Kuramoto Problem and Smale Institute
summary
Kuramoto with five oscillators reduces to a negative‑gradient flow on a 4‑torus and the low‑energy critical set breaks into finitely many saddles organized by 10 templates (and five 1‑D cycle boundaries); the top level Vmax is a two‑dimensional invariant manifold made from 24 pentagons (a genus‑4 surface) obtained by implicit‑function/Morse analysis and combinatorics, is normally hyperbolic so persists under perturbations, and is visualized via a 3D printed model and Poincaré‑plane tiling.
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