Structure of basins in high-dimensional Kuramoto ring oscillators

Determine the geometric and topological structure of the basins of attraction for q‑twisted states in the n‑node ring of identical Kuramoto oscillators with periodic boundary conditions governed by dot θ_i = sin(θ_{i+1} − θ_i) + sin(θ_{i−1} − θ_i), including the organization of basin volume into "tentacle" versus central regions, and characterize the boundary properties and their dependence on dimension and phase‑space slice selection.

Background

The paper examines basins of attraction in a high‑dimensional ring of identical Kuramoto oscillators, where q‑twisted states serve as attractors and exhibit an octopus‑like basin geometry with most volume concentrated in tentacles rather than the central head. Although basin size scaling with q is reported, the overall geometric and boundary structure across dimensions and slices is not fully characterized.

The authors note that intuition about topology breaks down beyond four dimensions, complicating visualization and analysis. They highlight the need for specialized sampling techniques to gain a clearer picture of the high‑dimensional phase space, motivating a precise paper of basin geometry and boundary properties.