更改

混沌最早出现在''N''=4的情况下。在某些<math>\omega_1, \omega_2, \omega_3, K</math>的设置下，系统存在[[奇异吸引子]]<ref>{{Cite journal |last1=Maistrenko |first1=Yuri L. |last2=Popovych |first2=Oleksandr V. |last3=Tass |first3=Peter A. |title=Kuramoto模型的混沌吸引子 |date=November 2005 |url=https://www.worldscientific.com/doi/abs/10.1142/S0218127405014155 |journal=国际双困杂志 |language=en |volume=15 |issue=11 |pages=3457–3466 |doi=10.1142/S0218127405014155 |bibcode=2005IJBC...15.3457M |issn=0218-1274}}</ref>。

混沌最早出现在''N''=4的情况下。在某些<math>\omega_1, \omega_2, \omega_3, K</math>的设置下，系统存在[[奇异吸引子]]<ref>{{Cite journal |last1=Maistrenko |first1=Yuri L. |last2=Popovych |first2=Oleksandr V. |last3=Tass |first3=Peter A. |title=Kuramoto模型的混沌吸引子 |date=November 2005 |url=https://www.worldscientific.com/doi/abs/10.1142/S0218127405014155 |journal=国际双困杂志 |language=en |volume=15 |issue=11 |pages=3457–3466 |doi=10.1142/S0218127405014155 |bibcode=2005IJBC...15.3457M |issn=0218-1274}}</ref>。

== Connection to Hamiltonian systems ==

== 与哈密顿系统的联系 ==

The dissipative Kuramoto model is contained<ref>{{cite journal | last1 = Witthaut | first1 = Dirk | last2 = Timme | first2 = Marc | title = Kuramoto Dynamics in Hamiltonian Systems | journal = Phys. Rev. E | volume=90 | issue = 3 | pages = 032917 | doi = 10.1103/PhysRevE.90.032917| pmid = 25314514 | year = 2014|arxiv = 1305.1742 |bibcode = 2014PhRvE..90c2917W | s2cid = 7510614 }}</ref> in certain conservative [[Hamiltonian mechanics|Hamiltonian systems]] with [[Hamilton function|Hamiltonian]] of the form

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：<ref>{{cite journal | last1 = Witthaut | first1 = Dirk | last2 = Timme | first2 = Marc | title = 哈密顿系统中的库拉莫托动力学 | journal = 物理评论E | volume=90 | issue = 3 | pages = 032917 | doi = 10.1103/PhysRevE.90.032917| pmid = 25314514 | year = 2014|arxiv = 1305.1742 |bibcode = 2014PhRvE..90c2917W | s2cid = 7510614 }}</ref>

:<math> \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}

:<math> \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>

      (q_i p_j - q_j p_i) ( q_j^2 + p_j^2 - q_i^2 - p_i^2 ) </math> 

              \sqrt{I_j I_i} (I_j  - I_i) \sin(\phi_j - \phi_i), </math>

Hamilton's equation of motion become

:<math>

:<math> \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>

    \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>

:<math>   \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>

:<math>

:<math> \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>

\frac{d \phi_i}{dt} = \frac{\partial \mathcal{H}'}{\partial I_i}

 

  = \omega_i + \frac{K}{N}  

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]]。

         \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>

So the manifold with <math> I_j = I </math> is invariant because <math> \frac{d I_i}{dt} = 0 </math> and the phase dynamics <math> \frac{d \phi_i}{dt} </math> becomes the dynamics of the Kuramoto model (with the same coupling constants for <math> I = 1/2 </math>). The class of Hamiltonian systems characterizes certain quantum-classical systems including [[Bose–Einstein condensate]]s.

== Variations of the models ==

== Variations of the models ==