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Dynamical Taxonomy: Some Taxonomic Ranks to Systematically Classify Every Chaotic Attractor

Rouen Normandie University — CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France

Laboratory of Topological Methods in Dynamics, National Research University Higher School of Economics, Bolshaya Pecherskaya Str., 25/12, Nizhny Novgorod 603155, Russia

E-mail Address: [email protected]

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Otto E. Rössler

Division of Theoretical Chemistry, University of Tübingen, D-72076 Tübingen, Germany

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https://doi.org/10.1142/S021812742230004XCited by:10 (Source: Crossref)

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 labeling. Addressing these problems corresponds 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 chaos. 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 Rössler and the Lorenz attractors, we extended the description of branched manifold — the highest known taxonomic rank for classifying chaotic attractor — by a linking matrix (or linker) to multicomponent attractors (bounded by a torus whose genus $g \geq 3$ ).

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