Templex-based dynamical units for a taxonomy of chaos

TL;DR

This paper introduces templex-based dynamical units, O-unit and S-unit, for classifying chaos across different dimensions, providing a unified framework for analyzing complex attractors beyond traditional three-dimensional limitations.

Contribution

It presents a novel, dimension-independent approach using templex to develop a taxonomy of chaos with minimal units, extending analysis capabilities to higher-dimensional systems.

Findings

Successfully applied to classical attractors like R"ossler and Lorenz

Extended to four-dimensional and toroidal chaos cases

Provides a synthetic, comprehensive view of chaotic properties

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 (R\"ossler, 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. R\"ossler.