2022/08/01 by Gisela D. Charó, Christophe Letellier, Denisse Sciamarella · 1 voice · 27 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Artificial intelligence #Attractor #Chaotic #Combinatorics #Computer science #Graph #Homology (biology) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Slime Mold and Myxomycetes Research #State space #Template #Theoretical computer science #Topological and Geometric Data Analysis #Topology (electrical circuits) #nlin.CD
paper · pdf · doi:10.1063/5.0092933
published in Chaos An Interdisciplinary Journal of Nonlinear Science 32(8), 083108 (American Institute of Physics)
openalex publication_date 2022/08/01 · openalex created_date 2025/10/10 · arxiv published 2026/02/02 · arxiv updated 2026/02/02 · openalex updated_date 2026/07/28
The theory of homologies introduces cell complexes to provide an algebraic description of spaces up to topological equivalence. Attractors in state space can be studied using Branched Manifold Analysis through Homologies: this strategy constructs a cell complex from a cloud of points in state space and uses homology groups to characterize its topology. The approach, however, does not consider the action of the flow on the cell complex. The procedure is here extended to take this fundamental property into account, as done with templates. The goal is achieved endowing the cell complex with a directed graph that prescribes the flow direction between its highest-dimensional cells. The tandem of cell complex and directed graph, baptized templex, is shown to allow for a sophisticated characterization of chaotic attractors and for an accurate classification of them. The cases of a few well-known chaotic attractors are investigated-namely, the spiral and funnel Rössler attractors, the Lorenz attractor, the Burke and Shaw attractor, and a four-dimensional system. A link is established with their description in terms of templates.