Functorial invariants for chaos topology from data

TL;DR

This paper introduces a category-theoretic framework for topological invariants of chaotic systems, enabling robust analysis of tipping points and physical mechanisms from finite data.

Contribution

It develops functorial invariants using directed path algebras and semigroup structures, providing a new approach to chaos topology analysis.

Findings

Invariants distinguish chaos from finite data.

Framework applied to R"ossler, Lorenz, climate, and speech data.

Semigroup structure explains topological modes of variability.

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 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 R\"ossler and Lorenz attractors are presented as paradigmatic examples, followed by the analysis of a climatic simulation and an experimental speech signal.