Kuramoto model

 ( Exactly Solvable Models ) 

$\backslash frac\{d\; \backslash theta\_i\}\{d\; t\}\; =\; \backslash omega\_i\; +\; \backslash frac\{1\}\{N\}\; \backslash sum\_\{j=1\}^\{N\}\; K\_\{ij\}\; \backslash sin(\backslash theta\_j\; -\; \backslash theta\_i),\backslash qquad\; i\; =\; 1\; \backslash ldots\; N$,

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

Define the "order" parameters r and ψ as

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

$\backslash frac\{d\; \backslash theta\_i\}\{d\; t\}\; =\; \backslash omega\_i\; +\; K\; r\; \backslash sin(\backslash psi-\backslash theta\_i)$.

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. $\backslash psi=0$). Finally, the governing equation becomes

$\backslash frac\{d\; \backslash theta\_i\}\{d\; t\}\; =\; \backslash omega\_i\; -\; K\; r\; \backslash sin(\backslash theta\_i)$.

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

The continuity equation for oscillator density will be

$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\backslash rho\; =\; 0,$

$\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash theta\}\backslash rho\; =\; 0.$

Finally, the definition of the order parameters must be rewritten for the continuum (infinite N) limit. $\backslash theta\_i$ must be replaced by its ensemble average (over all $\backslash omega$) and the sum must be replaced by an integral, to give

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

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

$\backslash rho\; =\; \backslash frac\{\backslash rm\{normalization\; \backslash ;\; constant\}\}\{(\backslash omega\; -\; K\; r\; \backslash sin(\backslash theta\; -\; \backslash psi))\}$

When $g$ is unimodal and symmetric, then a stable state solution for the system is $$r=rK\; \backslash int\_\{-\backslash pi\; /\; 2\}^\{\backslash pi\; /\; 2\}\; \backslash cos\; ^2\; \backslash theta\; g(K\; r\; \backslash sin\; \backslash theta)\; d\; \backslash theta$$As coupling increases, there is a critical value $K\_c\; =\; 2/\backslash pi\; g(0)$ such that when , the long-term average of $r=0$, but when $K=\; K\_c(1+\backslash mu)$, where $\backslash mu\; >\; 0$ is small, then $r\; \backslash approx\; \backslash sqrt\{\backslash frac\{16\}\{\backslash pi\; K\_\{\backslash mathrm\{c\}\}^3\}\}\; \backslash sqrt\{\backslash frac\{\backslash mu\}\{-g^\{\backslash prime\; \backslash prime\}(0)\}\}$.

The N=2 case is trivial. In the rotating frame $\backslash omega\_1\; =\; -\backslash omega\_2$, and so the system is described exactly by the angle between the two oscillators: $\backslash Delta\; \backslash theta\; =\; \backslash theta\_1\; -\; \backslash theta\_2$. When , the angle cycles around the circle (that is, the fast oscillator keeps lapping around the slow oscillator). When $K\; >\; K\_c$, 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 $\backslash omega\_1,\; \backslash omega\_2,\; \backslash omega\_3,\; K$, the system has a strange attractor.

A related case for N=2 is the circle map or phase-locked loop. In this model, one of the oscillators is driven at a fixed frequency (and thus no longer free to vary), while the other, weakly coupled to the driver, is free to spin arbitrarily.

$\backslash mathcal\{H\}(q\_1,\backslash ldots,\; q\_N,\; p\_1,\; \backslash ldots,\; p\_N)\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash frac\{\backslash omega\_i\}\{2\}\; (q\_i^2\; +\; p\_i^2)\; +\; \backslash frac\{K\}\{4N\}\; \backslash sum\_\{i,j\; =\; 1\}^\{N\}$

      (q_i p_j - q_j p_i) ( q_j^2 + p_j^2 - q_i^2 - p_i^2 )  
     

 \phi_i = \mathrm{arctan} \left( q_i/p_i \right) , exact Kuramoto dynamics emerges on invariant manifolds of constant $I\_i\; \backslash equiv\; I$. With the transformed Hamiltonian
     

$\backslash mathcal\{H\text{'}\}(I\_1,\; \backslash ldots\; I\_N,\; \backslash phi\_1\; \backslash ldots,\; \backslash phi\_N)\; =\; \backslash sum\_\{i\; =\; 1\}^N\; \backslash omega\_i\; I\_i$

      - \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

   \frac{d I_i}{dt} = - \frac{\partial \mathcal{H}'}{\partial \phi_i}
 = - \frac{2 K}{N} \sum_{k=1}^N \sqrt{I_k I_i} (I_k - I_i) \cos(\phi_k - \phi_i)
     

\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] .
     

Uniform synchrony, waves and spirals can readily be observed in two-dimensional Kuramoto networks with diffusive local coupling. The stability of waves in these models can be determined analytically using the methods of Turing stability analysis. Uniform synchrony tends to be stable when the local coupling is everywhere positive whereas waves arise when the long-range connections are negative (inhibitory surround coupling). Waves and synchrony are connected by a topologically distinct branch of solutions known as ripple. These are low-amplitude spatially-periodic deviations that emerge from the uniform state (or the wave state) via a Hopf bifurcation. The existence of ripple solutions was predicted (but not observed) by Wiley, Strogatz and Michelle Girvan, who called them multi-twisted q-states.

The topology on which the Kuramoto model is studied can be made adaptive by use of fitness model showing enhancement of synchronization and neuropercolation in a self-organised way.

A graph with the minimal degree at least $d\_\{min\}\backslash ge\; 0.5\backslash \; n$ will be connected nevertheless for a graph to synchronize a little more it is required for such case it is known that there is critical connectivity threshold $\backslash mu\_c$ such that any graph on $n$ nodes with minimum degree $d\_\{min\}\backslash ge\; \backslash mu\_c\; (n-1)$ must globally synchronise.for $n$ large enough. The minimum maximum are known to lie between $0.6875\backslash le\; \backslash mu\_c\backslash le\; 0.75$.

Similarly it is known that Erdős-Rényi graphs with edge probability precisely $p=(1+\backslash epsilon)\backslash ln\; (n)/n$ as $n$ goes to infinity will be connected and it has been conjectured that this value is too the number at which these random graphs undergo synchronization which a 2022 preprint claims to have proved.

$\backslash frac\{d\; \backslash theta\_i\}\{d\; t\}\; =\; \backslash omega\_i\; +\; \backslash sum\_\{j=1\}^\{N\}\; a\_\{ij\}\; \backslash sin(\backslash theta\_j\; -\; \backslash theta\_i),\; \backslash qquad\; i\; =\; 1\; \backslash ldots\; N$

$\backslash Gamma(\backslash phi)\; =\; \backslash sin(\backslash phi)\; +\; a\_1\; \backslash sin(2\backslash phi\; +\; b\_1)\; +\; ...\; +\; a\_n\; \backslash sin(2n\backslash phi\; +\; b\_n)$,

• 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.

• Circle map
• Master stability function
• Oscillatory neural network
• Phase-locked loop
• Swarmalators