Current behavior

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Environment

• Version : Boostnote 0.11.10
• OS Version and name : Linux Mint 18.1

My markdown file

Kuramoto Model

The model is consist of $N$ coupled oscillators, $\theta_i(t)$, with natural frequencies $\omega_i$ deduced from the distribution $g(\omega)$. $$\frac{d\theta_i}{dt}=\omega_i+\Sigma_{j=1}^N K_{ij}\sin(\theta_i-\theta_j)$$ Each oscillator tries to run independently at its own frequency while the coupling tends to synchronize it to all the others. By making a suitable choice of a rotating frame, $$\theta_i \rightarrow \theta_i-\Omega t$$ in which $\Omega$ is the first moment of $g(\omega)$, we can transform the system to system of phase oscillators whose natural frequencies have zero mean. Many different models for the coupling matrix $K_{ij}$ have been considered such as nearest-neighbor coupling, hierarchical coupling, random long-range coupling or even state dependent interactions.

Kuramoto model with mean-field coupling

For this model, synchronization is conveniently measured by an order parameter. In the limit of infinitely many oscillators, $N = \infty$, the modulus of the order parameter vanishes if the oscillators are out of synchrony, and it is positive in synchronized states.

The original analysis of synchronization by Kuramoto consider mean-field coupling, $K_{ij}=K/N>0$, where $K$ is coupling constant.

def Kuramoto_orig(t, theta, arg):
    omega, K, N = arg
    theta_t = theta[:,None]
    delta_theta = theta-theta_t
    dtheta = omega + (K/N)*np.sum(np.sin(delta_theta), axis=1)
    return dtheta

The complex valued order parameter can be defined as: $$re^{i\psi}=\frac{1}{N}\Sigma_{j=1}^N e^{i\theta_j}$$

• $0 \leq r(t) \leq 1$ is the phase coherence.
• $\psi(t)$ is the average pahse.

  def order_parameter(theta):
    theta = theta.T
    z = sum(np.exp(theta*1j))/len(theta)
    return np.array([np.absolute(z), np.angle(z)])

  Then the equation becomes, $$\frac{d\theta_i}{dt}=\omega_i+Kr\sin (\psi-\theta_i)$$

For $N=500, K=6$:

For $N=500, K=1$

# Original Kuramoto
# Fix random init state
np.random.seed(123)
# Network parameters
N = 500
K = 6
# time interval
t0 = 0
dt = 1e-3
tf = 5000*dt
# initial condition
theta_0 = np.random.uniform(0, 2.*np.pi, N)
# Natural frequency
omega = np.random.uniform(-np.pi, np.pi, N)
arg = [omega, K, N]
solver = ode(Kuramoto_orig)
solver.set_integrator('vode',
                       atol = 1e-8,
                       rtol = 1e-6)
solver.set_f_params(arg)
solver.set_initial_value(theta_0,t0)
tvals = []
x = []
i = 0
while solver.successful() and solver.t < tf:
    solver.integrate(solver.t+dt)
    tvals.append(solver.t)
    x.append(solver.y)
    i += 1 
x = np.asarray(x)
R, psi = order_parameter(x)

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