Robust Wada Boundaries and Entropy Scaling in pp-Wave Spacetimes

TL;DR

This paper analyzes geodesic dynamics in pp-wave spacetimes with polynomial profiles, showing that basin boundaries are fractal and that unpredictability increases with polynomial degree, confirmed by basin entropy measures.

Contribution

It demonstrates the robustness of Wada basin boundaries under polynomial degree variation and quantifies unpredictability using basin entropy metrics.

Findings

Wada property of escape basins is robust across polynomial degrees.

Basin entropy measures increase monotonically with polynomial degree.

Boundary basin entropy exceeds ln(2) for degree > 3, indicating fractal boundaries.

Abstract

We study the dynamics of the geodesics of pp-wave spacetimes with polynomial profiles, which are dynamically equivalent to the motion of a classical particle in a two-dimensional harmonic polynomial potential. We demonstrate that the Wada property of the escape basins is robust under variation of the polynomial degree, i.e., the basin boundaries remain maximally intermingled as the number of escape channels increases. We further provide a quantitative characterization of the degree of dynamical uncertainty by computing the basin entropy $S_{b}$ and the boundary basin entropy $S_{bb}$. We find that these measures increase monotonically with the polynomial degree, indicating enhanced unpredictability of the final state of the system. We also show that $S_{bb}$ is greater than $\ln(2)$ for $n>3$, and this confirms that the basin boundaries are fractal.