| Abstract: |
| Topological data analysis (TDA) is usually associated with point clouds and scalar filtrations, but many problems in dynamics naturally come with richer objects: finite trajectory segments, delay-coordinate reconstructions, or vector fields known either analytically or only through sampling. In this talk I will discuss a collection of recent TDA-inspired methods that shift the focus from the topology of static data to the comparison of dynamical systems themselves \cite{Carlsson2009,EdelsbrunnerHarer2010}. On the one hand, I will present tests for topological conjugacy constructed from finite orbit samples and time series, including a formulation based on Takens reconstructions, which makes it possible to assess whether two observed processes encode the same underlying dynamics \cite{Takens1981,DlotkoLipinskiSignerska2024}. On the other hand, I will describe new descriptors for sampled and continuous vector fields---including begin--end point embeddings, densities of directions, Euler characteristic curves, and Euler characteristic profiles---designed to compare dynamics directly at the level of trajectories or vector fields, detect qualitative changes such as bifurcations, and remain computationally feasible in data-driven settings \cite{MarszewskaSignerskaRynkowskaDlotkoInPrep}. Taken together, these methods show that ideas originating in TDA can be turned into practical tools for the quantitative and qualitative analysis of dynamics well beyond the standard point-cloud setting: from finite samples of trajectories to sampled or continuous vector fields, and from discrete observations to analytically defined systems. |
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