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

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 log2, the basin is fractal.

Chaotic dynamics and fractal structures in experiments with cold atoms

“…Some escape basins are not only fractal, but also posses the stronger property of Wada [8,10]. This means, that all the basins have a common boundary separating them.…”

“…In this respect, it has already been shown [8] for different launching distances x 0 , so that the atoms take a time t to reach the scattering center in x = 0. In our regime of parameters, the basin entropy saturates for a time t ≈ 500 regardless of the initial horizontal speed v x0 .…”

“…The idea of basin entropy [8] was introduced in order to answer a simple question: how is it possible to affirm that one basin is more unpredictable than another? To this aim, a new quantitative measure was defined.…”

“…The method presented here and based on the concept of basin entropy circumvents partially this difficulty [8] by defining an indicator at a given scale which detects faithfully fractality in most cases. We propose to use this concept to characterize the phase space fractal properties in a scattering problem.…”

“…One of these powerful tools is the basin entropy, introduced in [8]. The framework of the basin entropy allows to quantify the unpredictability associated to the different outcomes in a dynamical system.…”