Topological data analysis approach to time series and shape analysis of dynamical system

Area of Science:

• Dynamical Systems
• Topological Data Analysis
• Nonlinear Dynamics

Background:

• Time series and phase space are crucial for understanding dynamical systems.
• Topological Data Analysis (TDA) offers novel methods for characterizing complex systems.
• Existing TDA methods have limitations in analyzing dynamical system characteristics directly.

Purpose of the Study:

• To apply TDA to time series and phase space of dynamical systems.
• To introduce a comprehensive TDA pipeline for nonlinear dynamics.
• To enable automated analysis and classification of dynamical behaviors.

Main Methods:

• Utilized persistent homology on time series data converted to point clouds.
• Employed the Rips complex for homology computation.
• Applied cubical homology to phase-space images for the first time.
• Computed topological machine learning features like persistent landscapes and persistence images.

Main Results:

• Successfully measured topological features of system behavior from time series.
• Developed a novel image-based approach for phase portrait analysis using cubical homology.
• Generated machine learning features for automated classification of dynamical behaviors.
• Demonstrated the utility of TDA in analyzing the Rössler-like attractor.

Conclusions:

• TDA offers a powerful framework for analyzing dynamical systems.
• The developed pipeline makes TDA accessible for nonlinear dynamics research.
• Integrating TDA with machine learning facilitates automated behavior classification.