We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) which attract each other according to a function $\varphi$ of their inner products. When $\varphi$ is linear ($\varphi(t) = t$), the points converge to a common value (i.e., synchronize) in various connectivity scenarios: this is part of classical work on Kuramoto oscillator networks. When $\varphi$ is exponential ($\varphi(t) = e^{\beta t}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2024). Accordingly, they ask whether synchronization occurs for exponential $\varphi$. In the context of consensus for multi-agent control, Markdahl et al. (2018) show that for $d \geq 3$ (spheres), if the interaction graph is connected and $\varphi$ is increasing and convex, then the system synchronizes. What is the situation on circles ($d=2$)? First, we show that $\varphi$ being increasing and convex is no longer sufficient. Then we identify a new condition (that the Taylor coefficients of $\varphi'$ are decreasing) under which we do have synchronization on the circle. In so doing, we provide some answers to the open problems posed by Geshkovski et al. (2024).

翻译：我们研究了 $\mathbb{R}^d$（$d \geq 2$）空间中球面上 $n$ 个点的动力学行为，这些点根据其内积的函数 $\varphi$ 相互吸引。当 $\varphi$ 为线性函数（$\varphi(t) = t$）时，在各种连通性场景下，这些点会收敛到一个共同值（即同步）：这是关于仓本振荡器网络经典研究的一部分。当 $\varphi$ 为指数函数（$\varphi(t) = e^{\beta t}$）时，这些动力学对应于理想化 Transformer 处理数据的一种极限情况，如 Geshkovski 等人（2024）所述。因此，他们提出指数型 $\varphi$ 是否会导致同步的问题。在多智能体控制的一致性研究背景下，Markdahl 等人（2018）证明，对于 $d \geq 3$（球面），如果交互图是连通的且 $\varphi$ 是递增凸函数，则系统会同步。那么在圆（$d=2$）上的情况如何？首先，我们证明 $\varphi$ 为递增凸函数不再充分。随后，我们确定了一个新的条件（即 $\varphi'$ 的泰勒系数递减），在此条件下圆上确实可以实现同步。通过这项工作，我们为 Geshkovski 等人（2024）提出的开放性问题提供了一些解答。