Templex-based dynamical units for a taxonomy of chaos.
Caterina Mosto1,2, Gisela D Charó1,2,3, Christophe Letellier4
1CONICET-Universidad de Buenos Aires, Centro de Investigaciones del Mar y la Atmósfera (CIMA), C1428EGA CABA, Argentina.
Chaos (Woodbury, N.Y.)
|November 6, 2024
Summary
Researchers developed a new method to classify complex chaotic systems. This templex approach simplifies high-dimensional chaos into fundamental oscillating (O-units) and switching (S-units) components.
Area of Science:
- Dynamical Systems and Chaos Theory
- Complex Systems Analysis
- Computational Mathematics
Background:
- Classifying chaotic systems is challenging, especially in higher dimensions.
- Current methods like templates are limited to 3D systems and knot theory.
- A new approach is needed for analyzing high-dimensional chaotic dynamics.
Purpose of the Study:
- To introduce a novel method for analyzing and classifying high-dimensional chaotic attractors.
- To develop a taxonomy of chaos based on fundamental units.
- To demonstrate the versatility of the templex approach across various chaotic systems.
Main Methods:
- Introduction of the templex: a combination of a BraMAH cell complex and a directed graph.
- Automatic reduction of templex to a minimal form for synthetic analysis.
- Application of the templex reduction to known and novel chaotic attractors.
Main Results:
- The templex method successfully reduces complex chaotic attractors into a manageable form.
- A taxonomy of chaos is established using two elementary units: oscillating (O-units) and switching (S-units).
- The approach is validated on diverse attractors, including 3D (Rössler, Lorenz, Burke-Shaw) and 4D systems, and toroidal chaos.
Conclusions:
- The templex reduction provides a comprehensive and synthetic view of chaotic attractor properties.
- This method offers a dimension-independent framework for understanding and classifying chaos.
- The O-unit and S-unit taxonomy represents a significant advancement in chaos theory.
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