kuramoto_model

Python implementation of the Kuramoto model on graphs.

Features

• Graph is represented as an adjacency matrix A, a 2D numpy ndarray.
• Interactions between oscillators are symmetric (i.e., A = A^T)
• It is implemented with a graph in mind, but basically can be used to any system, provided its representation can be mapped onto a (symmetric) adjacency matrix.

Example:

import numpy as np
import matplotlib.pyplot as plt
import networkx as nx
import seaborn as sns

from kuramoto import Kuramoto

# Instantiate a random graph and transform into an adjacency matrix
graph_nx = nx.erdos_renyi_graph(n=100, p=1) # p=1 -> all-to-all connectivity
graph = nx.to_numpy_array(graph_nx)

# Instantiate model with parameters
model = Kuramoto(coupling=3, dt=0.01, T=10, n_nodes=len(graph))

# Run simulation - output is time series for all nodes (node vs time)
act_mat = model.run(adj_mat=graph)

# Plot all the time series
plt.figure(figsize=(12, 4))
plt.plot(np.sin(act_mat.T))
plt.xlabel('time', fontsize=25)
plt.ylabel(r'$\sin(\theta)$', fontsize=25)

# Plot evolution of order global order parameter R_t
plt.figure(figsize=(12, 4))
plt.plot(
    [model.phase_coherence(vec)
     for vec in act_mat.T],
    'o'
)
plt.ylabel('order parameter', fontsize=25)
plt.xlabel('time', fontsize=25)
plt.ylim((-0.01, 1))

# Plot oscillators in complex plane at times t = 0, 250, 500
fig, axes = plt.subplots(ncols=3, nrows=1, figsize=(14, 4))

times = [0, 200, 500]
for ax, time in zip(axes, times):
    ax.plot(np.cos(act_mat[:, time]), 
            np.sin(act_mat[:, time]), 
            'o', markersize=10)
    ax.set_title(f'time = {time}')
    ax.set_ylim((-1.1, 1.1))
    ax.set_xlim((-1.1, 1.1))
    ax.set_xlabel(r'$\cos(\theta)$', fontsize=25)
    ax.set_ylabel(r'$\sin(\theta)$', fontsize=25)
    ax.grid(True)
plt.tight_layout()

Requirements

• numpy
• scipy
• For the example:
  • matplotlib
  • networkx
  • seaborn

Kuramoto model 101

• The Kuramoto model is used to study a wide range of systems with synchronization behaviour.
• It is a system of N coupled periodic oscillators.
• Each oscillator has its own natural frequency omega_i, i.e., constant angular velocity.
• Usually, the distribution of natural frequencies is choosen to be a gaussian-like symmetric function.
• A random initial (angular) position theta_i is assigned to each oscillator.
• The oscillator's state (position) theta_i is governed by the following differential equation:

where K is the coupling parameter and M_i is the number of oscillators interacting with oscillator i. A is the adjacency matrix enconding the interactions - typically binary and undirected (symmetric), such that if node i interacts with node j, A_ij = 1, otherwise 0. The basic idea is that, given two oscillators, the one running ahead is encouraged to slow down while the one running behind to accelerate.

In particular, the classical set up has M = N, since the interactions are all-to-all (i.e., a complete graph). Otherwise, M_i is the degree of node i.

Kuramoto model 201

A couple of facts in order to gain intuition about the model's behaviour:

• If synchronization occurs, it happens abruptly.
• That is, synchronization might not occur.
• Partial synchronization is a possible outcome.
• The order parameter R_t measures global synchronization at time t. It is basically the normalized length of the sum of all vectors (oscillators in the complex plane).
• About the global order parameter R_t:
• constant, in the double limit N -> inf, t -> inf
• independent of the initial conditions
• denpends on coupling strength
• it shows a sharp phase transition (as function of coupling)
• Steady solutions can be computed assuming R_t constant. The result is basically that each oscillator responds to the mean field produced by the rest.
• The higher the dimension of the lattice on which the oscillators are embedded, the easier it is to synchronize. For example, there isn't any good synchronization in one dimension, even with strong coupling. In two dimensions it is not clear yet. From 3 dimensions on, the model starts behaving more like the mean field prediction.

For more and better details, this talk by the great Steven Strogatz is a nice primer.

References & links

• Math Insight - applet
• Kuramoto, Y. (1984). Chemical Oscillations, Waves, and Turbulence (Vol. 19).