发表机构
Starling Research Institute（星形研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究揭示N个脉冲耦合振荡器的同步结果由N的素因子分解控制，同步概率有算术结构，为奇异盆几何新增算术盆类型。
AI 中文摘要
N个相同的脉冲耦合振荡器群体最终会收敛到两种结果之一：完全同步或共存的同步簇状态。我们证明，出现哪种结果由N的素因子分解控制。在临界充电曲线（线性，同步与聚类区域的边界）处，同步盆呈现精确的算术结构。对所有N，同步概率为$\u03a8_{sync}=A_{N,1}/N^N$，其中$A_{N,1}$满足精确递推关系。对于素数N，$A_{N,1}=N^N-1$，给出闭式$\u03a8_{sync}=1-1/N^N$；对于合数N，观测到的渐近标度为$1-\u03a8_{sync}\u223c C_m N^{-(m-1)}$，其中m是最小素因子。该结果为奇异盆几何图集增添了新成员：除了分形、 riddled和触手状盆，现在还有一个算术盆。
英文摘要
A population of $N$ identical pulse-coupled oscillators ultimately settles into one of two outcomes: full synchrony or a state of co-existing synchronized clusters. We show that which outcome occurs is controlled by the prime factorization of $N$. At the critical charging curve --- linear, the boundary between the synchronizing and clustering regimes --- the synchronization basin acquires exact arithmetic structure. The synchronization probability is $\Psync=A_{N,1}/N^N$ for all $N$, where $A_{N,1}$ satisfies an exact recurrence relation. For prime $N$, $A_{N,1}=N^N-1$ giving the closed form $\Psync=1-1/N^N$; for composite $N$, the observed asymptotic scaling is $1-\Psync\sim C_m N^{-(m-1)}$, where $m$ is the smallest prime divisor. The result adds a new member to the atlas of exotic basin geometries: alongside fractal, riddled, and tentacled basins, we now have a basin that is arithmetic.
CommentsAdded explicit K=1 recurrence after eq. (3) per referee optional remark; Supplementary Material updated; fixed truncated title
