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.

Contents

1 Definition
2 Transformation
3 Large ''N'' limit
4 Solutions

Definition

In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency [\omega_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 = \omega_i + \frac \sum_^ \sin(\theta_j - \theta_i), \qquad i = 1 \ldots N],

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

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^ = \frac \sum_^ e^ ].

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 = \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 [\omega_i] over all oscillators is zero.

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 [\rho(\theta, \omega, t)]. Normalization requires that

[ \int_^ \rho(\theta, \omega, t) \, d \theta = 1. ]

The continuity equation for oscillator density will be

[ \frac + \frac[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 + \frac[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. [\theta_i] must be replaced by its ensemble average (over all ω) and the sum must be replaced by an integral, to give

[ r e^ = \int_^ e^ \int_^ \rho(\theta, \omega, t) g(\omega) \, d \omega \, d \theta.]

Solutions

The incoherent state with all oscillators drifting randomly corresponds to the solution [\rho = 1/(2\pi)]. 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. The fully synchronized state occurs when all oscillators share the same phase, i.e., [\rho = \delta(\theta-\theta_0)] (where δ represents the Dirac delta function). Note that this state can only occur in the finite N problem (unless K is allowed to diverge to infinity).

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\right)\right)]

for locked oscillators, and

[\rho = \frac}]

for drifting oscillators. The cutoff occurs when [|\omega| < K r ].

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