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

What is it about?

Understanding the qualitative behavior of nonlinear dynamical systems, such as time series and phase portraits, remains a challenging task. Topological data analysis offers a robust framework for uncovering the intrinsic structures of such complex systems by extracting shape-based features that are stable under noise and deformation. This study applied topological data analysis (TDA) tools, particularly persistent homology (PH), to analyze the time series and phase portrait images of a Rössler-like system. Persistence diagrams (PDs) capture the birth and death of topological features across scales, enabling differentiation between dynamic behaviors. To further analyze the geometrical shape of the phase space of the system, we utilized cubical homology (CH) to generate topological descriptors from phase portraits. These descriptors, including persistence landscapes, Betti curves, and persistence images, were also computed from a persistence diagram for use in machine learning (ML) to classify different dynamic regimes. The novelty of this work lies in presenting a comprehensive topological data-analysis-based pipeline tailored to dynamical systems, aiming to make these techniques accessible and applicable to researchers in nonlinear dynamics who may not have prior experience with topological tools.

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Why is it important?

This study provides a comprehensive overview of topological data analysis for dynamical systems. In this study, two powerful tools of topological data analysis were used to analyze the time series and phase portraits of the dynamics. A persistent homology pipeline is applied to the time series of a piecewise linear Rössler-like model. On the other hand, we also used cubical homology directly on the images of the phase space of the Rössler-like model. The idea of cubical homology to phase portraits emerges when there are only phase portraits, instead of time series. To the best of our knowledge, this is the first study to implement the cubical homology ideology of phase-space images to investigate them topologically. These topological data analysis methods allow us to detect and analyze the time-series shape and topological structure of the phase portrait trajectories of the system. These techniques provide a shape-based physical interpretation of the dynamics, which reveals shape changes in phase-space images and time series. Traditional dynamical tools, such as Fourier analysis, Lyapunov exponents, and Poincarè map, may not capture this information.

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