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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 AsGet 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

  1. J. A. Barmak. Algebraic Topology of Finite Topological Spaces and Applications. Lecture Notes in Mathematics 2032. Springer Verlag, Berlin - Heidelberg - New York, 2011. Google Scholar
  2. N. P. Bhatia and G. P. Szegö. Dynamical Systems: Stability Theory and Applications. Lecture Notes in Mathematics 35. Springer Verlag, Berlin - Heidelberg - New York, 1967. Google Scholar
  3. Gunnar Carlsson and Vin de Silva. Zigzag persistence. Foundations of Computational Mathematics, 10(4):367-405, August 2010. URL: https://doi.org/10.1007/s10208-010-9066-0.
  4. C. Conley. Isolated invariant sets and the Morse index. In CBMS Regional Conference Series 38, American Mathematical Society, 1978. Google Scholar
  5. Tamal K. Dey, Mateusz Juda, Tomasz Kapela, Jacek Kubica, Michal Lipinski, and Marian Mrozek. Persistent homology of Morse decompositions in combinatorial dynamics. SIAM Journal on Applied Dynamical Systems, 18(1):510-530, 2019. URL: https://doi.org/10.1137/18M1198946.
  6. R. Forman. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, 228:629-681, 1998. URL: https://doi.org/10.1007/PL00004638.
  7. R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90-145, 1998. URL: https://doi.org/10.1006/aima.1997.1650.
  8. J. Hale and H. Koçak. Dynamics and Bifurcations. Texts in Applied Mathematics 3. Springer-Verlag, 1991. Google Scholar
  9. Allen Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002. Google Scholar
  10. Michał Lipiński, Jacek Kubica, Marian Mrozek, and Thomas Wanner. Conley-Morse-Forman theory for generalized combinatorial multivector fields on finite topological spaces. arXiv:1911.12698 [math.DS], 2019. URL: http://arxiv.org/abs/1911.12698.
  11. Edward N. Lorenz. Deterministic Nonperiodic Flow, pages 25-36. Springer New York, New York, NY, 2004. URL: https://doi.org/10.1007/978-0-387-21830-4_2.
  12. K. Mischaikow and M. Mrozek. The Conley Index. Handbook of Dynamical Systems II: Towards Applications. (B. Fiedler, ed.) North-Holland, 2002. Google Scholar
  13. K. Mischaikow, M. Mrozek, J. Reiss, and A. Szymczak. Construction of symbolic dynamics from experimental time series. Physical Review Letters, 82:1144-1147, February 1999. URL: https://doi.org/10.1103/PhysRevLett.82.1144.
  14. Konstantin Mischaikow and Marian Mrozek. Chaos in the Lorenz equations: a computer-assisted proof. Bulletin of the American Mathematical Society, 33:66-72, 1995. URL: https://doi.org/10.1090/S0273-0979-1995-00558-6.
  15. Marian Mrozek. Conley-Morse-Forman theory for combinatorial multivector fields on Lefschetz complexes. Foundations of Computational Mathematics, 17(6):1585-1633, December 2017. URL: https://doi.org/10.1007/s10208-016-9330-z.
  16. J.R. Munkres. Topology. Featured Titles for Topology Series. Prentice Hall, Incorporated, 2000. Google Scholar
  17. H.J. Poincaré. Sur le probleme des trois corps et les équations de la dynamique. Acta Mathematica, 13:1-270, 1890. Google Scholar
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