A large, well-mixed population of Belousov-Zhabotinsky oscillator droplets is coupled…

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 large, well-mixed population of Belousov-Zhabotinsky oscillator droplets is coupled through the mean field of the Kuramoto model,

\dot\theta_i=\omega_i+Kr\sin(\psi-\theta_i)

, where

r

and

\psi

are the Kuramoto order parameter's magnitude and mean phase. The droplets' natural angular frequencies are spread uniformly (a box distribution),

g(\omega) = \begin{cases}\dfrac{1}{2\Delta}, & |\omega|\le\Delta\\ 0, & |\omega|\gt\Delta\end{cases}

with

\Delta = 2.0\,\mathrm{rad\,s^{-1}}

. ((a)) Starting from the self-consistency equation for

r

(splitting the population into locked and drifting oscillators, and using that the drifting ones contribute nothing to

r

in the stationary state), derive

K_{\rm c} = 2/[\pi g(0)]

in the limit

r\to0^+

. ((b)) Evaluate

K_{\rm c}

for this population.

Browse the problem library