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Tracking Dynamical Features via Continuation and Persistence

Authors Tamal K. Dey , Michał Lipiński , Marian Mrozek , Ryan Slechta

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Author Details

Tamal K. Dey
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Michał Lipiński
  • Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Marian Mrozek
  • Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Ryan Slechta
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA

Funding

This work is partially supported by NSF grants CCF-2049010, CCF-1839252, Polish National Science Center under Maestro Grant 2014/14/A/ST1/00453, Opus Grant 2019/35/B/ST1/ 00874 and Preludium Grant 2018/29/N/ST1/00449

Cite AsGet BibTex

Tamal K. Dey, Michał Lipiński, Marian Mrozek, and Ryan Slechta. Tracking Dynamical Features via Continuation and Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.35

Abstract

Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set - a salient feature of a combinatorial dynamical system - across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • combinatorial dynamical systems
  • continuation
  • index pair
  • Conley index
  • persistent homology

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