A fleet of signal buoys flashes in a mean-field Kuramoto coupling

PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Complex and collective systems · Complex and collective systems

Many Parts, One Behaviour, Ch Synchronisation, cascades and the resilience of networks

Problem

A fleet of signal buoys flashes in a mean-field Kuramoto coupling, as in the Belousov-Zhabotinsky droplet population. Their natural angular frequencies now follow a parabolic (once-peaked, once-differentiable) density, g(ω)={34Δ(1−ω2Δ2),∣ω∣≤Δ0,∣ω∣>Δg(\omega) = \begin{cases}\dfrac{3}{4\Delta}\left(1-\dfrac{\omega^2}{\Delta^2}\right), & |\omega|\le\Delta\\ 0, & |\omega|\gt\Delta\end{cases} with Δ=1.5 rad s−1\Delta = 1.5\,\mathrm{rad\,s^{-1}} (this gg is normalised: ∫−ΔΔg(ω) dω=1\int_{-\Delta}^{\Delta}g(\omega)\,\mathrm{d}\omega=1). ((a)) Using the self-consistency argument for the Kuramoto order parameter (splitting locked from drifting oscillators, taking r→0+r\to0^+), obtain Kc=2/[πg(0)]K_{\rm c}=2/[\pi g(0)]. ((b)) Evaluate KcK_{\rm c} for this fleet.

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