Kuramoto Model Simulations

These simulations investigate a Kuramoto model system of oscillators. All oscillators interact via a phase-minimizing coupling term.

This project aims to replicate and expand on the simulations performed in the research paper “Physics of the Rhythmic Applause” by Néda, Z., E. Ravasz, T. Vicsek, Y. Brechet, and A. L. Barabási. This review on Kurthe amoto model is also a useful resource.

Numerical simulations were implemented using the Euler method for time integration.

Simulation Outline

Parameters:

• K coupling strength
• D variance in natural frequency (omegaNtr) distribution
• N number of oscillators

Initial Conditions in most cases are a uniform distribution in initial oscillator phase,

for i = 1:N;
    theta(i) = rand*2.0*pi;
end

and a normal distribution in oscillator natural frequency

for i = 1:N;
    omegaNtr(i) = sqrt(D)*randn + omegaMean;
end

Files

kura1.m

Outputs the time evolution of the order parameter r.

plot_kura1.m

Must run kura1 before this script. Plots the simulation results from kura1.m alongside figure 10 from the aforementioned Kuramoto review for comparison.

kura2.m

Run a simulation which is an attempt to recreate the behavior seen in the simulations performed in the Physics of Rythmic Applause paper.

Useful Outputs:

• momega oscillator frequency at each recorded point in time
• int mean oscillator frequency, also referred to as intensity
• r order parameter
• mi, mr time-averaged intensity and order parameter corresponding to times mt

ordertest1.m

Parameter scan over different values of the coupling strength. For each value of K the order parameter is calculated.

Useful Outputs:

• Kmat all scanned K values
• rRun order parameter values corresponding to each K