Synchronization on circles and spheres with nonlinear interactions

We consider the dynamics of n points on a sphere in ℝd (d ≥ 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) = eβ 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 ≥ 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 ∈ (0, 1]. The proofs are based on nonconvex landscape analysis.