Complete set of representations for dissipative chaotic three-dimensional dynamical systems.

Area of Science:

• Dynamical Systems Theory
• Topology
• Data Representation

Background:

• Embeddings serve as diffeomorphisms mapping dynamical phase spaces to reconstructed images.
• The equivalence of different embeddings under isotopy is a key consideration.
• Embeddings are treated as fundamental representations of dynamical phase space.

Purpose of the Study:

• To determine the necessary topological labels for distinguishing inequivalent representations.
• Focuses on three-dimensional dissipative dynamical systems.
• Investigates embeddings into Euclidean spaces R(k), where k=3, 4, and 5.

Main Methods:

• Analysis of diffeomorphisms between dynamical phase space and reconstructed images.
• Topological classification of embeddings based on isotopy equivalence.
• Determination of labeling requirements for different embedding dimensions (R³, R⁴, R⁵).

Main Results:

• Three topological labels are required to distinguish inequivalent representations when embedding into R³.
• Only one topological label is sufficient for distinguishing representations when embedding into R⁴.
• A single, universal representation exists for embeddings into R⁵.

Conclusions:

• The dimensionality of the embedding space significantly impacts the complexity of representing dynamical systems.
• Higher-dimensional embeddings (R⁴, R⁵) offer more simplified and universal representations.
• Topological labeling provides a robust method for classifying inequivalent dynamical system representations.