The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本 由紀, Kuramoto Yoshiki), is a mathematical model used in describing synchronization. More specifically, it is a model for the behavior of a large set of coupled oscillators. Its formulation was motivated by the behavior of systems of chemical and biological oscillators, and it has found widespread applications in areas such as neuroscience and oscillating flame dynamics. Kuramoto was quite surprised when the behavior of some physical systems, namely coupled arrays of Josephson junctions, followed his model. The model makes several assumptions, including that there is weak coupling, that the oscillators are identical or nearly identical, and that interactions depend sinusoidally on the phase difference between each pair of objects.
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 {\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 → ∞; alternatively, using self-consistency arguments, one may obtain steady-state solutions of the order parameter. The most popular form of the model has the following governing equations:
d θ i d t = ω i + 1 N ∑ j = 1 N K i j sin ( θ j − θ i ) , i = 1 … N , {\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+{\frac {1}{N}}\sum _{j=1}^{N}K_{ij}\sin(\theta _{j}-\theta _{i}),\qquad i=1\ldots N,}
where the system is composed of N limit-cycle oscillators, with phases θ i {\displaystyle \theta _{i}} and coupling constant K. Noise can be added to the system. In that case, the original equation is altered to
d θ i d t = ω i + ζ i + K N ∑ j = 1 N sin ( θ j − θ i ) , {\displaystyle {\frac {d\theta _{i}}{dt}}=\omega _{i}+\zeta _{i}+{\dfrac {K}{N}}\sum _{j=1}^{N}\sin(\theta _{j}-\theta _{i}),}
where ζ i {\displaystyle \zeta _{i}} is the fluctuation and a function of time. If the noise is considered to be white noise, then
⟨ ζ i ( t ) ⟩ = 0 , {\displaystyle \langle \zeta _{i}(t)\rangle =0,}
⟨ ζ i ( t ) ζ j ( t ′ ) ⟩ = 2 D δ i j δ ( t − t ′ ) , {\displaystyle \langle \zeta _{i}(t)\zeta _{j}(t')\rangle =2D\delta _{ij}\delta (t-t'),} with D {\displaystyle D} denoting 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
r e i ψ = 1 N ∑ j = 1 N e i θ j {\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. Substituting in the equation gives
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