Unpredictability, Uncertainty and Fractal Structures in Physics

TL;DR

This paper discusses how fractal structures and basin entropy in chaotic and multi-stable systems influence predictability in physics, highlighting new quantitative measures for unpredictability.

Contribution

It introduces the concept of basin entropy as a novel quantitative measure of unpredictability in systems with fractal basin boundaries.

Findings

Fractal and Wada basin boundaries complicate prediction.

Small initial uncertainties lead to unpredictable outcomes.

Basin entropy quantifies unpredictability in physical systems.

Abstract

In Physics, we have laws that determine the time evolution of a given physical system, depending on its parameters and its initial conditions. When we have multi-stable systems, many attractors coexist so that their basins of attraction might possess fractal or even Wada boundaries in such a way that the prediction becomes more complicated depending on the initial conditions. Chaotic systems typically present fractal basins in phase space. A small uncertainty in the initial conditions gives rise to a certain unpredictability of the final state behavior. The new notion of basin entropy provides a new quantitative way to measure the unpredictability of the final states in basins of attraction. Simple methods from chaos theory can contribute to a better understanding of fundamental questions in physics as well as other scientific disciplines.