Kuramoto model

In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic natural frequency ${\displaystyle \omega _{i}}$, and each is coupled equally to all other oscillators. Surprisingly, this fully nonlinear model can be solved exactly in the limit of infinite oscillators, N→ ∞;[12] alternatively, using self-consistency arguments one may obtain steady-state solutions of the order parameter.[13]

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

${\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {K}{N}}\sum _{j=1}^{N}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N}$,

where the system is composed of N limit-cycle oscillators, with phases ${\displaystyle \theta _{i}}$ and coupling constant K.

Noise can be added to the system. In that case, the original equation is altered to

${\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+\zeta _{i}+{\dfrac {K}{N}}\sum _{j=1}^{N}\sin(\theta _{j}-\theta _{i})}$,

where ${\displaystyle \zeta _{i}}$ is the fluctuation and a function of time. If we consider the noise to be white noise, then

${\displaystyle \langle \zeta _{i}(t)\rangle =0}$ ,

${\displaystyle \langle \zeta _{i}(t)\zeta _{j}(t')\rangle =2D\delta _{ij}\delta (t-t')}$

with ${\displaystyle D}$ denoting the strength of noise.

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

${\displaystyle re^{i\psi }={\frac {1}{N}}\sum _{j=1}^{N}e^{i\theta _{j}}}$.

Here r represents the phase-coherence of the population of oscillators and ψ indicates the average phase. Multiplying this equation with ${\displaystyle e^{-i\theta _{i}}}$ and only considering the imaginary part gives

${\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+Kr\sin(\psi -\theta _{i})}$.

Thus the oscillators' equations are no longer explicitly coupled; instead the order parameters govern the behavior. A further transformation is usually done, to a rotating frame in which the statistical average of phases over all oscillators is zero (i.e. ${\displaystyle \psi =0}$). Finally, the governing equation becomes

${\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}-Kr\sin(\theta _{i})}$.

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 ${\displaystyle \rho (\theta ,\omega ,t)}$. Normalization requires that

${\displaystyle \int _{-\pi }^{\pi }\rho (\theta ,\omega ,t)\,d\theta =1.}$

The continuity equation for oscillator density will be

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

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

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

Finally, we must rewrite the definition of the order parameters for the continuum (infinite N) limit. ${\displaystyle \theta _{i}}$ must be replaced by its ensemble average (over all ${\displaystyle \omega }$) and the sum must be replaced by an integral, to give

${\displaystyle re^{i\psi }=\int _{-\pi }^{\pi }e^{i\theta }\int _{-\infty }^{\infty }\rho (\theta ,\omega ,t)g(\omega )\,d\omega \,d\theta .}$

The incoherent state with all oscillators drifting randomly corresponds to the solution ${\displaystyle \rho =1/(2\pi )}$. In that case ${\displaystyle r=0}$, 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 can be 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

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

for locked oscillators, and

${\displaystyle \rho ={\frac {\rm {normalization\;constant}}{(\omega -Kr\sin(\theta -\psi ))}}}$

for drifting oscillators. The cutoff occurs when ${\displaystyle |\omega |<Kr}$.

When ${\displaystyle g}$ is unimodal and symmetric, then a stable state solution for the system is

$${\displaystyle r=rK\int _{-\pi /2}^{\pi /2}\cos ^{2}\theta g(Kr\sin \theta )d\theta }$$

As coupling increases, there is a critical value ${\displaystyle K_{c}=2/\pi g(0)}$ such that when ${\displaystyle K<K_{c}}$, the long-term average of ${\displaystyle r=0}$, but when ${\displaystyle K=K_{c}(1+\mu )}$, where ${\displaystyle \mu >0}$ is small, then ${\displaystyle r\approx {\sqrt {\frac {16}{\pi K_{\mathrm {c} }^{3}}}}{\sqrt {\frac {\mu }{-g^{\prime \prime }(0)}}}}$.[14][15]

When N is small, the solutions given above breaks down, as we cannot use the continuum approximation.

The N=2 case is trivial. In the rotating frame, we have ${\displaystyle \omega _{1}=-\omega _{2}}$, and so the system is described exactly by the angle between the two oscillators: ${\displaystyle \Delta \theta =\theta _{1}-\theta _{2}}$. When ${\displaystyle K<K_{c}=2|\omega _{1}|}$, the angle cycles around the circle (that is, the fast oscillator keeps lapping around the slow oscillator). When ${\displaystyle K>K_{c}}$, the angle falls into a stable attractor (that is, the two oscillators lock in phase). Similarly, the state space of the N=3 case is a 2-dimensional torus, and so the system evolves as a flow on the 2-torus, which cannot be chaotic.

Chaos first occurs when N=4. For some settings of ${\displaystyle \omega _{1},\omega _{2},\omega _{3},K}$, the system has a strange attractor.[16]

The dissipative Kuramoto model is contained[17] in certain conservative Hamiltonian systems with Hamiltonian of the form

${\displaystyle {\mathcal {H}}(q_{1},\ldots ,q_{N},p_{1},\ldots ,p_{N})=\sum _{i=1}^{N}{\frac {\omega _{i}}{2}}(q_{i}^{2}+p_{i}^{2})+{\frac {K}{4N}}\sum _{i,j=1}^{N}(q_{i}p_{j}-q_{j}p_{i})(q_{j}^{2}+p_{j}^{2}-q_{i}^{2}-p_{i}^{2})}$

After a canonical transformation to action-angle variables with actions ${\displaystyle I_{i}=\left(q_{i}^{2}+p_{i}^{2}\right)/2}$ and angles (phases) ${\displaystyle \phi _{i}=\mathrm {arctan} \left(q_{i}/p_{i}\right)}$, exact Kuramoto dynamics emerges on invariant manifolds of constant ${\displaystyle I_{i}\equiv I}$. With the transformed Hamiltonian

${\displaystyle {\mathcal {H'}}(I_{1},\ldots I_{N},\phi _{1}\ldots ,\phi _{N})=\sum _{i=1}^{N}\omega _{i}I_{i}-{\frac {K}{N}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\sqrt {I_{j}I_{i}}}(I_{j}-I_{i})\sin(\phi _{j}-\phi _{i}),}$

Hamilton's equation of motion become

${\displaystyle {\frac {dI_{i}}{dt}}=-{\frac {\partial {\mathcal {H}}'}{\partial \phi _{i}}}=-{\frac {2K}{N}}\sum _{k=1}^{N}{\sqrt {I_{k}I_{i}}}(I_{k}-I_{i})\cos(\phi _{k}-\phi _{i})}$

and

${\displaystyle {\frac {d\phi _{i}}{dt}}={\frac {\partial {\mathcal {H}}'}{\partial I_{i}}}=\omega _{i}+{\frac {K}{N}}\sum _{k=1}^{N}\left[2{\sqrt {I_{i}I_{k}}}\sin(\phi _{k}-\phi _{i})\right.\left.+{\sqrt {I_{k}/I_{i}}}(I_{k}-I_{i})\sin(\phi _{k}-\phi _{i})\right].}$

So the manifold with ${\displaystyle I_{j}=I}$ is invariant because ${\displaystyle {\frac {dI_{i}}{dt}}=0}$ and the phase dynamics ${\displaystyle {\frac {d\phi _{i}}{dt}}}$ becomes the dynamics of the Kuramoto model (with the same coupling constants for ${\displaystyle I=1/2}$). The class of Hamiltonian systems characterizes certain quantum-classical systems including Bose–Einstein condensates.

Distinct synchronization patterns in a two-dimensional array of Kuramoto-like oscillators with differing phase interaction functions and spatial coupling topologies. (A) Pinwheels. (B) Waves. (C) Chimeras. (D) Chimeras and waves combined. Color scale indicates oscillator phase.

There are a number of types of variations that can be applied to the original model presented above. Some models change to topological structure, others allow for heterogeneous weights, and other changes are more related to models that are inspired by the Kuramoto model but do not have the same functional form.

• pyclustering library includes a Python and C++ implementation of the Kuramoto model and its modifications. Also the library consists of oscillatory networks (for cluster analysis, pattern recognition, graph coloring, image segmentation) that are based on the Kuramoto model and phase oscillator.

• Master stability function
• Oscillatory neural network
• Phase-locked loop