Contractibility of a persistence map preimage
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Contractibility of a persistence map preimage Jacek Cyranka1,2
· Konstantin Mischaikow1 · Charles Weibel1
Received: 29 October 2018 / Accepted: 14 August 2020 © The Author(s) 2020
Abstract This work is motivated by the following question in data-driven study of dynamical systems: given a dynamical system that is observed via time series of persistence diagrams that encode topological features of snapshots of solutions, what conclusions can be drawn about solutions of the original dynamical system? We address this challenge in the context of an N dimensional system of ordinary differential equation defined in R N . To each point in R N (e.g. an initial condition) we associate a persistence diagram. The main result of this paper is that under this association the preimage of every persistence diagram is contractible. As an application we provide conditions under which multiple time series of persistence diagrams can be used to conclude the existence of a fixed point of the differential equation that generates the time series. Keywords Topological data analysis · Persistent homology · Dynamical systems · Fixed point theorem Mathematics Subject Classification 37C25 · 55N31 · 06B35 · 55-08 · 06F30
1 Introduction Topological data analysis (TDA), especially in the form of persistent homology, is rapidly developing into a widely used tool for the analysis of high dimensional data associated with nonlinear structures (Edelsbrunner and Harer 2010; Zomorodian and
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Jacek Cyranka [email protected] Konstantin Mischaikow [email protected] Charles Weibel [email protected]
1
Department of Mathematics, Rutgers, The State University of New Jersey, 110 Frelinghusen Rd., Piscataway, NJ 08854-8019, USA
2
Institute of Informatics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
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J. Cyranka et al.
Carlsson 2005; Oudot 2015). That topological tools can play a role in this subject should not be unexpected, given the central role of nonlinear functional analysis in the study of geometry, analysis, and differential equations, for example. What is perhaps surprising is that, to the best of our knowledge, there have been no systematic attempts to rigorously analyze the dynamics of differential equations using persistent homology. Persistent homology is often used as a means of data reduction. A typical example takes the form of a complicated scalar function defined over a fixed domain, where the geometry of the sub-(super)-level sets is encoded via homology. Of particular interest to us are settings in which the scalar function arises as a solution to a partial differential equation (PDE); we are interested in tracking the evolution of the function, but experimental data only provides information on the level of digital images of the process. Furthermore, capturing the dynamics of a PDE often requires a long time series of rather large digital images. Thus, rather than storing the full images, one can hope to work with a time series of persistence diagrams. Our aim is to draw conclusions about the dynamic
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