更改

因此，当<math> I_j = I </math>时，流形是守恒的，因为<math> \frac{d I_i}{dt} = 0 </math>，而相位动力学<math> \frac{d \phi_i}{dt} </math>成为库拉莫托模型的动力学（当<math> I = 1/2 </math>时，耦合常数相同）。这一类哈密顿系统包括某些量子-经典系统，例如[[玻色-爱因斯坦凝聚态]]。

因此，当<math> I_j = I </math>时，流形是守恒的，因为<math> \frac{d I_i}{dt} = 0 </math>，而相位动力学<math> \frac{d \phi_i}{dt} </math>成为库拉莫托模型的动力学（当<math> I = 1/2 </math>时，耦合常数相同）。这一类哈密顿系统包括某些量子-经典系统，例如[[玻色-爱因斯坦凝聚态]]。

== Variations of the models ==

==模型的变体==

=== Variations of network topology ===

===网络拓扑结构的变体===

Beside the original model, which has an all-to-all topology, a sufficiently dense [[Complex Network|complex network]]-like topology is amenable to the mean-field treatment used in the solution of the original model<ref>{{cite journal | last1 = Rodrigues | first1 = F. A. | last2 = Peron | first2 = T.K. | last3 = Jie | first3 = P.|last4 = Kurths | first4 = J.| title = The Kuramoto model in complex networks | journal = Physics Reports | volume=610 | issue=1 | pages = 1–98 | doi = 10.1016/j.physrep.2015.10.008 | year = 2016| arxiv = 1511.07139 | bibcode = 2016PhR...610....1R | s2cid = 119290926 }}</ref> (see [[#Transformation|Transformation]] and [[#Large N limit|Large ''N'' limit]] above for more info). Network topologies such as rings and coupled populations support chimera states.<ref>{{cite journal | last1 = Abrams | first1 = D.M. | last2 = Strogatz | first2 = S.H. | title = Chimera states for coupled oscillators | journal = Physical Review Letters | volume = 93 |issue = 17 | pages = 174102 | year = 2004|doi=10.1103/physrevlett.93.174102 | pmid = 15525081 |arxiv = nlin/0407045 |bibcode = 2004PhRvL..93q4102A | s2cid = 8615112 }}</ref> One also may ask for the behavior of models in which there are intrinsically local, like one-dimensional topologies which the chain and the ring are prototypical examples. In such topologies, in which the coupling is not scalable according to 1/''N'', it is not possible to apply the canonical mean-field approach, so one must rely upon case-by-case analysis, making use of symmetries whenever it is possible, which may give basis for abstraction of general principles of solutions.

除了原来的全连接（all-to-all）拓扑结构外，类似于[[复杂网络]]的足够密集的拓扑结构也可以使用原模型解中所用的平均场方法<ref>{{cite journal | last1 = Rodrigues | first1 = F. A. | last2 = Peron | first2 = T.K. | last3 = Jie | first3 = P.|last4 = Kurths | first4 = J.| title = The Kuramoto model in complex networks | journal = Physics Reports | volume=610 | issue=1 | pages = 1–98 | doi = 10.1016/j.physrep.2015.10.008 | year = 2016| arxiv = 1511.07139 | bibcode = 2016PhR...610....1R | s2cid = 119290926 }}</ref>（见上文中的[[#Transformation|变换]]和[[#Large N limit|大''N''极限]]部分了解更多信息）。环形和耦合群体等网络拓扑结构支持[[奇美拉态]]<ref>{{cite journal | last1 = Abrams | first1 = D.M. | last2 = Strogatz | first2 = S.H. | title = Chimera states for coupled oscillators | journal = Physical Review Letters | volume = 93 |issue = 17 | pages = 174102 | year = 2004|doi=10.1103/physrevlett.93.174102 | pmid = 15525081 |arxiv = nlin/0407045 |bibcode = 2004PhRvL..93q4102A | s2cid = 8615112 }}</ref>。此外，还可以研究在一维拓扑结构（如链和环）中模型的行为。在这种拓扑结构中，耦合无法根据1/''N''进行缩放，因此无法应用标准的平均场方法，因此必须依赖针对每种情况的分析，尽可能利用对称性，这可能为抽象出一般的解原则提供依据。

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.<ref>{{cite journal | last1 = Kazanci | first1 = F. | last2 = Ermentrout | first2 = B. | title = Pattern formation in an array of oscillators with electrical and chemical coupling | journal = SIAM J Appl Math | volume=67 | issue = 2 | pages = 512–529 | doi = 10.1137/060661041 | year = 2006| citeseerx = 10.1.1.140.1020 }}</ref> 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.<ref>{{cite journal | last1 = Heitmann | first1 = S. | last2 = Gong | first2 = P. | last3 = Breakspear | first3 = M | title = A computational role for bistability and traveling waves in motor cortex | journal = Front Comput Neurosci | volume=6 | issue=67 | pages = 67 | doi = 10.3389/fncom.2012.00067 | pmid = 22973223 | year = 2012| pmc = 3438483 | doi-access = free }}</ref> These are low-amplitude spatially-periodic deviations that emerge from the uniform state (or the wave state) via a [[Hopf bifurcation]].<ref>{{cite journal | last1 = Heitmann | first1 = S. | last2 = Ermentrout | first2 = B. |title = Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity | journal = Biological Cybernetics | pages = 1–15 | doi = 10.1007/s00422-015-0646-6 | pmid = 25677527 | year = 2015 | volume=109| issue = 3 | s2cid = 18561153 }}</ref> The existence of ripple solutions was predicted (but not observed) by Wiley, Strogatz and [[Michelle Girvan|Girvan]],<ref name=":0">{{cite journal | last1 = Wiley | first1 = D. | last2 = Strogatz | first2 = S. | last3 = Girvan | first3 = M|author3-link= Michelle Girvan | title = The size of the sync basin | volume = 16 | issue = 1 | journal = Chaos | pages = 015103 | doi = 10.1063/1.2165594 | pmid = 16599769 | year = 2006|bibcode = 2006Chaos..16a5103W | s2cid = 21173189 }}</ref> who called them multi-twisted q-states.

在具有扩散局部耦合的二维库拉莫托网络中，可以轻松观察到均匀同步、波和螺旋。使用[[Turings稳定性分析]]的方法可以通过分析方法确定这些波的稳定性<ref>{{cite journal | last1 = Kazanci | first1 = F. | last2 = Ermentrout | first2 = B. | title = 数组中带有电气和化学耦合的振荡器的模式形成 | journal = SIAM J Appl Math | volume=67 | issue = 2 | pages = 512–529 | doi = 10.1137/060661041 | year = 2006| citeseerx = 10.1.1.140.1020 }}</ref>。当局部耦合在所有地方为正时，均匀同步往往是稳定的；而当负的长程连接（抑制性环绕耦合）存在时，会产生波。波和同步通过一种名为纹波（ripple）的不同拓扑分支的解决方案相连<ref>{{cite journal | last1 = Heitmann | first1 = S. | last2 = Gong | first2 = P. | last3 = Breakspear | first3 = M | title = 计算运动皮层中的双稳态和旅行波的角色 | journal = Front Comput Neurosci | volume=6 | issue=67 | pages = 67 | doi = 10.3389/fncom.2012.00067 | pmid = 22973223 | year = 2012| pmc = 3438483 | doi-access = free }}</ref>。这些是从均匀状态（或波状态）通过[[霍夫二倍分支]]出现的低幅度空间周期偏差<ref>{{cite journal | last1 = Heitmann | first1 = S. | last2 = Ermentrout | first2 = B. |title = 空间耦合库拉莫托振荡器中的同步、波和纹波（Synchrony, waves and ripple in spatially coupled Kuramoto oscillators with Mexican hat connectivity） | journal = Biological Cybernetics | pages = 1–15 | doi = 10.1007/s00422-015-0646-6 | pmid = 25677527 | year = 2015 | volume=109| issue = 3 | s2cid = 18561153 }}</ref>。Wiley、Strogatz和[[Michelle Girvan|姬]]预测了纹波解的存在（但未观察到），并称其为多重曲线q态<ref name=":0">{{cite journal | last1 = Wiley | first1 = D. | last2 = Strogatz | first2 = S. | last3 = Girvan | first3 = M|author3-link= Michelle Girvan | title = 同步 Sink的大小 | volume = 16 | issue = 1 | journal = Chaos | pages = 015103 | doi = 10.1063/1.2165594 | pmid = 16599769 | year = 2006|bibcode = 2006Chaos..16a5103W | s2cid = 21173189 }}</ref>。

The topology on which the Kuramoto model is studied can be made adaptive<ref>{{cite journal | last1 = Eom | first1 = Y.-H. | last2 = Boccaletti | first2 = S. | last3 = Caldarelli | first3 = G | title = Concurrent enhancement of percolation and synchronization in adaptive networks | volume = 7 | journal = Scientific Reports | pages = 015103 | doi = 10.1038/srep27111 | pmid = 27251577 | pmc = 4890019 | year = 2016 | arxiv = 1511.05468 | bibcode = 2016NatSR...627111E }}</ref> by use of [[Fitness model (network theory)|fitness model]] showing enhancement of synchronization and [[neuropercolation|percolation]] in a self-organised way.

可以通过使用[[适应性模型（网络理论）|适应性模型]]使库拉莫托模型的拓扑结构自适应<ref>{{cite journal | last1 = Eom | first1 = Y.-H. | last2 = Boccaletti | first2 = S. | last3 = Caldarelli | first3 = G | title = 适应性网络的同步和连通性的同时增强 | volume = 7 | journal = Scientific Reports | pages = 015103 | doi = 10.1038/srep27111 | pmid = 27251577 | pmc = 4890019 | year = 2016 | arxiv = 1511.05468 | bibcode = 2016NatSR...627111E }}</ref>，从而以自组织方式增强同步和[[神经连通性|连通性]]。

A graph with the minimal degree at least <math>d_{min}\ge 0.5\ n</math> 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 <math>\mu_c</math> such that any graph on <math>n</math> nodes with minimum degree <math>d_{min}\ge \mu_c (n-1)</math> must globally synchronise.for <math>n</math> large enough. The minimum<ref name=":0" /><ref>{{Cite arXiv |last=Canale |first=Eduardo A. |date=2022 |title=From weighted to unweighted graphs in Synchronizing Graph Theory |class=math.CO |eprint=2209.06362}}</ref> maximum<ref>{{Cite journal |last1=Kassabov |first1=Martin |last2=Strogatz |first2=Steven H. |last3=Townsend |first3=Alex |date=2021-07-01 |title=Sufficiently dense Kuramoto networks are globally synchronizing |url=https://pubs.aip.org/cha/article/31/7/073135/342231/Sufficiently-dense-Kuramoto-networks-are-globally |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |language=en |volume=31 |issue=7 |page=073135 |doi=10.1063/5.0057659 |pmid=34340322 |arxiv=2105.11406 |bibcode=2021Chaos..31g3135K |s2cid=235166315 |issn=1054-1500}}</ref> are known to lie between <math>0.6875\le \mu_c\le 0.75</math>.

具有最小度数至少为<math>d_{min}\ge 0.5\ n</math>的图将是连通的。然而，为了使图同步得更好，需要满足一个临界连接度阈值<math>\mu_c</math>，使得任何具有<math>n</math>个节点且最小度数<math>d_{min}\ge \mu_c (n-1)</math>的图必须实现全局同步（当<math>n</math>足够大时）。已知<math>\mu_c</math>的最小值和最大值介于<math>0.6875\le \mu_c\le 0.75</math>之间<ref name=":0" /><ref>{{Cite arXiv |last=Canale |first=Eduardo A. |date=2022 |title=From weighted to unweighted graphs in Synchronizing Graph Theory |class=math.CO |eprint=2209.06362}}</ref><ref>{{Cite journal |last1=Kassabov |first1=Martin |last2=Strogatz |first2=Steven H. |last3=Townsend |first3=Alex |date=2021-07-01 |title=Sufficiently dense Kuramoto networks are globally synchronizing |url=https://pubs.aip.org/cha/article/31/7/073135/342231/Sufficiently-dense-Kuramoto-networks-are-globally |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |language=en |volume=31 |issue=7 |page=073135 |doi=10.1063/5.0057659 |pmid=34340322 |arxiv=2105.11406 |bibcode=2021Chaos..31g3135K |s2cid=235166315 |issn=1054-1500}}</ref>。

Similarly it is known that [[Erdős–Rényi model|Erdős-Rényi graphs]] with edge probability precisely <math>p=(1+\epsilon)\ln (n)/n</math> as <math>n</math> goes to infinity will be connected and it has been conjectured<ref>{{Cite journal |last1=Ling |first1=Shuyang |last2=Xu |first2=Ruitu |last3=Bandeira |first3=Afonso S. |date=January 2019 |title=On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization |url=https://epubs.siam.org/doi/10.1137/18M1217644 |journal=SIAM Journal on Optimization |language=en |volume=29 |issue=3 |pages=1879–1907 |arxiv=1809.11083 |doi=10.1137/18M1217644 |issn=1052-6234 |s2cid=53067184}}</ref> that this value is too the number at which these random graphs undergo synchronization which a 2022 preprint claims to have proved.<ref>{{Cite arXiv |last1=Abdalla |first1=Pedro |last2=Bandeira |first2=Afonso S. |last3=Kassabov |first3=Martin |last4=Souza |first4=Victor |last5=Strogatz |first5=Steven H. |last6=Townsend |first6=Alex |date=2022 |title=Expander graphs are globally synchronising |class=math.CO |eprint=2210.12788}}</ref><ref>{{Cite web |last=Sloman |first=Leila |date=24 July 2023 |title=New Proof Shows That 'Expander' Graphs Synchronize |url=https://www.quantamagazine.org/new-proof-shows-that-expander-graphs-synchronize-20230724/ |website=[[Quanta Magazine]]}}</ref>

类似地，已知[[Erdős-Rényi随机图|Erdős-Rényi随机图]]在节点数趋近于无穷大时，边概率为<math>p=(1+\epsilon)\ln (n)/n</math>时将是连通的<ref>{{Cite journal |last1=Ling |first1=Shuyang |last2=Xu |first2=Ruitu |last3=Bandeira |first3=Afonso S. |date=January 2019 |title=On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization |url=https://epubs.siam.org/doi/10.1137/18M1217644 |journal=SIAM Journal on Optimization |language=en |volume=29 |issue=3 |pages=1879–1907 |arxiv=1809.11083 |doi=10.1137/18M1217644 |issn=1052-6234 |s2cid=53067184 }}</ref>，并且已经提出猜想认为此值也是这些随机图实现同步的节点数，2022年的一篇预印本声称已证明了这一点<ref>{{Cite arXiv |last1=Abdalla |first1=Pedro |last2=Bandeira |first2=Afonso S. |last3=Kassabov |first3=Martin |last4=Souza |first4=Victor |last5=Strogatz |first5=Steven H. |last6=Townsend |first6=Alex |date=2022 |title=Expander graphs are globally synchronising |class=math.CO |eprint=2210.12788}}</ref><ref>{{Cite web |last=Sloman |first=Leila |date=24 July 2023 |title=New Proof Shows That 'Expander' Graphs Synchronize |url=https://www.quantamagazine.org/new-proof-shows-that-expander-graphs-synchronize-20230724/ |website=[[Quanta Magazine]]}}</ref>。

=== Variations of network topology and network weights: from vehicle coordination to brain synchronization ===

===网络拓扑和网络权重的变体：从车辆协调到脑部同步===

[[File:Wesphysdemo - Synchronized Metronomes.webm|thumb|upright=1.1|[[Metronome]]s, initially out of phase, synchronize through small motions of the base on which they are placed. This system has been shown to be equivalent to the Kuramoto model.<ref>{{cite journal |last1=Pantaleone |first1=James |title=Synchronization of metronomes |journal=American Journal of Physics |date=October 2002 |volume=70 |issue=10 |pages=992–1000 |doi=10.1119/1.1501118 |bibcode=2002AmJPh..70..992P |url=http://www.math.pitt.edu/~bard/classes/mth3380/syncpapers/metronome.pdf}}</ref>]]

[[文件:Wesphysdemo - Synchronized Metronomes.webm|thumb|upright=1.1|[[节拍器]]，最初相位不同，通过它们放置的底座的微小运动实现同步。此系统已被证明与库拉莫托模型等效<ref>{{cite journal |last1=Pantaleone |first1=James |title=节拍器的同步 |journal=American Journal of Physics |date=October 2002 |volume=70 |issue=10 |pages=992–1000 |doi=10.1119/1.1501118 |bibcode=2002AmJPh..70..992P |url=http://www.math.pitt.edu/~bard/classes/mth3380/syncpapers/metronome.pdf}}</ref>]]

Some works in the control community have focused on the Kuramoto model on networks and with heterogeneous weights (i.e. the interconnection strength between any two oscillators can be arbitrary). The dynamics of this model reads as follows:

控制领域的一些研究集中于具有异构权重的网络中的库拉莫托模型（即任意两个振荡器之间的互连强度可以是任意值）。此模型的动力学如下所示:

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

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

where <math>a_{ij}</math> is a nonzero positive real number if oscillator <math>j</math> is connected to oscillator <math>i</math>. Such model allows for a more realistic study of, e.g., power grids,<ref>{{Cite journal |last=Filatrella |first=G. |last2=Nielsen |first2=A.H. |last3=Pedersen |first3=N.F. |title=Analysis of a power grid using a Kuramoto-like model |url=https://link.springer.com/article/10.1140/epjb/e2008-00098-8 |journal=The European Physical Journal B |publication-date=2008 |volume=61 |pages=485-491}}</ref> flocking, schooling, and vehicle coordination.<ref>{{cite journal | last1 = Dorfler | first1 = F. | last2 = Bullo | first2 = F. | issue = 6 | journal = Automatica | pages = 1539–1564 | title = Synchronization in complex networks of phase oscillators: A survey | volume = 50 | year = 2014 |doi = 10.1016/j.automatica.2014.04.012 }}</ref> In the work from Dörfler and colleagues, several theorems provide rigorous conditions for phase and frequency synchronization of this model. Further studies, motivated by experimental observations in neuroscience, focus on deriving analytical conditions for cluster synchronization of heterogeneous Kuramoto oscillators on arbitrary network topologies.<ref>{{cite journal | last1 = Menara | first1 = T. | last2 = Baggio | first2 = G. | last3 = Bassett | first3 = D. | last4 = Pasqualetti | first4 = F. | journal = IEEE Transactions on Control of Network Systems | pages = 302–314 | volume = 7 | issue = 1 | title = Stability Conditions for Cluster Synchronizations in Networks of Heterogeneous Kuramoto Oscillators | year = 2020 | doi = 10.1109/TCNS.2019.2903914 | arxiv = 1806.06083 | s2cid = 73729229 }}</ref> Since the Kuramoto model seems to play a key role in assessing synchronization phenomena in the brain,<ref>{{cite journal | last1 = Cabral | first1 = J. | last2 = Hugues | first2 = E. | last3 = Sporns | first3 = O. | last4 = Deco | first4 = G. | journal = NeuroImage | volume = 57 | issue = 1| pages = 130–139| title = Role of local network oscillations in resting-state functional connectivity | year = 2011 |doi = 10.1016/j.neuroimage.2011.04.010 | pmid = 21511044 | s2cid = 13959959 }}</ref> theoretical conditions that support empirical findings may pave the way for a deeper understanding of neuronal synchronization phenomena.

其中<math>a_{ij}</math>是非零正实数，当振荡器<math>j</math>连接到振荡器<math>i</math>时。这种模型允许对例如[[电网]]<ref>{{Cite journal |last=Filatrella |first=G. |last2=Nielsen |first2=A.H. |last3=Pedersen |first3=N.F. |title=Using a Kuramoto-like model to analyze a power grid |url=https://link.springer.com/article/10.1140/epjb/e2008-00098-8 |journal=The European Physical Journal B |publication-date=2008 |volume=61 |pages=485–491 }}</ref>、成群行为、鱼群和车辆协调等更为现实的研究。Dörfler及其同事的工作中，多个定理提供了严格的条件，以实现此模型的相位和频率同步<ref>{{cite journal | last1 = Dorfler | first1 = F. | last2 = Bullo | first2 = F. | issue = 6 | journal = Automatica | pages = 1539–1564 | title = 复杂相位振荡器网络的同步：综述 | volume = 50 | year = 2014 |doi = 10.1016/j.automatica.2014.04.012 }}</ref>。此外，受神经科学实验观察的启发，进一步的研究侧重于在任意网络拓扑上推导异构库拉莫托振荡器的集群同步的分析条件<ref>{{cite journal | last1 = Menara | first1 = T. | last2 = Baggio | first2 = G. | last3 = Bassett | first3 = D. | last4 = Pasqualetti | first4 = F. | journal = IEEE Transactions on Control of Network Systems | pages = 302–314 | volume = 7 | issue = 1 | title = 稀疏库拉莫托振荡器的网络集群同步稳定性条件 | year = 2020 | doi = 10.1109/TCNS.2019.2903914 | arxiv = 1806.06083 | s2cid = 73729229 }}</ref>。由于库拉莫托模型似乎在评估大脑中同步现象方面起着关键作用<ref>{{cite journal | last1 = Cabral | first1 = J. | last2 = Hugues | first2 = E. | last3 = Sporns | first3 = O. | last4 = Deco | first4 = G. | journal = NeuroImage | volume = 57 | issue = 1| pages = 130–139| title = Local network oscillations in resting-state functional connectivity中的作用 | year = 2011 |doi = 10.1016/j.neuroimage.2011.04.010 | pmid = 21511044 | s2cid = 13959959 }}</ref>，支持实证结果的理论条件可能有助于深入理解神经同步现象。

=== Variations of the phase interaction function ===

===相位交互函数的变体===

Kuramoto approximated the phase interaction between any two oscillators by its first Fourier component, namely <math>\Gamma(\phi) = \sin(\phi)</math>, where <math>\phi = \theta_j - \theta_i</math>. Better approximations can be obtained by including higher-order Fourier components,

库拉莫托用两个振荡器之间的相位相互作用的第一个[[傅里叶分析|傅里叶分量]]进行了近似，即<math>\Gamma(\phi) = \sin(\phi)</math>，其中<math>\phi = \theta_j - \theta_i</math>。通过包含更高阶的傅里叶分量，可以获得更好的近似：

:<math>\Gamma(\phi) = \sin(\phi) + a_1 \sin(2\phi + b_1) + ... + a_n \sin(2n\phi + b_n)</math>,

:<math>\Gamma(\phi) = \sin(\phi) + a_1 \sin(2\phi + b_1) + ... + a_n \sin(2n\phi + b_n)</math>，

where parameters <math>a_i</math> and <math>b_i</math> must be estimated. For example, synchronization among a network of weakly-coupled [[Hodgkin–Huxley model|Hodgkin–Huxley neurons]] can be replicated using coupled oscillators that retain the first four Fourier components of the interaction function.<ref>{{cite journal | last1 = Hansel | first1 = D. | last2 = Mato | first2 = G. | last3 = Meunier | first3 = C | issue = 5 | journal = Europhysics Letters | pages = 367–372 | title = Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons | volume = 23 | year = 1993|bibcode = 1993EL.....23..367H |doi = 10.1209/0295-5075/23/5/011 | s2cid = 250884499 }}</ref> The introduction of higher-order phase interaction terms can also induce interesting dynamical phenomena such as partially synchronized states,<ref>{{Cite journal|last1=Clusella|first1=Pau|last2=Politi|first2=Antonio|last3=Rosenblum|first3=Michael|date=2016|title=A minimal model of self-consistent partial synchrony|journal=New Journal of Physics|language=en|volume=18|issue=9|pages=093037|doi=10.1088/1367-2630/18/9/093037|issn=1367-2630|bibcode=2016NJPh...18i3037C|arxiv=1607.07178|s2cid=126227664 }}</ref> [[heteroclinic cycle]]s,<ref>{{cite journal | last1 = Hansel | first1 = D. | last2 = Mato | first2 = G. | last3 = Meunier | first3 = C | title = Clustering and slow switching in globally coupled phase oscillators | journal = Physical Review E | volume = 48 | issue = 5 | pages = 3470–3477 |doi=10.1103/physreve.48.3470 | pmid = 9961005 | year = 1993|bibcode = 1993PhRvE..48.3470H | url = https://hal.science/hal-02387163/file/Hansel1993.pdf }}</ref> and [[Chaos theory|chaotic dynamics]].<ref>{{cite journal | last1 = Bick | first1 = C. | last2 = Timme | first2 = M. | last3 = Paulikat | first3 = D. | last4 = Rathlev | first4 = D. | last5 = Ashwin | first5 = P. | title = Chaos in Symmetric Phase Oscillator Networks | journal = Physical Review Letters | volume = 107 |issue = 24 | pages = 244101 | year = 2011 | doi = 10.1103/PhysRevLett.107.244101 | pmid = 22243002 | arxiv = 1105.2230 | bibcode = 2011PhRvL.107x4101B | s2cid = 16144737 }}</ref>

其中参数<math>a_i</math>和<math>b_i</math>必须估计。例如，可以使用保留交互函数的前四个傅里叶分量的耦合振荡器来复制一组弱耦合的[[Hodgkin–Huxley模型|Hodgkin–Huxley神经元]]的同步<ref>{{cite journal | last1 = Hansel | first1 = D. | last2 = Mato | first2 = G. | last3 = Meunier | first3 = C | issue = 5 | journal = Europhysics Letters | pages = 367–372 | title = 弱耦合Hodgkin-Huxley神经元的相位动力学 | volume = 23 | year = 1993|bibcode = 1993EL.....23..367H |doi = 10.1209/0295-5075/23/5/011 | s2cid = 250884499 }}</ref>。引入更高阶的相位交互项也会引发有趣的动态现象，例如部分同步状态<ref>{{Cite journal|last1=Clusella|first1=Pau|last2=Politi|first2=Antonio|last3=Rosenblum|first3=Michael|date=2016|title=自洽部分同步的最小模型|journal=New Journal of Physics|language=en|volume=18|issue=9|pages=093037|doi=10.1088/1367-2630/18/9/093037|issn=1367-2630|bibcode=2016NJPh...18i3037C|arxiv=1607.07178|s2cid=126227664 }}</ref>、[[异离心循环]]<ref>{{cite journal | last1 = Hansel | first1 = D. | last2 = Mato | first2 = G. | last3 = Meunier | first3 = C | title = 全球耦合相位振荡器中的聚类和缓慢切换 | journal = Physical Review E | volume = 48 | issue = 5 | pages = 3470–3477 |doi=10.1103/physreve.48.3470 | pmid = 9961005 | year = 1993|bibcode = 1993PhRvE..48.3470H | url = https://hal.science/hal-02387163/file/Hansel1993.pdf }}</ref>，以及[[混沌理论|混沌动力学]]<ref>{{cite journal | last1 = Bick | first1 = C. | last2 = Timme | first2 = M. | last3 = Paulikat | first3 = D. | last4 = Rathlev | first4 = D. | last5 = Ashwin | first5 = P. | title = 对称相位振荡器网络中的混沌 | journal = Physical Review Letters | volume = 107 |issue = 24 | pages = 244101 | year = 2011 | doi = 10.1103/PhysRevLett.107.244101 | pmid = 22243002 | arxiv = 1105.2230 | bibcode = 2011PhRvL.107x4101B | s2cid = 16144737 }}</ref>。

== Availability ==

== Availability ==