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Functorial invariants for chaos topology from data
1Institut Franco-Argentin d'Études sur le Climat et ses Impacts, CONICET, CNRS-Centre National de la Recherche Scientifique, 75016 Paris, France; , -Universidad de Buenos Aires, Centro de Investigaciones del Mar y la Atmósfera (CIMA), C1428EGA Buenos Aires, Argentina; and , (CNRS-IRD-CONICET-UBA) (IRL 3351 IFAECI), C1428EGA Ciudad Autónoma de Buenos Aires, Argentina.
Abstract:
The templex is a topological object bridging homologies and templates for chaotic dynamics. This article places the templex within category theory, introducing a directed path algebra, an edge operator on directed paths, and an equivalence relation for directed cycles that is distinct from directed homologies. The resulting functorial invariants are of two kinds: Abelian-group invariants, namely the homology groups, and semigroup invariants, namely the generatex semigroups. These invariants are separable through forgetful functors and constitute a robust framework for identifying tipping points, disambiguating physical mechanisms, and benchmarking data-driven models against observations or simulations. The formulation sets forth a nonmetric criterion for chaos from finite-time data and reveals that the concatenable nature of topological modes of variability is a direct consequence of the semigroup structure of the directed path algebra. Two applications are presented: an experimental speech signal and a climatic numerical simulation.
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