A small Kuramoto simulation records eight oscillator phases (in radians) at one…

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Problem

A small Kuramoto simulation records eight oscillator phases (in radians) at one instant:

\theta = \{0,\ 0.3,\ -0.2,\ 0.5,\ -0.4,\ 0.1,\ 0.6,\ -0.1\}.

((a)) Using the definition

re^{i\psi} = N^{-1}\sum_j e^{i\theta_j}

, compute the instantaneous coherence

r

and mean phase

\psi

(in radians) from these eight phases. ((b)) The oscillators obey the mean-field Kuramoto equation

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

with coupling

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

. For the oscillator at

\theta = 0.5\ \mathrm{rad}

with natural frequency

\omega = 0.5\ \mathrm{rad\,s^{-1}}

, use the

r

and

\psi

found in part (a) to find its instantaneous phase velocity

\dot\theta

. ((c)) In a separate, larger run,

N=1000

oscillators with Lorentzian-distributed natural frequencies of half-width

\Delta = 1.2\ \mathrm{rad\,s^{-1}}

are integrated with a fourth-order Runge-Kutta scheme, step

0.01

(dimensionless time units), seed

42

, for

10

independent realisations, discarding a transient of

200

time units, at coupling

K = 5.0\ \mathrm{rad\,s^{-1}}

. The steady-state order parameter measured is

r_{\rm sim} = 0.72 \pm 0.02

. Compute the theoretical coherence for a Lorentzian frequency density at this coupling, and comment on the agreement.

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