Basins with tentacles

TL;DR

This paper investigates the complex geometry of basins of attraction in high-dimensional dynamical systems, revealing that they resemble octopus-like structures with most volume in tentacle-like regions, especially in systems like coupled oscillators.

Contribution

It uncovers the geometric structure of basin boundaries in high-dimensional systems, showing they are octopus-like with tentacles containing most of the basin volume.

Findings

Basins are octopus-like with tentacles containing most of the volume.

Basin size scales as e^{-kq^2} with winding number q.

The geometry explains the prevalence of tentacle-like basins in high dimensions.

Abstract

To explore basin geometry in high-dimensional dynamical systems, we consider a ring of identical Kuramoto oscillators. Many attractors coexist in this system; each is a twisted periodic orbit characterized by a winding number $q$, with basin size proportional to $e^{-kq^2}.$ We uncover the geometry behind this size distribution and find the basins are octopus-like, with nearly all their volume in the tentacles, not the head of the octopus (the ball-like region close to the attractor). We present a simple geometrical reason why basins with tentacles should be common in high-dimensional systems.