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Kuromoto模型(或者'''Kuramoto–Daido模型'''), 最早是由{[[Yoshiki Kuramoto]]|蔵本 由紀}提出的<ref>{{cite book |last = Kuramoto | first = Yoshiki |title = Lecture Notes in Physics, International Symposium on Mathematical Problems in Theoretical Physics |volume = 39 | publisher = Springer-Verlag, New York | page = 420 |year = 1975 |editor= H. Araki}}</ref><ref>{{cite book | author = Kuramoto Y | title = Chemical Oscillations, Waves, and Turbulence | publisher = New York, NY: Springer-Verlag | year = 1984 }}</ref> 这是一个用以描述[[同步]]现象的模型。更具体的，这是一种描述大量耦合在一起的[[振子]]产生同步振动现象的模型.<ref name=Strogatz-2000>{{cite journal | last = Strogatz | first = Steven H. | author-link = Steven Strogatz| title = From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators | journal = Physica D | volume = 143 | issue = 1–4 | pages = 1–20 | year = 2000 | bibcode = 2000PhyD..143....1S | doi = 10.1016/S0167-2789(00)00094-4 | s2cid = 16668746 | url = https://www.stevenstrogatz.com/articles/from-kuramoto-to-crawford-exploring-the-onset-of-synchronization-in-populations-of-coupled-oscillators-pdf

Kuromoto模型(或者'''Kuramoto–Daido模型'''), 最早是由[[Yoshiki Kuramoto|蔵本 由紀]]提出的<ref>{{cite book |last = Kuramoto | first = Yoshiki |title = Lecture Notes in Physics, International Symposium on Mathematical Problems in Theoretical Physics |volume = 39 | publisher = Springer-Verlag, New York | page = 420 |year = 1975 |editor= H. Araki}}</ref><ref>{{cite book | author = Kuramoto Y | title = Chemical Oscillations, Waves, and Turbulence | publisher = New York, NY: Springer-Verlag | year = 1984 }}</ref> 这是一个用以描述[[同步]]现象的模型。更具体的，这是一种描述大量耦合在一起的[[振子]]产生同步振动现象的模型<ref name=Strogatz-2000>{{cite journal | last = Strogatz | first = Steven H. | author-link = Steven Strogatz| title = From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators | journal = Physica D | volume = 143 | issue = 1–4 | pages = 1–20 | year = 2000 | bibcode = 2000PhyD..143....1S | doi = 10.1016/S0167-2789(00)00094-4 | s2cid = 16668746 | url = https://www.stevenstrogatz.com/articles/from-kuramoto-to-crawford-exploring-the-onset-of-synchronization-in-populations-of-coupled-oscillators-pdf

| access-date = 2024-01-04| doi-access = free }}</ref><ref>{{cite journal | last1 = Acebrón | first1 = Juan A. | last2 = Bonilla | first2 = L. L. | last3 = Vicente | first3 = Pérez | last4 = Conrad | first4 = J. | last5 = Ritort | first5 = Félix | last6 = Spigler | first6 = Renato | journal = Reviews of Modern Physics | pages = 137–185 | title = The Kuramoto model: A simple paradigm for synchronization phenomena | url = http://scala.uc3m.es/publications_MANS/PDF/finalKura.pdf | volume = 77 | issue = 1 | year = 2005|bibcode = 2005RvMP...77..137A |doi = 10.1103/RevModPhys.77.137 | hdl = 2445/12768 | hdl-access = free }}</ref> 最早的模型是为了描述化学与生物学过程中的震荡现象而提出的，后来该模型又被用到了包括[[神经科学]]、<ref name="Understanding the dynamics of biolo">{{cite journal | last1 = Bick | first1 = Christian | last2 = Goodfellow | first2 = Marc | last3 = Laing | first3 = Carlo R. | last4 = Martens | first4 = Erik A. | journal = Journal of Mathematical Neuroscience | pages = 9 | title = Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review | volume = 10 | issue = 1 | year = 2020 | doi = 10.1186/s13408-020-00086-9 |arxiv=1902.05307 | pmid = 32462281 | pmc = 7253574 | doi-access = free }}</ref><ref>{{cite journal | last1 = Cumin | first1 = D. | last2 = Unsworth | first2 = C. P. | doi = 10.1016/j.physd.2006.12.004 | issue = 2 | journal = Physica D | pages = 181–196 | title = Generalising the Kuromoto model for the study of neuronal synchronisation in the brain | volume = 226 | year = 2007 | bibcode = 2007PhyD..226..181C | hdl = 2292/2666 | hdl-access = free }}</ref><ref>{{cite journal | doi = 10.3389/fnhum.2010.00190 |pmid=21151358 |pmc=2995481 |vauthors=Breakspear M, Heitmann S, Daffertshofer A | title = Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model | journal = Front Hum Neurosci | volume = 4 | issue = 190 | year = 2010 |page=190 |doi-access=free }}</ref><ref>{{cite journal | doi = 10.1016/j.neuroimage.2013.11.047 |vauthors=Cabral J, Luckhoo H, Woolrich M, Joensson M, Mohseni H, Baker A, Kringelbach ML, Deco G | title = Exploring mechanisms of spontaneous functional connectivity in MEG: How delayed network interactions lead to structured amplitude envelopes of band-pass filtered oscillations | journal = NeuroImage | volume = 90 | pages = 423–435 | year = 2014 | pmid=24321555| doi-access = free | hdl = 10230/23081 | hdl-access = free }}</ref> 火焰震荡动力学等更广泛的领域.<ref>{{cite journal | last1 = Sivashinsky | first1 = G.I. | doi = 10.1080/00102207708946779 |issue = 3–4 | journal = Combust. Sci. Technol. | pages = 137–146 | title = Diffusional-thermal theory of cellular flames | volume = 15 | year = 1977 }}</ref><ref>{{cite journal | last1 = Forrester | first1 = D.M. | doi = 10.1038/srep16994 | pmid = 26582365 | pmc = 4652215 | journal = Scientific Reports | pages = 16994  | title = Arrays of coupled chemical oscillators | volume = 5 | year = 2015 |bibcode = 2015NatSR...516994F | arxiv = 1606.01556 }}</ref> Kuramoto也诧异于该模型也能够用来描述一些物理现象，例如一组[[约瑟夫·森结]]。<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>

| access-date = 2024-01-04| doi-access = free }}</ref><ref>{{cite journal | last1 = Acebrón | first1 = Juan A. | last2 = Bonilla | first2 = L. L. | last3 = Vicente | first3 = Pérez | last4 = Conrad | first4 = J. | last5 = Ritort | first5 = Félix | last6 = Spigler | first6 = Renato | journal = Reviews of Modern Physics | pages = 137–185 | title = The Kuramoto model: A simple paradigm for synchronization phenomena | url = http://scala.uc3m.es/publications_MANS/PDF/finalKura.pdf | volume = 77 | issue = 1 | year = 2005|bibcode = 2005RvMP...77..137A |doi = 10.1103/RevModPhys.77.137 | hdl = 2445/12768 | hdl-access = free }}</ref>。 最早的模型是为了描述化学与生物学过程中的震荡现象而提出的，后来该模型又被用到了包括[[神经科学]]<ref name="Understanding the dynamics of biolo">{{cite journal | last1 = Bick | first1 = Christian | last2 = Goodfellow | first2 = Marc | last3 = Laing | first3 = Carlo R. | last4 = Martens | first4 = Erik A. | journal = Journal of Mathematical Neuroscience | pages = 9 | title = Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review | volume = 10 | issue = 1 | year = 2020 | doi = 10.1186/s13408-020-00086-9 |arxiv=1902.05307 | pmid = 32462281 | pmc = 7253574 | doi-access = free }}</ref><ref>{{cite journal | last1 = Cumin | first1 = D. | last2 = Unsworth | first2 = C. P. | doi = 10.1016/j.physd.2006.12.004 | issue = 2 | journal = Physica D | pages = 181–196 | title = Generalising the Kuromoto model for the study of neuronal synchronisation in the brain | volume = 226 | year = 2007 | bibcode = 2007PhyD..226..181C | hdl = 2292/2666 | hdl-access = free }}</ref><ref>{{cite journal | doi = 10.3389/fnhum.2010.00190 |pmid=21151358 |pmc=2995481 |vauthors=Breakspear M, Heitmann S, Daffertshofer A | title = Generative models of cortical oscillations: Neurobiological implications of the Kuramoto model | journal = Front Hum Neurosci | volume = 4 | issue = 190 | year = 2010 |page=190 |doi-access=free }}</ref><ref>{{cite journal | doi = 10.1016/j.neuroimage.2013.11.047 |vauthors=Cabral J, Luckhoo H, Woolrich M, Joensson M, Mohseni H, Baker A, Kringelbach ML, Deco G | title = Exploring mechanisms of spontaneous functional connectivity in MEG: How delayed network interactions lead to structured amplitude envelopes of band-pass filtered oscillations | journal = NeuroImage | volume = 90 | pages = 423–435 | year = 2014 | pmid=24321555| doi-access = free | hdl = 10230/23081 | hdl-access = free }}</ref>、 火焰震荡动力学等更广泛的领域<ref>{{cite journal | last1 = Sivashinsky | first1 = G.I. | doi = 10.1080/00102207708946779 |issue = 3–4 | journal = Combust. Sci. Technol. | pages = 137–146 | title = Diffusional-thermal theory of cellular flames | volume = 15 | year = 1977 }}</ref><ref>{{cite journal | last1 = Forrester | first1 = D.M. | doi = 10.1038/srep16994 | pmid = 26582365 | pmc = 4652215 | journal = Scientific Reports | pages = 16994  | title = Arrays of coupled chemical oscillators | volume = 5 | year = 2015 |bibcode = 2015NatSR...516994F | arxiv = 1606.01556 }}</ref>。 Kuramoto自己也诧异于该模型也能够用来描述一些物理现象，例如一组[[约瑟夫·森结]]。<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>

该模型有很多的假设，包括弱耦合性，振子的同一或弱同一性，每一对振子之间的交互强度依赖于它们在相位差上的正弦值等。

该模型有很多的假设，包括弱耦合性，振子的同一或弱同一性，每一对振子之间的交互强度依赖于它们在相位差上的正弦值等。

:<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>，

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''。

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

可以在系统中添加[[噪声]]. 在这种情况下，原方程修改为:<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>

其中<math>D</math>表示噪声的强度。

其中<math>D</math>表示噪声的强度。

== Transformation ==

== 变换 ==

Define the "order" parameters ''r'' and ''ψ'' as

定义有序参数''r''和''ψ''，分别表示相位共质和平均相位，如下所示：

:<math>re^{i \psi} = \frac{1}{N} \sum_{j=1}^{N} e^{i \theta_j}  </math>.

:<math>re^{i \psi} = \frac{1}{N} \sum_{j=1}^{N} e^{i \theta_j}  </math>。

Here ''r'' represents the phase-[[coherence (physics)|coherence]] of the population of oscillators and ''ψ'' indicates the average phase. Substituting in the equation gives

这里''r''表示振荡器群的相位共质，''ψ''表示平均相位。将该变换代入方程后，得到：

:<math> \frac{d \theta_i}{d t} = \omega_i + K r \sin(\psi-\theta_i) </math>.

:<math> \frac{d \theta_i}{d t} = \omega_i + K r \sin(\psi-\theta_i) </math>。

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. <math>\psi=0</math>). Finally, the governing equation becomes

这样，振荡器的方程不再显式耦合；相反，有序参数管理着行为。通常会进行进一步的变换，进入一个旋转坐标系，其中所有振荡器的相位统计平均值为零（即<math>\psi=0</math>）。最后，控制方程变为：

:<math> \frac{d \theta_i}{d t} = \omega_i - K r \sin(\theta_i) </math>.

:<math> \frac{d \theta_i}{d t} = \omega_i - K r \sin(\theta_i) </math>。

== Large ''N'' limit ==

== Large ''N'' limit ==