# The Size of the Sync Basin Revisited

## TL;DR

This paper introduces an efficient numerical method to estimate the size of basins of attraction in high-dimensional dynamical systems, and applies it to the Kuramoto model, revealing new scaling laws for basin volumes.

## Contribution

The authors develop a novel algorithm for accurately estimating basin volumes and demonstrate its effectiveness on complex oscillator models, revisiting previous assumptions about basin size distributions.

## Key findings

- Basin volumes scale as (1-4q/n)^n, differing from previous Gaussian assumptions.
- The method successfully applies to models with diverse natural frequencies and network structures.
- Revisits and refines understanding of basin size distributions in the Kuramoto model.

## Abstract

In dynamical systems, the full stability of fixed point solutions is determined by their basin of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006), P. J. Menck {\it et al.} Nat. Phys. {\bf 9}, 89 (2013)]. Here we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006)] that inspired the title of the present manuscript, and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number $q$ and the number $n$ of oscillators. We find that the basin volumes scale as $(1-4q/n)^n$, contrasting with the Gaussian behavior postulated in Wiley et al.'s paper. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.