更改

[[File:KuramotoModelPhaseLocking.ogv|thumb|right|Phase locking in the Kuramoto model|upright=1.1]]

[[File:KuramotoModelPhaseLocking.ogv|thumb|right|Phase locking in the Kuramoto model|upright=1.1]]

In the most popular version of the Kuramoto model, each of the oscillators is considered to have its own intrinsic [[natural frequency]] <math>\omega_i</math>, and each is coupled equally to all other oscillators. Surprisingly, this fully [[Nonlinear system|nonlinear]] model can be solved exactly in the limit of infinite oscillators, ''N''→ ∞;<ref name="Understanding the dynamics of biolo"/> alternatively, using self-consistency arguments one may obtain steady-state solutions of the order parameter.<ref name=Strogatz-2000 />  

在Kuramoto模型最流行的版本中，每个[[振荡器]]都被认为具有其自身的固有[[自然频率]]<math>\omega_i</math>，并且每个振荡器都与其他所有振荡器以相同的方式耦合。惊人的是，这个完全[[非线性系统]]的模型在振荡器数量趋近于无穷大（''N''→ ∞）时可以精确求解;<ref name="Understanding the dynamics of biolo"/>另一种方法是，通过自洽性论证，可以得到[[极大值参数]]的稳态解。<ref name=Strogatz-2000 />

:<math> \frac{d \theta_i}{d t} = \omega_i + \frac{1}{N} \sum_{j=1}^{N} K_{ij} \sin(\theta_j - \theta_i), \qquad i = 1 \ldots N</math>,

:<math> \frac{d \theta_i}{d t} = \omega_i + \frac{1}{N} \sum_{j=1}^{N} K_{ij} \sin(\theta_j - \theta_i), \qquad i = 1 \ldots N</math>，

where the system is composed of ''N'' limit-cycle oscillators, with phases <math> \theta_i </math> and coupling constant ''K''.

其中系统由''N''个[[Limit Cycle|极限周期]]振荡器组成，各自具有相位<math> \theta_i </math>和耦合常数''K''。

 

可以在系统中添加[[噪声]]. 在这种情况下，原方程修改为

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

:<math>

:<math>

\frac{d \theta_i}{d t} = \omega_{i}+\zeta_{i}+\dfrac{K}{N}\sum_{j=1}^N\sin(\theta_{j}-\theta_{i})

\frac{d \theta_i}{d t} = \omega_{i}+\zeta_{i}+\dfrac{K}{N}\sum_{j=1}^N\sin(\theta_{j}-\theta_{i})

</math>,

</math>

where <math>\zeta_{i}</math> is the fluctuation and a function of time. If the noise is considered to be white noise, then

，其中<math>\zeta_{i}</math>是[[Fluctuation|波动]]，且是时间的函数。如果将噪声视为[[White Noise|白噪声]]，则

:<math>

:<math>

\langle\zeta_{i}(t)\rangle=0

\langle\zeta_{i}(t)\rangle=0

</math> ,

</math> ，

:<math>

:<math>

\langle\zeta_{i}(t)\zeta_{j}(t')\rangle=2D\delta_{ij}\delta(t-t')

\langle\zeta_{i}(t)\zeta_{j}(t')\rangle=2D\delta_{ij}\delta(t-t')

</math>  

</math>

with <math>D</math> denoting the strength of noise.

其中<math>D</math>表示噪声的强度。

== Transformation ==

== Transformation ==