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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.