Templex: A bridge between homologies and templates for chaotic attractors

Area of Science:

• Dynamical Systems Theory
• Algebraic Topology
• Chaos Theory

Background:

• Homology theory uses cell complexes for algebraic descriptions of spaces.
• Branched Manifold Analysis through Homologies (BMH) characterizes state space attractors via cell complexes and homology groups.
• Existing BMH methods do not account for the flow's action on the cell complex.

Purpose of the Study:

• To extend BMH by incorporating flow dynamics into cell complex analysis.
• To develop a new framework, templex, for a more comprehensive study of chaotic attractors.
• To accurately classify and characterize chaotic attractors using the novel templex approach.

Main Methods:

• Endowing cell complexes with a directed graph to represent flow direction between highest-dimensional cells.
• Developing the templex framework by combining cell complexes and directed graphs.
• Investigating well-known chaotic attractors, including Rössler, Lorenz, and Burke-Shaw attractors.

Main Results:

• The templex framework successfully incorporates flow action into the topological analysis of attractors.
• Templex allows for sophisticated characterization and accurate classification of chaotic attractors.
• A link was established between templex descriptions and existing template-based analyses of chaotic systems.

Conclusions:

• Templex provides a powerful new tool for analyzing and classifying chaotic attractors.
• The integration of flow dynamics significantly enhances the descriptive power of cell complex methods.
• This approach offers a robust method for understanding complex dynamical systems.