Templex-based dynamical units for a taxonomy of chaos

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.

💡 Research Summary

The paper addresses the long‑standing limitation of template‑based classification of chaotic attractors, which is confined to three‑dimensional systems because it relies on knot theory. To overcome this restriction, the authors introduce the concept of a “templex,” a mathematical object that couples a BraMAH cell complex with a directed graph (digraph) that records how the flow visits the cells. The cell complex provides a combinatorial representation of the underlying manifold (points, edges, faces, volumes, etc.), while the digraph encodes the deterministic sequence of transitions between the highest‑dimensional cells. By orienting cells around joining and splitting loci, the templex captures both the topological structure (homology groups, torsion coefficients) and the dynamical organization of the flow.

A central contribution is an automatic reduction algorithm that merges cells and nodes without altering topological invariants, yielding a minimal templex. In this reduced form, two elementary dynamical units emerge naturally: the oscillating unit (O‑unit) and the switching unit (S‑unit). An O‑unit corresponds to at least a two‑dimensional oscillatory subsystem, manifesting as a closed loop or torus‑like structure in the phase space. An S‑unit is a one‑dimensional switching mechanism that implements the “stretch‑and‑fold” process by connecting joining and splitting loci, thereby generating sensitivity to initial conditions. The number and arrangement of O‑units and S‑units constitute a “templex signature,” which the authors relate to the Lyapunov spectrum: each positive Lyapunov exponent implies the presence of an S‑unit, while each zero or negative exponent corresponds to an O‑unit.

The authors demonstrate the method on a diverse set of attractors: the spiral and funnel versions of the Rössler system, the Lorenz attractor, the Burke‑Shaw system, a non‑trivial four‑dimensional chaotic system, and Deng’s toroidal chaos. For each case they construct the templex from trajectory point clouds, apply the reduction, and identify the O‑ and S‑units. The Rössler attractors reduce to a single O‑unit plus one S‑unit; the Lorenz attractor decomposes into two O‑units linked by a single S‑unit, reflecting its classic “butterfly” geometry; the Burke‑Shaw system exhibits multiple S‑units due to richer splitting structures; the 4‑D system requires several O‑units and multiple S‑units, illustrating how templexes handle higher‑dimensional holes and multiple tori; and Deng’s toroidal chaos is captured as a torus‑shaped O‑unit surrounded by an S‑unit, correctly preserving its toroidal topology.

Beyond classification, the templex framework provides a unified language that bridges topological invariants (homology, torsion) with dynamical descriptors (Lyapunov exponents, return maps). It eliminates the need for unstable periodic orbit extraction, which is essential for template construction, and works directly with raw data, making it suitable for experimental time series as well as numerical simulations. The authors argue that this approach can be extended to complex real‑world systems—such as climate models, neural networks, and high‑dimensional engineering systems—where traditional knot‑based templates are inapplicable.

In conclusion, the paper presents templexes as a dimension‑agnostic, mathematically rigorous tool for dissecting chaotic attractors into two fundamental units. This yields a new taxonomy of chaos that is both topologically complete and dynamically informative, offering a promising avenue for future research in high‑dimensional nonlinear dynamics and control.