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.

arXiv comment6 pages, 3 figures

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• Collective mode reductions for populations of coupled noisy oscillators
• Low-dimensional behavior of Kuramoto model with inertia in complex networks
• The Brownian Mean Field model
• Activation process in excitable systems with multiple noise sources: Large number of units
• Collective dynamics in two populations of noisy oscillators with asymmetric interactions
• Ott-Antonsen ansatz is the only admissible truncation of a circular cumulant series
• Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise
• 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
• Thermodynamics aspects of noise-induced phase synchronization
• Circular Cumulant Reductions for Macroscopic Dynamics of Kuramoto Ensemble with Multiplicative Intrinsic Noise
• Circular cumulant reductions for macroscopic dynamics of oscillator populations with non-Gaussian noise
• 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
• Information-theoretic description of a feedback-control Kuramoto model
• Two-community noisy Kuramoto model with general interaction strengths: Part I
• Gaussian theory for spatially distributed self-propelled particles
• Vortex with four-fold defect lines in a simple model of self-propelled particles
• The role of phase dynamics in a stochastic model of a passively advected scalar
• Macroscopic Models for Networks of Coupled Biological Oscillators
• Mean-field theory of vector spin models on networks with arbitrary degree distributions
• Bounds for the asymptotic order parameter of the stochastic Kuramoto model
• Two-community noisy Kuramoto model
• Self-organisation of random oscillators with Lévy stable distributions