Synchronization on circles and spheres with nonlinear interactions
<|reference_start|>Synchronization on circles and spheres with nonlinear interactions: We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) 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 ($\varphi(t) = e^{\beta t}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2024). Accordingly, they ask whether synchronization occurs for exponential $\varphi$.   In the context of consensus for multi-agent control, Markdahl et al. (2018) show that for $d \geq 3$ (spheres), if the interaction graph is connected and $\varphi$ is increasing and convex, then the system synchronizes. What is the situation on circles ($d=2$)? First, we show that $\varphi$ being increasing and convex is no longer sufficient. Then we identify a new condition (that the Taylor coefficients of $\varphi'$ are decreasing) under which we do have synchronization on the circle. In so doing, we provide some answers to the open problems posed by Geshkovski et al. (2024).<|reference_end|>
@article{criscitiello2024synchronization,
    title={Synchronization on circles and spheres with nonlinear interactions},
    author={Christopher Criscitiello, Quentin Rebjock, Andrew D. McRae, Nicolas
  Boumal},
    journal={arXiv preprint arXiv:2405.18273},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.18273},
    primaryClass={math.OC cs.LG math.DS}
}