Kuramoto model

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The Kuramoto model, first proposed by Yoshiki Kuramoto (蔵本由紀 Kuramoto Yoshiki), is a mathematical model for the behavior of a large set of coupled oscillators, and synchronization in general. Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications.

1 Definition
2 Transformation
3 Large N limit
4 Solutions

[edit] Definition

In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency $ω i$ , and each is coupled equally to all other oscillators. Surprisingly, this fully nonlinear model can be solved exactly, in the infinite-N limit, with a clever transformation and the application of self-consistency arguments.

The most popular form of the model has the following governing equations:

$\frac{\partial \theta_i}{\partial t} = \omega_i + \frac{K}{N} \sum_{j=1}^{N} \sin(\theta_j - \theta_i), \qquad i = 1 \ldots N$ ,

where the system is composed of N limit-cycle oscillators.

[edit] Transformation

The transformation that allows this model to be solved exactly (at least in the N → ∞ limit) is as follows. Define the "order" parameters r and ψ as

$re^{i \psi} = \frac{1}{N} \sum_{j=1}^{N} e^{i \theta_j}$ .

Here r represents the phase-coherence of the population of oscillators, and ψ indicates the average phase. Applying this transformation, the governing equation becomes

$\frac{\partial \theta_i}{\partial t} = \omega_i + K r \sin(\psi-\theta_i)$ .

Thus the oscillators' equations are no longer explicitly coupled; instead the order parameters govern behavior. A further transformation is usually done, to a rotating frame in which the statistical average of $ω i$ over all oscillators is zero.

[edit] Large N limit

Now consider the case as N tends to infinity. Take the distribution of intrinsic natural frequencies as g(ω) (assumed normalized). Then assume that the density of oscillators at a given phase θ, with given natural frequency ω, at time t is $ρ(θ,ω, t)$ . Normalization requires that

$\int_{-\infty}^{\infty} \rho(\theta, \omega, t) \, d \theta = 1.$

The continuity equation for oscillator density will be

$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \theta}[\rho v] = 0,$

where v is the drift velocity of the oscillators given by taking the infinite-N limit in the transformed governing equation, i.e.,

$\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \theta}[\rho \omega + \rho K r \sin(\psi-\theta)] = 0.$

Finally, we must rewrite the definition of the order parameters for the continuum (infinite N) limit. $θ i$ must be replaced by its ensemble average (over all ω) and the sum must be replaced by an integral, to give

$r e^{i \psi} = \int_{-\pi}^{\pi} e^{i \theta} \int_{-\infty}^{\infty} \rho(\theta, \omega, t) g(\omega) \, d \omega \, d \theta.$

[edit] Solutions

The incoherent state with all oscillators drifting randomly corresponds to the solution $ρ = 1 / (2π)$ . In that case $r = 0$ , and there is no coherence among the oscillators. They are uniformly distributed across all possible phases, and the population is in a statistical steady-state (although individual oscillators continue to change phase in accordance with their intrinsic ω).

When coupling K is sufficiently strong, a fully synchronized solution is possible. In the fully synchronized state, all the oscillators share a common frequency, although their phases are different.

A solution for the case of partial synchronization yields a state in which only some oscillators (those near the ensemble's mean natural frequency) synchronize; other oscillators drift incoherently. Mathematically, the state has

$\rho = \delta\left(\theta - \psi - \arcsin\left(\frac{\omega}{K r}\right)\right)$