Synchronization on Circles and Spheres with Nonlinear Interactions

Abstract. We consider the dynamics of [Formula: see text] points on a sphere in [Formula: see text] ([Formula: see text]) which attract each other according to a function [Formula: see text] of their inner products. When [Formula: see text] is linear ([Formula: see text]), 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 [Formula: see text] is exponential ([Formula: see text]), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. [ Bull. Amer. Math. Soc., 62 (2025), pp. 427–479]. Accordingly, they ask whether synchronization occurs for exponential [Formula: see text]. The answer depends on the dimension [Formula: see text]. In the context of consensus for multiagent control, Markdahl et al. [ IEEE Trans. Automat. Control, 63 (2018), pp. 1664–1675] show that for [Formula: see text] (spheres), if the interaction graph is connected and [Formula: see text] is increasing and convex, then the system synchronizes. We give a separate proof of this result. What is the situation on circles ([Formula: see text])? First, we show that [Formula: see text] 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 [Formula: see text] are decreasing). As a corollary, this provides synchronization for exponential [Formula: see text] with [Formula: see text]. The proofs are based on nonconvex landscape analysis.