Templex-based dynamical units for a taxonomy of chaos

TL;DR

Templex provides a dimension-free topological framework that unifies the attractor's geometric structure (via a BraMAH complex) with the flow dynamics (via a directed graph). A reduction procedure merges cells while preserving homology and torsion, revealing two elementary dynamical units—O-units (oscillators) and S-units (switches)—connected by bonds that encode the skeleton of chaotic motion. Through detailed analyses of Lorenz, spiral and funnel Rössler, Burke–Shaw, a four-dimensional attractor, and Deng’s toroidal chaos, the authors demonstrate that chaos can be synthesized from simple unit interactions and mapped to first-return structures without relying on three-dimensional templates. The templex approach advances chaos taxonomy by offering a scalable, data-agnostic pathway to classify and compare chaotic attractors, with promising applications to experiments and higher-dimensional systems.

Abstract

Discriminating different types of chaos is still a very challenging topic, even for dissipative three-dimensional systems for which the most advanced tool is the template. Nevertheless, getting a template is, by definition, limited to three-dimensional objects, since based on knot theory. To deal with higher-dimensional chaos, we recently introduced the templex combining a flow-oriented {\sc BraMAH} cell complex and a directed graph (a digraph). There is no dimensional limitation in the concept of templex. Here, we show that a templex can be automatically reduced into a ``minimal'' form to provide a comprehensive and synthetic view of the main properties of chaotic attractors. This reduction allows for the development of a taxonomy of chaos in terms of two elementary units: the oscillating unit (O-unit) and the switching unit (S-unit). We apply this approach to various well-known attractors (Rössler, Lorenz, and Burke-Shaw) as well as a non-trivial four-dimensional attractor. A case of toroidal chaos (Deng) is also treated. This work is dedicated to Otto E. Rössler.

Figures (16)

• Figure 1: Three ingoing strips --- the flow is from the top to the bottom --- are joined into a single outgoing strip in (a). An ingoing strip is split into three outgoing strips according to the two critical points P$_1$ and P$_2$ in (b). In (a), boundaries are in solid thick lines. The thick line in (a) represents the joining locus and the dashed thick line in (b) corresponds to the splitting locus. In (c), joining subgraph. In (d), splitting subgraph.
• Figure 2: Original generating complex $K_{\rm L}$ (a) from Ref. Cha22 with oriented components of the joining locus as thick arrows and components of the splitting locus as dashed lines; reduced cell complex $\overline{K}_{\rm L}$ (b); the template of the Lorenz attractor in which cells of $\overline{K}_{\rm L}$ are reported (c); reduced digraph $\overline{G}_{\rm L}$ with bonds in color (d).
• Figure 3: Generating complex $K_{\rm R_s}$ from Fig. 12 in Ref. Cha22 for the spiral Rössler attractor and its reduced version $\overline{K}_{\rm R_s}$ (b). The splitting lines are dashed and the joining locus are thick. The corresponding reduced digraph $\overline{G}_{\rm R_s}$ with its dynamical units is drawn in (d). The bond $\underline{1} \rightarrow \overline{2}$ is colored in blue.
• Figure 4: Reduced templex for the funnel Rössler attractor $\overline{T}_{\rm R_f} = (\overline{K}_{\rm R_f}, \overline{G}_{\rm R_f})$ (corresponding to Fig. 15 in Ref. Cha22).
• Figure 5: Reduced templex $\overline{T}_{\rm BS} = (\overline{K}_{\rm BS}, \overline{G}_{\rm BS})$ for the Burke and Shaw attractor. $\overline{K}_{\rm BS}$ is shown in (a), followed by the template representation with the cell numbers on the strips (b). The reduced digraph $\overline{G}_{\rm BS}$ with the colored bond, and the fundamental units are shown in (c).

Theorems & Definitions (11)

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