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Persistence of the Conley Index in Combinatorial Dynamical Systems

Authors Tamal K. Dey, Marian Mrozek , Ryan Slechta

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Author Details

Tamal K. Dey
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Marian Mrozek
  • Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Ryan Slechta
  • Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA

Funding

  • Dey, Tamal K.: Partially supported by NSF grants CCF-1740761 and DMS-1547357.
  • Mrozek, Marian: Partially supported by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453.
  • Slechta, Ryan: Supported by NSF grant DMS-1547357.

Cite As Get BibTex

Tamal K. Dey, Marian Mrozek, and Ryan Slechta. Persistence of the Conley Index in Combinatorial Dynamical Systems. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 37:1-37:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.37

Abstract

A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman [R. Forman, 1998; R. Forman, 1998] and their recent generalization to multivector fields [Mrozek, 2017] have provided a starting point for building such a connection. In this work, we strengthen this relationship by placing the Conley index in the persistent homology setting. Conley indices are homological features associated with so-called isolated invariant sets, so a change in the Conley index is a response to perturbation in an underlying multivector field. We show how one can use zigzag persistence to summarize changes to the Conley index, and we develop techniques to capture such changes in the presence of noise. We conclude by developing an algorithm to "track" features in a changing multivector field.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • Dynamical systems
  • combinatorial vector field
  • multivector
  • Conley index
  • persistence

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References

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