Analysis of luminosity measurements of the pre-white dwarf PG 1159-035

TL;DR

This paper analyzes luminosity data of the pre-white dwarf PG 1159-035, revealing its underlying non-linear physical processes through advanced time series analysis including embedding and Lyapunov exponent estimation.

Contribution

It introduces a novel analysis approach for the luminosity data, confirming the optimal embedding dimension and estimating the Lyapunov exponent to demonstrate non-linearity.

Findings

Optimal embedding dimension m0=10 confirmed by Cao's method

Lyapunov exponent λ ≈ 0.092 indicating non-linear dynamics

Results consistent across different distance norms

Abstract

The study of the luminosity measurements of the pre-white dwarf PG 1159-035 has established the properties of the rich power spectrum of the detected radiation and, derived thereof, the physical properties of this celestial body. Those of the measurements which are available online are analysed in this work from a different perspective. After the measurements were band-passed, they were split into two parts (of comparable sizes), one yielding the training (learning) set (i.e., the database of embedding vectors and associated predictions), the other the test set. The optimal embedding dimension $m_0=10$ was obtained using Cao's method; this result was confirmed by an analysis of the correlation dimension. Subsequently, the extraction of the maximal Lyapunov exponent $\lambda$ was pursued for embedding dimensions $m$ between $3$ and $12$; results were obtained after removing the prominent undulations of the out-of-sample prediction-error arrays $S (k)$ by fitting a monotonic function to the data. The grand mean of the values, obtained for sufficient embedding dimensions ($10 \leq m \leq 12$), was: $\lambda = (9.2 \pm 1.0 ({\rm stat.}) \pm 2.7 ({\rm syst.})) \cdot 10^{-2}~\Delta \tau^{-1}$, where $\Delta \tau=10$ s is the sampling interval in the measurements. On the basis of this significantly non-zero result, it may be concluded that the physical processes, underlying the variation of the luminosity of PG 1159-035, are non-linear. The aforementioned result for $\lambda$ was obtained using the $L^\infty$-norm distance; a larger, yet not incompatible, result was extracted with the Euclidean ($L^2$-norm) distance.