Basin entropy: a new tool to analyze uncertainty in dynamical systems

TL;DR

The paper introduces basin entropy, a new measure to quantify unpredictability in dynamical systems' basins of attraction, aiding in understanding and classifying uncertainty across various scientific fields.

Contribution

It presents basin entropy as a novel tool for analyzing uncertainty in dynamical systems and offers a criterion for identifying fractal basin boundaries.

Findings

Basin entropy effectively quantifies unpredictability.

Higher basin entropy indicates fractal basin boundaries.

Application examples demonstrate its utility across disciplines.

Abstract

In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than $\log 2 $, the basin is fractal.