- Kuramoto model
The

**Kuramoto model**, first proposed byYoshiki Kuramoto (蔵本 由紀 "Kuramoto Yoshiki"), is amathematical model used to describesynchronization . More specifically, it is a model for the behavior of a large set of coupledoscillators . Its formulation was motivated by the behavior of systems ofchemical and biological oscillators, and it has found widespread applications.Fact|date=March 2008The model makes several assumptions, including that there is weak coupling, identical or nearly identical oscillators, and that interactions depend sinusoidally on the phase difference between the two objects.

**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 fullynonlinear 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::$frac\{partial\; heta\_i\}\{partial\; t\}\; =\; omega\_i\; +\; frac\{K\}\{N\}\; sum\_\{j=1\}^\{N\}\; sin(\; heta\_j\; -\; heta\_i),\; qquad\; i\; =\; 1\; ldots\; N$,where the system is composed of "N" limit-cycle oscillators.

Noise can be added to the system. In that case, the original equation is altered to::$frac\{partial\; heta\_i\}\{partial\; t\}\; =\; omega\_\{i\}+zeta\_\{i\}+dfrac\{K\}\{N\}sum\_\{j=1\}^Nsin(\; heta\_\{j\}-\; heta\_\{i\})$,where $zeta\_\{i\}$ is the fluctuation and a function of time. If we consider the noise to be white noise, then:$langlezeta\_\{i\}(t)\; angle=0$ ,:$langlezeta\_\{i\}(t)zeta\_\{j\}(t\text{'})\; angle=2Ddelta\_\{ij\}delta(t-t\text{'})$

with $D$ denotes the strength of noise.**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:$re^\{i\; psi\}\; =\; frac\{1\}\{N\}\; sum\_\{j=1\}^\{N\}\; e^\{i\; heta\_j\}$.Here "r" represents the phase-

coherence of the population of oscillators, and "ψ" indicates the average phase. Applying this transformation, the governing equation becomes:$frac\{partial\; heta\_i\}\{partial\; t\}\; =\; omega\_i\; +\; K\; r\; sin(psi-\; heta\_i)$.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 phases over all oscillators is zero. That is, $psi=0$. Finally, the governing equation becomes:$frac\{partial\; heta\_i\}\{partial\; t\}\; =\; omega\_i\; -\; K\; r\; sin(\; heta\_i)$.**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 $ho(\; heta,\; omega,\; t)$. Normalization requires that :$int\_\{-pi\}^\{pi\}\; ho(\; heta,\; omega,\; t)\; ,\; d\; heta\; =\; 1.$

The

continuity equation for oscillator density will be :$frac\{partial\; ho\}\{partial\; t\}\; +\; frac\{partial\}\{partial\; heta\}\; [\; ho\; v]\; =\; 0,$where "v" is the drift velocity of the oscillators given by taking the infinite-"N" limit in the transformed governing equation, i.e., :$frac\{partial\; ho\}\{partial\; t\}\; +\; frac\{partial\}\{partial\; heta\}\; [\; ho\; omega\; +\; ho\; K\; r\; sin(psi-\; heta)]\; =\; 0.$Finally, we must rewrite the definition of the order parameters for the continuum (infinite "N") limit. $heta\_i$ must be replaced by its ensemble average (over all ω) and the sum must be replaced by an integral, to give:$r\; e^\{i\; psi\}\; =\; int\_\{-pi\}^\{pi\}\; e^\{i\; heta\}\; int\_\{-infty\}^\{infty\}\; ho(\; heta,\; omega,\; t)\; g(omega)\; ,\; d\; omega\; ,\; d\; heta.$

**Solutions**The incoherent state with all oscillators drifting randomly corresponds to the solution $ho\; =\; 1/(2pi)$. 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:$ho\; =\; deltaleft(\; heta\; -\; psi\; -\; arcsinleft(frac\{omega\}\{K\; r\}\; ight)\; ight)$ for locked oscillators, and:$ho\; =\; frac\{\; m\{normalization\; ;\; constant\{(omega\; -\; K\; r\; sin(\; heta\; -\; psi))\}$for drifting oscillators. The cutoff occurs when $|omega|\; <\; K\; r$.

**References**Acebrón, Juan A.; Bonilla, L. L.; Pérez Vicente, Conrad J.; Ritort, Félix; Spigler, Renato.

**The Kuramoto model: a simple paradigm for synchronization phenomena**[*http://scala.uc3m.es/publications_MANS/PDF/finalKura.pdf*]Strogatz, S.

**From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators**, Physica D 143 (2000) 1–20

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