Potentials and Limits to Basin Stability Estimation

TL;DR

This paper evaluates the effectiveness and limitations of basin stability estimation in dynamical systems, especially those with complex basin geometries like fractal, riddled, or intermingled basins, using Monte Carlo methods.

Contribution

It analyzes the applicability of basin stability estimation to systems with challenging basin geometries, highlighting its potential and limitations.

Findings

Numerical basin stability is meaningful for fractal basin boundaries.

Estimation reaches limits for riddled basins with holes.

Monte Carlo sampling can be effective within certain geometric constraints.

Abstract

Stability assessment methods for dynamical systems have recently been complemented by basin stability and derived measures, i.e. probabilistic statements whether systems remain in a basin of attraction given a distribution of perturbations. This requires numerical estimation via Monte-Carlo sampling and integration of differential equations. Here, we analyze the applicability of basin stability to systems with basin geometries challenging for this numerical method, having fractal basin boundaries and riddled or intermingled basins of attraction. We find that numerical basin stability estimation is still meaningful for fractal boundaries but reaches its limits for riddled basins with holes.