“Kuromoto模型”的版本间的差异

2025年3月17日 (一) 15:02的版本

Kuromoto模型(或者Kuramoto–Daido模型), 最早是由蔵本由紀提出的^[1]^[2] 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^[3]^[4]。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^[5]^[6]^[7]^[8]、火焰震荡动力学等更广泛的领域^[9]^[10]。 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。^[11]

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。

定义

文件:KuramotoModelPhaseLocking.ogv

Phase locking in the Kuramoto model

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]\displaystyle{ \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;^[5]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。^[3] 该模型最流行的形式具有以下定律方程：

[math]\displaystyle{ \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个极限周期振荡器组成，各自具有相位[math]\displaystyle{ \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:[math]\displaystyle{ \frac{d \theta_i}{d t} = \omega_{i}+\zeta_{i}+\dfrac{K}{N}\sum_{j=1}^N\sin(\theta_{j}-\theta_{i}) }[/math] ，其中[math]\displaystyle{ \zeta_{i} }[/math]是波动，且是时间的函数。如果将噪声视为白噪声，则

[math]\displaystyle{ \langle\zeta_{i}(t)\rangle=0 }[/math] ，

[math]\displaystyle{ \langle\zeta_{i}(t)\zeta_{j}(t')\rangle=2D\delta_{ij}\delta(t-t') }[/math]

其中[math]\displaystyle{ D }[/math]表示噪声的强度。

变换

允许该模型（至少在振荡器数量趋于无穷大时）精确求解的变换如下：

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]\displaystyle{ re^{i \psi} = \frac{1}{N} \sum_{j=1}^{N} e^{i \theta_j} }[/math]。

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]\displaystyle{ \frac{d \theta_i}{d t} = \omega_i + K r \sin(\psi-\theta_i) }[/math]。

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]\displaystyle{ \psi=0 }[/math]）。最后，控制方程变为：

[math]\displaystyle{ \frac{d \theta_i}{d t} = \omega_i - K r \sin(\theta_i) }[/math]。

N很大时的极限情况

当振荡器数量趋于无穷大时，考虑此情况。假设固有自然频率的分布为g(ω（已归一化）。然后假设在给定相位θ和给定自然频率ω时，振荡器的密度在时间t为[math]\displaystyle{ \rho(\theta, \omega, t) }[/math]。归一化要求

[math]\displaystyle{ \int_{-\pi}^{\pi} \rho(\theta, \omega, t) , d \theta = 1. }[/math]

振荡器密度的连续性方程为

[math]\displaystyle{ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \theta}[\rho v] = 0, }[/math]

其中v是振荡器的漂移速度，通过在变换后的控制方程中取无穷-N极限得到，使得

[math]\displaystyle{ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial \theta}[\rho \omega + \rho K r \sin(\psi-\theta)] = 0. }[/math]

最后，必须将有序参数的定义改写为连续（无穷N）极限。[math]\displaystyle{ \theta_i }[/math]必须替换为其在所有[math]\displaystyle{ \omega }[/math]上的合集平均值，并用积分替换求和，得到

[math]\displaystyle{ r e^{i \psi} = \int_{-\pi}^{\pi} e^{i \theta} \int_{-\infty}^{\infty} \rho(\theta, \omega, t) g(\omega) , d \omega , d \theta. }[/math]

大N极限下的解

当所有振荡器以随机漂移的方式呈现不相干（incoherent）状态时，对应的解为[math]\displaystyle{ \rho = 1/(2\pi) }[/math]。此时[math]\displaystyle{ r = 0 }[/math]，振荡器之间没有相位共质（coherence）。它们均匀分布在所有可能的相位中，整体处于统计稳态（尽管单个振荡器仍根据其固有ω改变相位）。

当耦合强度K足够大时，可能出现全同步（fully synchronized）的解。在全同步状态下，所有振荡器共享相同的频率，尽管它们的相位可能不同。

部分同步的解对应的状态是只有部分振荡器（靠近集合平均自然频率的振荡器）实现同步，其余振荡器则以不相干的方式漂移。从数学上来说，这种状态下，锁定（locked）振荡器的密度为： [math]\displaystyle{ \rho = \delta\left(\theta - \psi - \arcsin\left(\frac{\omega}{K r}\right)\right) }[/math]，

而漂移振荡器的密度为： [math]\displaystyle{ \rho = \frac{\text{归一化常数}}{\omega - K r \sin(\theta - \psi)}。 }[/math] 截止条件为[math]\displaystyle{ |\omega| \lt K r }[/math]。

当g（自然频率分布）为单峰且对称时，系统的稳定状态解为： [math]\displaystyle{ r=rK \int_{-\pi / 2}^{\pi / 2} \cos ^2 \theta g(K r \sin \theta) d \theta。 }[/math]

随着耦合强度K的增加，当K达到临界值K_c = 2/\pi g(0)时，系统的长期平均[math]\displaystyle{ r=0 }[/math]；但当K稍大于临界值（即K= K_c(1+\mu)，其中μ > 0是一个小量）时，解近似为： [math]\displaystyle{ r \approx \sqrt{\frac{16}{\pi K_{\mathrm{c}}^3}} \sqrt{\frac{\mu}{-g^{\prime \prime}(0)}}。 }[/math]^[12]^[3]

小N情况

当N较小时，上述解失效，因为无法使用连续近似。

对于N=2的情况是平凡的。在旋转框架中，[math]\displaystyle{ \omega_1 = -\omega_2 }[/math]，因此系统的描述完全由两振荡器之间的角度差[math]\displaystyle{ \Delta \theta = \theta_1 - \theta_2 }[/math]给出。当[math]\displaystyle{ K \lt K_c = 2|\omega_1| }[/math]时，角度会在圆周上循环（即快振荡器会不断绕慢振荡器环行）。当[math]\displaystyle{ K \gt K_c }[/math]时，角度会进入一个稳定的吸引子（即两振荡器的相位锁定）。类似地，N=3的情况下，状态空间是二维环面，系统在二维环面上演化，演化过程无法出现混沌。

混沌最早出现在N=4的情况下。在某些[math]\displaystyle{ \omega_1, \omega_2, \omega_3, K }[/math]的设置下，系统存在奇异吸引子^[13]。

与哈密顿系统的联系

耗散型库拉莫托模型包含在某些具有以下形式的哈密顿函数的保守哈密顿系统中：^[14]

[math]\displaystyle{ \mathcal{H}(q_1,\ldots, q_N, p_1, \ldots, p_N) = \sum_{i = 1}^N \frac{\omega_i}{2} (q_i^2 + p_i^2) + \frac{K}{4N} \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 ) }[/math]

在将正则变换应用于作用-角度变量（以作用[math]\displaystyle{ I_i = \left( q_i^2 + p_i^2 \right)/2 }[/math]和角度（相位）[math]\displaystyle{ \phi_i = \mathrm{arctan} \left( q_i/p_i \right) }[/math]为变量）后，库拉莫托动力学在常数[math]\displaystyle{ I_i \equiv I }[/math]的不变流形上出现。经过变换的哈密顿函数为：

[math]\displaystyle{ \mathcal{H'}(I_1, \ldots I_N, \phi_1 \ldots, \phi_N) = \sum_{i = 1}^N \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), }[/math]

哈密顿运动方程变为：

[math]\displaystyle{ \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) }[/math]

和

[math]\displaystyle{ \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] . }[/math]

因此，当[math]\displaystyle{ I_j = I }[/math]时，流形是守恒的，因为[math]\displaystyle{ \frac{d I_i}{dt} = 0 }[/math]，而相位动力学[math]\displaystyle{ \frac{d \phi_i}{dt} }[/math]成为库拉莫托模型的动力学（当[math]\displaystyle{ 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。

模型的变体

在库拉莫托模型中存在多种类型的变体。有些模型改变了拓扑结构，有些允许使用异构权重，而其他一些变体则与受库拉莫托模型启发但功能形式不同的模型更加相关。

网络拓扑结构的变体

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^[15]（见上文中的变换和大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^[16]。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。

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^[17]。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^[18]。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^[19]。Wiley、Strogatz和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^[20]。

可以通过使用适应性模型使库拉莫托模型的拓扑结构自适应^[21]，从而以自组织方式增强同步和连通性。

具有最小度数至少为[math]\displaystyle{ 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]\displaystyle{ \mu_c }[/math]，使得任何具有[math]\displaystyle{ n }[/math]个节点且最小度数[math]\displaystyle{ d_{min}\ge \mu_c (n-1) }[/math]的图必须实现全局同步（当[math]\displaystyle{ n }[/math]足够大时）。已知[math]\displaystyle{ \mu_c }[/math]的最小值和最大值介于[math]\displaystyle{ 0.6875\le \mu_c\le 0.75 }[/math]之间^[20]^[22]^[23]。

类似地，已知Erdős-Rényi随机图在节点数趋近于无穷大时，边概率为[math]\displaystyle{ p=(1+\epsilon)\ln (n)/n }[/math]时将是连通的^[24]，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^[25]^[26]。

网络拓扑和网络权重的变体：从车辆协调到脑部同步

文件:Wesphysdemo - Synchronized Metronomes.webm

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^[27]

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]\displaystyle{ \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]，

其中[math]\displaystyle{ a_{ij} }[/math]是非零正实数，当振荡器[math]\displaystyle{ j }[/math]连接到振荡器[math]\displaystyle{ i }[/math]时。这种模型允许对例如电网^[28]、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^[29]。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^[30]。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^[31]，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。

相位交互函数的变体

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量进行了近似，即[math]\displaystyle{ \Gamma(\phi) = \sin(\phi) }[/math]，其中[math]\displaystyle{ \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]\displaystyle{ \Gamma(\phi) = \sin(\phi) + a_1 \sin(2\phi + b_1) + ... + a_n \sin(2n\phi + b_n) }[/math]，

其中参数[math]\displaystyle{ a_i }[/math]和[math]\displaystyle{ 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神经元的同步^[32]。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^[33]、异离心循环^[34]，以及混沌动力学^[35]。

Availability

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.

References

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↑ Kassabov, Martin; Strogatz, Steven H.; Townsend, Alex (2021-07-01). "Sufficiently dense Kuramoto networks are globally synchronizing". Chaos: An Interdisciplinary Journal of Nonlinear Science (in English). 31 (7): 073135. arXiv:2105.11406. Bibcode:2021Chaos..31g3135K. doi:10.1063/5.0057659. ISSN 1054-1500. PMID 34340322. S2CID 235166315.
↑ Ling, Shuyang; Xu, Ruitu; Bandeira, Afonso S. (January 2019). "On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization". SIAM Journal on Optimization (in English). 29 (3): 1879–1907. arXiv:1809.11083. doi:10.1137/18M1217644. ISSN 1052-6234. S2CID 53067184.
↑ Abdalla, Pedro; Bandeira, Afonso S.; Kassabov, Martin; Souza, Victor; Strogatz, Steven H.; Townsend, Alex (2022). "Expander graphs are globally synchronising". arXiv:2210.12788 [math.CO].
↑ Sloman, Leila (24 July 2023). "New Proof Shows That 'Expander' Graphs Synchronize". Quanta Magazine.
↑ Pantaleone, James (October 2002). "节拍器的同步" (PDF). American Journal of Physics. 70 (10): 992–1000. Bibcode:2002AmJPh..70..992P. doi:10.1119/1.1501118.
↑ Filatrella, G.; Nielsen, A.H.; Pedersen, N.F. (2008). "Using a Kuramoto-like model to analyze a power grid". The European Physical Journal B. 61: 485–491.
↑ Dorfler, F.; Bullo, F. (2014). "复杂相位振荡器网络的同步：综述". Automatica. 50 (6): 1539–1564. doi:10.1016/j.automatica.2014.04.012.
↑ Menara, T.; Baggio, G.; Bassett, D.; Pasqualetti, F. (2020). "稀疏库拉莫托振荡器的网络集群同步稳定性条件". IEEE Transactions on Control of Network Systems. 7 (1): 302–314. arXiv:1806.06083. doi:10.1109/TCNS.2019.2903914. S2CID 73729229.
↑ Cabral, J.; Hugues, E.; Sporns, O.; Deco, G. (2011). "Local network oscillations in resting-state functional connectivity中的作用". NeuroImage. 57 (1): 130–139. doi:10.1016/j.neuroimage.2011.04.010. PMID 21511044. S2CID 13959959.
↑ Hansel, D.; Mato, G.; Meunier, C (1993). "弱耦合Hodgkin-Huxley神经元的相位动力学". Europhysics Letters. 23 (5): 367–372. Bibcode:1993EL.....23..367H. doi:10.1209/0295-5075/23/5/011. S2CID 250884499. {{cite journal}}: Check |s2cid= value (help)
↑ Clusella, Pau; Politi, Antonio; Rosenblum, Michael (2016). "自洽部分同步的最小模型". New Journal of Physics (in English). 18 (9): 093037. arXiv:1607.07178. Bibcode:2016NJPh...18i3037C. doi:10.1088/1367-2630/18/9/093037. ISSN 1367-2630. S2CID 126227664.
↑ Hansel, D.; Mato, G.; Meunier, C (1993). "全球耦合相位振荡器中的聚类和缓慢切换" (PDF). Physical Review E. 48 (5): 3470–3477. Bibcode:1993PhRvE..48.3470H. doi:10.1103/physreve.48.3470. PMID 9961005.
↑ Bick, C.; Timme, M.; Paulikat, D.; Rathlev, D.; Ashwin, P. (2011). "对称相位振荡器网络中的混沌". Physical Review Letters. 107 (24): 244101. arXiv:1105.2230. Bibcode:2011PhRvL.107x4101B. doi:10.1103/PhysRevLett.107.244101. PMID 22243002. S2CID 16144737.

[1] Kuramoto, Yoshiki (1975). H. Araki. ed. Lecture Notes in Physics, International Symposium on Mathematical Problems in Theoretical Physics. 39. Springer-Verlag, New York. p. 420.

[2] Kuramoto Y (1984). Chemical Oscillations, Waves, and Turbulence. New York, NY: Springer-Verlag.

[Strogatz-2000-3] 3.0 ^3.1 ^3.2 Strogatz, Steven H. (2000). "From Kuramoto to Crawford: Exploring the onset of synchronization in populations of coupled oscillators". Physica D. 143 (1–4): 1–20. Bibcode:2000PhyD..143....1S. doi:10.1016/S0167-2789(00)00094-4. S2CID 16668746. Retrieved 2024-01-04.

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[Understanding_the_dynamics_of_biolo-5] 5.0 ^5.1 Bick, Christian; Goodfellow, Marc; Laing, Carlo R.; Martens, Erik A. (2020). "Understanding the dynamics of biological and neural oscillator networks through exact mean-field reductions: a review". Journal of Mathematical Neuroscience. 10 (1): 9. arXiv:1902.05307. doi:10.1186/s13408-020-00086-9. PMC 7253574. PMID 32462281.

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[8] Cabral J, Luckhoo H, Woolrich M, Joensson M, Mohseni H, Baker A, Kringelbach ML, Deco G (2014). "Exploring mechanisms of spontaneous functional connectivity in MEG: How delayed network interactions lead to structured amplitude envelopes of band-pass filtered oscillations". NeuroImage. 90: 423–435. doi:10.1016/j.neuroimage.2013.11.047. hdl:10230/23081. PMID 24321555.

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[11] Steven Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003.

[12] Strogatz, Steven H. (1994). "Norbert Wiener's Brain Waves". In Levin, Simon A. (in en). Frontiers in Mathematical Biology. Lecture Notes in Biomathematics. 100. Berlin, Heidelberg: Springer. pp. 122–138. doi:10.1007/978-3-642-50124-1_7. ISBN 978-3-642-50124-1. https://link.springer.com/chapter/10.1007/978-3-642-50124-1_7.

[13] Maistrenko, Yuri L.; Popovych, Oleksandr V.; Tass, Peter A. (November 2005). "Kuramoto模型的混沌吸引子". 国际双困杂志 (in English). 15 (11): 3457–3466. Bibcode:2005IJBC...15.3457M. doi:10.1142/S0218127405014155. ISSN 0218-1274.

[14] Witthaut, Dirk; Timme, Marc (2014). "哈密顿系统中的库拉莫托动力学". 物理评论E. 90 (3): 032917. arXiv:1305.1742. Bibcode:2014PhRvE..90c2917W. doi:10.1103/PhysRevE.90.032917. PMID 25314514. S2CID 7510614.

[15] Rodrigues, F. A.; Peron, T.K.; Jie, P.; Kurths, J. (2016). "The Kuramoto model in complex networks". Physics Reports. 610 (1): 1–98. arXiv:1511.07139. Bibcode:2016PhR...610....1R. doi:10.1016/j.physrep.2015.10.008. S2CID 119290926.

[16] Abrams, D.M.; Strogatz, S.H. (2004). "Chimera states for coupled oscillators". Physical Review Letters. 93 (17): 174102. arXiv:nlin/0407045. Bibcode:2004PhRvL..93q4102A. doi:10.1103/physrevlett.93.174102. PMID 15525081. S2CID 8615112.

[17] Kazanci, F.; Ermentrout, B. (2006). "数组中带有电气和化学耦合的振荡器的模式形成". SIAM J Appl Math. 67 (2): 512–529. CiteSeerX 10.1.1.140.1020. doi:10.1137/060661041.

[18] Heitmann, S.; Gong, P.; Breakspear, M (2012). "计算运动皮层中的双稳态和旅行波的角色". Front Comput Neurosci. 6 (67): 67. doi:10.3389/fncom.2012.00067. PMC 3438483. PMID 22973223.

[19] Heitmann, S.; Ermentrout, B. (2015). "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）". Biological Cybernetics. 109 (3): 1–15. doi:10.1007/s00422-015-0646-6. PMID 25677527. S2CID 18561153.

[:0-20] 20.0 ^20.1 Wiley, D.; Strogatz, S.; Girvan, M (2006). "同步 Sink的大小". Chaos. 16 (1): 015103. Bibcode:2006Chaos..16a5103W. doi:10.1063/1.2165594. PMID 16599769. S2CID 21173189.

[21] Eom, Y.-H.; Boccaletti, S.; Caldarelli, G (2016). "适应性网络的同步和连通性的同时增强". Scientific Reports. 7: 015103. arXiv:1511.05468. Bibcode:2016NatSR...627111E. doi:10.1038/srep27111. PMC 4890019. PMID 27251577.

[22] Canale, Eduardo A. (2022). "From weighted to unweighted graphs in Synchronizing Graph Theory". arXiv:2209.06362 [math.CO].

[23] Kassabov, Martin; Strogatz, Steven H.; Townsend, Alex (2021-07-01). "Sufficiently dense Kuramoto networks are globally synchronizing". Chaos: An Interdisciplinary Journal of Nonlinear Science (in English). 31 (7): 073135. arXiv:2105.11406. Bibcode:2021Chaos..31g3135K. doi:10.1063/5.0057659. ISSN 1054-1500. PMID 34340322. S2CID 235166315.

[24] Ling, Shuyang; Xu, Ruitu; Bandeira, Afonso S. (January 2019). "On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization". SIAM Journal on Optimization (in English). 29 (3): 1879–1907. arXiv:1809.11083. doi:10.1137/18M1217644. ISSN 1052-6234. S2CID 53067184.

[25] Abdalla, Pedro; Bandeira, Afonso S.; Kassabov, Martin; Souza, Victor; Strogatz, Steven H.; Townsend, Alex (2022). "Expander graphs are globally synchronising". arXiv:2210.12788 [math.CO].

[26] Sloman, Leila (24 July 2023). "New Proof Shows That 'Expander' Graphs Synchronize". Quanta Magazine.

[27] Pantaleone, James (October 2002). "节拍器的同步" (PDF). American Journal of Physics. 70 (10): 992–1000. Bibcode:2002AmJPh..70..992P. doi:10.1119/1.1501118.

[28] Filatrella, G.; Nielsen, A.H.; Pedersen, N.F. (2008). "Using a Kuramoto-like model to analyze a power grid". The European Physical Journal B. 61: 485–491.

[29] Dorfler, F.; Bullo, F. (2014). "复杂相位振荡器网络的同步：综述". Automatica. 50 (6): 1539–1564. doi:10.1016/j.automatica.2014.04.012.

[30] Menara, T.; Baggio, G.; Bassett, D.; Pasqualetti, F. (2020). "稀疏库拉莫托振荡器的网络集群同步稳定性条件". IEEE Transactions on Control of Network Systems. 7 (1): 302–314. arXiv:1806.06083. doi:10.1109/TCNS.2019.2903914. S2CID 73729229.

[31] Cabral, J.; Hugues, E.; Sporns, O.; Deco, G. (2011). "Local network oscillations in resting-state functional connectivity中的作用". NeuroImage. 57 (1): 130–139. doi:10.1016/j.neuroimage.2011.04.010. PMID 21511044. S2CID 13959959.

[32] Hansel, D.; Mato, G.; Meunier, C (1993). "弱耦合Hodgkin-Huxley神经元的相位动力学". Europhysics Letters. 23 (5): 367–372. Bibcode:1993EL.....23..367H. doi:10.1209/0295-5075/23/5/011. S2CID 250884499. {{cite journal}}: Check |s2cid= value (help)

[33] Clusella, Pau; Politi, Antonio; Rosenblum, Michael (2016). "自洽部分同步的最小模型". New Journal of Physics (in English). 18 (9): 093037. arXiv:1607.07178. Bibcode:2016NJPh...18i3037C. doi:10.1088/1367-2630/18/9/093037. ISSN 1367-2630. S2CID 126227664.

[34] Hansel, D.; Mato, G.; Meunier, C (1993). "全球耦合相位振荡器中的聚类和缓慢切换" (PDF). Physical Review E. 48 (5): 3470–3477. Bibcode:1993PhRvE..48.3470H. doi:10.1103/physreve.48.3470. PMID 9961005.

[35] Bick, C.; Timme, M.; Paulikat, D.; Rathlev, D.; Ashwin, P. (2011). "对称相位振荡器网络中的混沌". Physical Review Letters. 107 (24): 244101. arXiv:1105.2230. Bibcode:2011PhRvL.107x4101B. doi:10.1103/PhysRevLett.107.244101. PMID 22243002. S2CID 16144737.

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@@ 第87行： / 第87行： @@
 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==
-[[File:Spatial synchronization patterns in variants of the Kuramoto Model.png|thumb|upright=1.1|right|Distinct synchronization patterns in a two-dimensional array of Kuramoto-like oscillators with differing phase interaction functions and spatial coupling topologies.  (A) Pinwheels. (B) Waves. (C) Chimeras. (D) Chimeras and waves combined. Color scale indicates oscillator phase.]]
+在库拉莫托模型中存在多种类型的变体。有些模型改变了拓扑结构，有些允许使用异构权重，而其他一些变体则与受库拉莫托模型启发但功能形式不同的模型更加相关。
-There are a number of types of variations that can be applied to the original model presented above. Some models change the topological structure, others allow for heterogeneous weights, and other changes are more related to models that are inspired by the Kuramoto model but do not have the same functional form.
-=== 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 ==