Synchronization on Circles and Spheres with Nonlinear Interactions

We consider the dynamics of n points on a sphere in ℝ^d (d ≥ 2) which attract each other according to a function ρ of their inner products. When ρ is linear (ρ(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 ρ is exponential (ρ(t) = e^βt ), 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 ρ. The answer depends on the dimension d. In the context of consensus for multiagent control, Markdahl et al. [IEEE Trans. Automat. Control, 63 (2018), pp. 1664-1675] show that for d ≥ 3 (spheres), if the interaction graph is connected and ρ 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 ρ 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 ρ^' are decreasing). As a corollary, this provides synchronization for exponential ρ with β ∈ (0, 1]. The proofs are based on nonconvex landscape analysis.