Synchronization on circles and spheres with nonlinear interactions

Abstract

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. (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 $\beta \in (0, 1]$. The proofs are based on nonconvex landscape analysis.