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 ($\varphi(t) = e^{βt}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2025). Accordingly, they ask whether synchronization occurs for exponential $\varphi$. The answer depends on the dimension $d$. 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. We give a separate proof of this result. What is the situation on circles ($d=2$)? First, we show that $\varphi$ being increasing and convex is no longer sufficient (even for complete graphs). Then we identify a new condition under which we do have synchronization on the circle (namely, if the Taylor coefficients of $\varphi'$ are decreasing). As a corollary, this provide synchronization for exponential $\varphi$ with $β\in (0, 1]$. The proofs are based on nonconvex landscape analysis.

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，在不同连通性场景下这些点会收敛至共同值（即同步）：这是仓本振子网络经典研究的一部分。当$\varphi$为指数函数（$\varphi(t) = e^{βt}$）时，该动力学对应于理想化Transformer处理数据的极限情况，如Geshkovski等人（2025）所述。因此，他们提出指数型$\varphi$是否会导致同步的问题。答案取决于维度$d$。在多智能体控制的共识问题背景下，Markdahl等人（2018）证明对于$d \geq 3$（球面），若交互图连通且$\varphi$为递增凸函数，则系统达到同步。我们给出了该结论的独立证明。在圆（$d=2$）上的情况如何？首先，我们证明$\varphi$的递增凸性条件不再充分（即使在完全图中）。随后我们提出了保证圆上同步的新条件（即$\varphi'$的泰勒系数递减）。作为推论，这为$β\in (0, 1]$的指数型$\varphi$提供了同步保证。证明基于非凸势景分析。