Three invariants of strange attractors derived through hypergeometric entropy

TL;DR

This paper introduces a new invariant called correlation concentration for strange attractors, along with correlation dimension and entropy, providing a comprehensive framework to analyze their scaling behaviors using hypergeometric entropy.

Contribution

The study presents the correlation concentration as a novel invariant and develops a unified method to estimate all three invariants simultaneously through hypergeometric entropy.

Findings

Correlation concentration exhibits a new scaling behavior in the macroscopic limit.

All three invariants can be estimated simultaneously via nonlinear regression.

The method is validated on both discrete and continuous systems.

Abstract

A new description of strange attractor systems through three geometrical and dynamical invariants is provided. They are the correlation dimension ($\mathcal{D}$) and the correlation entropy ($\mathcal{K}$), both having attracted attention over the past decades, and a new invariant called the correlation concentration ($\mathcal{A}$) introduced in the present study. The correlation concentration is defined as the normalised mean distance between the reconstruction vectors, evaluated by the underlying probability measure on the infinite-dimensional embedding space. These three invariants determine the scaling behaviour of the system's R\'{e}nyi-type extended entropy, modelled by Kummer's confluent hypergeometric function, with respect to the gauge parameter ($\rho$) coupled to the distance between the reconstruction vectors. The entropy function reproduces the known scaling behaviours of $\mathcal{D}$ and $\mathcal{K}$ in the 'microscopic' limit $\rho\to\infty$ while exhibiting a new scaling behaviour of $\mathcal{A}$ in the other, 'macroscopic' limit $\rho\to 0$. The three invariants are estimated simultaneously via nonlinear regression analysis without needing separate estimations for each invariant. The proposed method is verified through simulations in both discrete and continuous systems.