Data and graphics from: "Three invariants of strange attractors derived through hypergeometric entropy"

This package contains data and graphics related to the publication:

Keisuke Okamura, "Three invariants of chaotic attractors derived through hypergeometric entropy", Chaos, Solitons & Fractals 170 (2023) 113392; https://doi.org/10.1016/j.chaos.2023.113392

Abstract of the original article

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 Rényi-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.

Data files

The following files are provided (see the original article for the notation):

• The comma-separated values (CSV) file named 'est_sum_all.csv' summarises the results of the correlation dimension (\(\mathcal{D}\)), correlation entropy (\(\mathcal{K}\)) and correlation concentration (\(\mathcal{A}\)) estimates for two discrete (logistic and Hénon maps) and four continuous (Lorenz, Rössler, Duffing-Ueda and Langford attractors) chaotic systems in tabular form.
• The six CSV files whose file names begin with 'H_' record the extended Rényi entropy values calculated for each \(\sigma\) value (in increments of 0.1) for the chaotic system whose name follows the prefix, for each embedding dimension (\(m\)). The column 'mN' shows the results for the embedding dimension \(N\).
• The six PDF files whose file names begin with 'gr_' illustrate, for the named chaotic systems following their prefixes, their i) appearance (left), ii) graph of the extended Rényi entropy as a function of \(\sigma\) (centre) and iii) the chaotic invariants (\(\mathcal{D}, \mathcal{K}, \mathcal{A}\)) estimation results (right).