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第48行： − + − The [[Coherence (physics)|incoherent]] state with all oscillators drifting randomly corresponds to the solution <math>\rho = 1/(2\pi)</math>. In that case <math>r = 0</math>, and there is no coherence among the oscillators. They are uniformly distributed across all possible phases, and the population is in a statistical [[steady-state]] (although individual oscillators continue to change phase in accordance with their intrinsic ''ω'').+ − When coupling ''K'' is sufficiently strong, a fully synchronized solution is possible. In the fully synchronized state, all the oscillators share a common frequency, although their phases can be different.+ − A solution for the case of partial synchronization yields a state in which only some oscillators (those near the ensemble's mean natural frequency) synchronize; other oscillators drift incoherently. Mathematically, the state has+ − :<math>\rho = \delta\left(\theta - \psi - \arcsin\left(\frac{\omega}{K r}\right)\right)</math> + − for locked oscillators, and − :<math>\rho = \frac{\rm{normalization \; constant}}{(\omega - K r \sin(\theta - \psi))}</math> − for drifting oscillators. The cutoff occurs when <math>|\omega| < K r </math>. − When <math>g</math> is unimodal and symmetric, then a stable state solution for the system is <math display="block">r=rK \int_{-\pi / 2}^{\pi / 2} \cos ^2 \theta g(K r \sin \theta) d \theta</math>As coupling increases, there is a critical value <math>K_c = 2/\pi g(0)</math> such that when <math>K < K_c</math>, the long-term average of <math>r=0</math>, but when <math>K= K_c(1+\mu)</math>, where <math>\mu > 0</math> is small, then <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/>+ + + + + + + + +
→Solutions for the large N limit
</math>
</math>
== Solutions for the large ''N'' limit ==
== 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>\rho = 1/(2\pi)</math>。此时<math>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>\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>\rho = \frac{\text{归一化常数}}{\omega - K r \sin(\theta - \psi)}。</math>
截止条件为<math>|\omega| < 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 display="block">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>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>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 ==
== Small ''N'' cases ==