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DOI: 10.4230/LIPIcs.SoCG.2022.35

Tracking Dynamical Features via Continuation and Persistence

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

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

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.

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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)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2022.35,
  author =	{Dey, Tamal K. and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Slechta, Ryan},
  title =	{{Tracking Dynamical Features via Continuation and Persistence}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{35:1--35:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.35},
  URN =		{urn:nbn:de:0030-drops-160439},
  doi =		{10.4230/LIPIcs.SoCG.2022.35},
  annote =	{Keywords: combinatorial dynamical systems, continuation, index pair, Conley index, persistent homology}
}

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DOI: 10.4230/LIPIcs.SoCG.2020.37

Persistence of the Conley Index in Combinatorial Dynamical Systems

Authors: Tamal K. Dey, Marian Mrozek, and Ryan Slechta

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

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.

Cite as

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)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2020.37,
  author =	{Dey, Tamal K. and Mrozek, Marian and Slechta, Ryan},
  title =	{{Persistence of the Conley Index in Combinatorial Dynamical Systems}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{37:1--37:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.37},
  URN =		{urn:nbn:de:0030-drops-121958},
  doi =		{10.4230/LIPIcs.SoCG.2020.37},
  annote =	{Keywords: Dynamical systems, combinatorial vector field, multivector, Conley index, persistence}
}

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Documents authored by Slechta, Ryan

Tracking Dynamical Features via Continuation and Persistence

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Persistence of the Conley Index in Combinatorial Dynamical Systems

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