更改

<math>r \approx \sqrt{\frac{16}{\pi K_{\mathrm{c}}^3}} \sqrt{\frac{\mu}{-g^{\prime \prime}(0)}}。</math><ref>{{Cite book |last=Strogatz |first=Steven H. |title=Frontiers in Mathematical Biology |chapter=Norbert Wiener's Brain Waves |date=1994 |editor-last=Levin |editor-first=Simon A. |chapter-url=https://link.springer.com/chapter/10.1007/978-3-642-50124-1_7 |series=Lecture Notes in Biomathematics |volume=100 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=122–138 |doi=10.1007/978-3-642-50124-1_7 |isbn=978-3-642-50124-1}}</ref><ref name=Strogatz-2000/>

<math>r \approx \sqrt{\frac{16}{\pi K_{\mathrm{c}}^3}} \sqrt{\frac{\mu}{-g^{\prime \prime}(0)}}。</math><ref>{{Cite book |last=Strogatz |first=Steven H. |title=Frontiers in Mathematical Biology |chapter=Norbert Wiener's Brain Waves |date=1994 |editor-last=Levin |editor-first=Simon A. |chapter-url=https://link.springer.com/chapter/10.1007/978-3-642-50124-1_7 |series=Lecture Notes in Biomathematics |volume=100 |language=en |location=Berlin, Heidelberg |publisher=Springer |pages=122–138 |doi=10.1007/978-3-642-50124-1_7 |isbn=978-3-642-50124-1}}</ref><ref name=Strogatz-2000/>

== Small ''N'' cases ==

== 小''N''情况 ==

When N is small, the solutions given above breaks down, as the continuum approximation cannot be used.

当''N''较小时，上述解失效，因为无法使用连续近似。

The N=2 case is trivial. In the rotating frame <math>\omega_1 = -\omega_2</math>, and so the system is described exactly by the angle between the two oscillators: <math>\Delta \theta = \theta_1 - \theta_2</math>. When <math>K < K_c = 2|\omega_1|</math>, the angle cycles around the circle (that is, the fast oscillator keeps lapping around the slow oscillator). When <math>K > K_c</math>, the angle falls into a stable attractor (that is, the two oscillators lock in phase). Similarly, the state space of the N=3 case is a 2-dimensional torus, and so the system evolves as a flow on the 2-torus, which cannot be chaotic.

对于''N''=2的情况是平凡的。在旋转框架中，<math>\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>\Delta \theta = \theta_1 - \theta_2</math>给出。当<math>K < 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>K > 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。

Chaos first occurs when N=4. For some settings of <math>\omega_1, \omega_2, \omega_3, K</math>, the system has a [[strange attractor]].<ref>{{Cite journal |last1=Maistrenko |first1=Yuri L. |last2=Popovych |first2=Oleksandr V. |last3=Tass |first3=Peter A. |title=Chaotic Attractor in the Kuramoto Model |date=November 2005 |url=https://www.worldscientific.com/doi/abs/10.1142/S0218127405014155 |journal=International Journal of Bifurcation and Chaos |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 ==

== Connection to Hamiltonian systems ==