Functorial invariants for chaos topology from data

TL;DR

This work extends chaos topology by embedding BraMAH cell complexes into a directed-topology framework, yielding a functorial description of chaotic dynamics. By pairing a BraMAH complex with a flow-directed digraph, the templex supports two coupled invariant structures: classical homology and generatex semigroups, connected via the Poincaré-edge operator. The generatex quotient captures causal equivalence classes of directed cycles, enabling a category-theoretic treatment with forgetful functors and pushouts that model how trajectories transition through joining loci. Illustrative applications to a speech signal and a wind-driven double-gyre demonstrate that directed invariants reveal finite-time chaos and topological tipping in nonautonomous settings, expanding the toolkit for data-driven chaos analysis. The framework clarifies the coexistence of time and topological structure and provides a rigorous, operational pathway from data to invariant chaos descriptors.

Abstract

The templex is a recently introduced topological object bridging homologies and templates for chaotic attractors: its cell complex encodes the directionless properties of the attractor's branched manifold in phase space, and its directed graph captures the flow-compatible paths starting and ending in joining loci. Algebraic topology is deeply connected to category theory because it studies spaces by translating them into algebraic objects through structure-preserving mappings. The homology functor translates structural properties into a set of layered invariants called homology groups. The templex is shown here to play the same role for directed spaces that cell complexes play for spaces. The directed properties of a templex are found therewith to admit a functorial formulation. This formulation provides a rigorous foundation for a theory of chaos topology developed so far algorithmically, and establishes operationally a topological criterion for finite-time chaos. A climatic simulation and an experimental speech signal are analyzed as illustrative applications.

Figures (3)

• Figure 1: The time series of the pressure fluctuation values (in arbitrary units) as a function of their position in the data file. (a) The entire time series for $t = 0.147~\text{sec}$. (b) The three--dimensional time--delay embedding of the time series, with $\tau=5$.
• Figure 2: Forgetful functor analysis of the time series in Fig. \ref{['data']}. (a) Point cloud segmented into the $2$-cells of the BraMAH complex (faces are omitted for clarity). (b) Directed multigraph obtained as the union of the three generatexes $G_1$ in green, $G_2$ in blue, and $G_3$ in red; parallel edges indicate bonds — the triple one $B_{123}$ and the double one $B_{12}$. Removing parallel edges yields the digraph of the templex. (c) Generatex visitation sequence extracted from the time series [$\mu$sec].
• Figure 3: (a) Directed multigraph obtained as the union of the six generatexes in red, blue, green, magenta, orange and gray for the templex of the aperiodically forced wind--driven double gyre example. (b) Generatex visiting sequence extracted from the time series (in years) of an individual solution, shown in Fig. 15 of Ref. charo2025topological.