The Tentacles Landscape

TL;DR

This paper rigorously proves the octopus-like basin of attraction structure in high-dimensional Kuramoto models, confirming previous conjectures based on simulations.

Contribution

It provides a rigorous mathematical proof of the octopus-shaped basins of attraction in a class of high-dimensional oscillator models.

Findings

Confirmed the octopus-like basin structure in the Kuramoto ring model.

Showed that basin volume scales as e^{-kq^2} in the winding number q.

Validated the conjecture that such geometry is common in high dimensions.

Abstract

Zhang and Strogatz [Phys. Rev. Lett. 127, 194101 (2021)] used high-dimensional simulations to argue that basins of attraction in the Kuramoto ring are octopus-like: their volume scales as $e^{-kq^2}$ in the winding number $q$, nearly all of it concentrated in filamentary tentacles rather than near the attractor. They conjecture this geometry to be common in high dimensions but note that high-dimensional simulations are unreliable. We prove every feature of the octopus picture rigorously for identical oscillators on a ring coupled by any smooth odd function strictly increasing on $(-\pi,\pi)$.