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 φ of their inner products. When φ is linear (φ(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 φ is exponential (φ(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 φ. The answer depends on the dimension d. In the context of consensus for multi-agent 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 φ 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 φ being increasing and convex is no longer sufficient (even for complete graphs). Then we identify a new condition under which synchronization does occur on the circle (namely, if the Taylor coefficients of φ′ are decreasing). As a corollary, this provides synchronization for exponential φ with β ∈ (0, 1]. The proofs are based on nonconvex landscape analysis.