Synchronization on circles and spheres with nonlinear interactions

TL;DR

This paper investigates the conditions under which points on spheres and circles synchronize when attracted by nonlinear functions of their inner products, revealing new criteria for synchronization especially on circles with exponential interactions.

Contribution

It provides new synchronization conditions on circles, especially for exponential functions, and offers a separate proof for known results on higher-dimensional spheres.

Findings

Synchronization occurs on spheres for convex, increasing interactions when dimension is at least 3.

On circles, convexity and monotonicity are insufficient for synchronization.

Exponential interactions synchronize on circles if the parameter is in (0, 1].

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.