Contractibility of a persistence map preimage
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 $${\mathbb {R}}^N$$ R N . To each point in $${\mathbb {R}}^N$$ 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.
Document Details
- Document Type
- Pub Defense Publication
- Publication Date
- Aug 28, 2020
- Source ID
- 10.1007/s41468-020-00059-7
Entities
People
- Charles Weibel
- Jacek Cyranka
- Konstantin Mischaikow
Organizations
- Defense Advanced Research Projects Agency
- National Institutes of Health
- National Science Foundation
- National Science Foundation Division of Mathematical Sciences
- Polish National Agency for Academic Exchange