Functorial invariants for chaos topology from data
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917431560110080 |
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| author | Sciamarella, Denisse |
| author_facet | Sciamarella, Denisse |
| contents | 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 non-metric 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. The Rössler and Lorenz attractors are presented as paradigmatic examples, followed by the analysis of a climatic simulation and an experimental speech signal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18463 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Functorial invariants for chaos topology from data Sciamarella, Denisse Dynamical Systems Chaotic Dynamics 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 non-metric 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. The Rössler and Lorenz attractors are presented as paradigmatic examples, followed by the analysis of a climatic simulation and an experimental speech signal. |
| title | Functorial invariants for chaos topology from data |
| topic | Dynamical Systems Chaotic Dynamics |
| url | https://arxiv.org/abs/2602.18463 |