two-community noisy Kuramoto model with general interaction strengths: Part II
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• Vortex with four-fold defect lines in a simple model of self-propelled particles
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• Mean-field theory of vector spin models on networks with arbitrary degree distributions
• Bounds for the asymptotic order parameter of the stochastic Kuramoto model
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• Self-organisation of random oscillators with Lévy stable distributions