Synchronization on circles and spheres with nonlinear interactions

• DOI: 10.48550/arxiv.2405.18273
• Publication Date: 2024-05-28
• Authors: Christopher Crisc...; Quentin Rebjock; Andrew D. McRae; Nicolas Boumal
• Keywords: Separate spheres

We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) which attract each other according to function $\varphi$ their inner products. When is linear ($\varphi(t) = t$), converge common value (i.e., synchronize) various connectivity scenarios: this part classical work Kuramoto oscillator networks. exponential e^{\beta t}$), these correspond limit how idealized transformers process data, as described by Geshkovski et al. (2024). Accordingly, they ask whether synchronization occurs for $\varphi$. In context consensus multi-agent control, Markdahl (2018) show that $d 3$ (spheres), if interaction graph connected and increasing convex, then system synchronizes. What situation circles ($d=2$)? First, we being convex no longer sufficient. Then identify new condition (that Taylor coefficients $\varphi'$ are decreasing) under do have circle. so doing, provide some answers open problems posed