Characterizing accurately chaotic behaviors is not a trivial problem and must allow to determine what are 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 Otto E. Rössler [1], we systematized the description of chaotic regimes observed in three- and four-dimensional spaces, covering 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...) [2]
By treating extensively the Rössler 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. Templex [3] is a promising tool for completing this taxonomy.
[1] O. E. Rössler, The chaotic hierarchy, Zeitschrift für Naturforschung A, 38, 788-801, 1983.
[2] C. Letellier, N. Stankevich & O. E. Rössler, Dynamical Taxonomy : some taxonomic ranks to systematically classify every chaotic attractor, International Journal of Bifurcation & Chaos, 32 (2), 2230004, 2022. ArXiv
[3] G. D. Charó, C. Letellier & D. Sciamarella, Templex : a bridge between homologies and templates for chaotic attractors, Chaos, 32 (8), 083108, 2022. Online