Dynamical Taxonomy: some taxonomic ranks to systematically classify every chaotic attractor

TL;DR

This paper develops a systematic dynamical taxonomy for classifying chaotic attractors in three- and four-dimensional systems, extending existing hierarchies with topological and Lyapunov-based ranks, demonstrated on R"ossler and Lorenz attractors.

Contribution

It introduces a comprehensive taxonomy framework that combines Lyapunov exponents and topological concepts to classify chaotic regimes in higher-dimensional systems.

Findings

Extended the hierarchy of chaos to include higher topological ranks.

Applied the taxonomy to R"ossler and Lorenz attractors, demonstrating its effectiveness.

Linked the branched manifold description with a linking matrix for multi-component attractors.

Abstract

Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine the properties that two given chaotic invariant sets share or not. The underlying problem is the classification of chaotic regimes, and their labelling. Addressing these problems correspond to the development of a dynamical taxonomy, exhibiting the key properties discriminating the variety of chaotic behaviors discussed in the abundant literature. Starting from the hierarchy of chaos initially proposed by one of us, we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, which cover a large variety of known (and less known) examples of chaotic systems. Starting with the spectrum of Lyapunov exponents as the first taxonomic ranks, we extended the description to higher ranks with some concepts inherited from topology (bounding torus, surface of section, first-return map...). By treating extensively the R\"ossler and the Lorenz attractors, we extended the description of branched manifold -- the highest taxonomic rank for classifying chaotic attractor -- by a linking matrix (or linker) to multi-component attractors (bounded by a torus whose genus g <= 3