A test for fractal boundaries based on the basin entropy

TL;DR

This paper introduces the $S_{bb}$ fractality test, an improved method based on basin entropy for detecting fractal basin boundaries in dynamical systems, surpassing the traditional $ ext{ln} 2$ criterion in sensitivity.

Contribution

The paper develops the $S_{bb}$ fractality test, enhancing the detection of fractal boundaries using basin entropy, applicable to high-dimensional and experimental systems.

Findings

The $S_{bb}$ test detects fractal boundaries more effectively than the $ ext{ln} 2$ criterion.

Application to the driven Duffing oscillator demonstrates the test's improved sensitivity.

The method is suitable for complex, high-dimensional dynamical systems.

Abstract

In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the $\ln 2$ criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the \textit{$S_{bb}$ fractality test}. Using the same single scale information, the $S_{bb}$ fractality test allows for the detection of fractal boundaries in many more cases than the $\ln 2$ criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.