Conley index approach to sampled dynamics

Bogdan Batko, Konstantin Mischaikow, Marian Mrozek, Mateusz Przybylski

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

The topological method for the reconstruction of dynamics from time series [K. Mischaikow et al., Phys. Rev. Lett., 82 (1999), pp. 1144{1147] is reshaped to improve its range of applicability, particularly in the presence of sparse data and strong expansion. The improvement is based on a multivalued map representation of the data. However, unlike the previous approach, it is not required that the representation has a continuous selector. Instead of a selector, a recently developed new version of Conley index theory for multivalued maps [B. Batko, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1587{1617; B. Batko and M. Mrozek, SIAM J. Appl. Dyn. Syst., 15 (2016), pp. 1143{ 1162] is used in computations. The existence of a continuous, single valued generator of the relevant dynamics is guaranteed in the vicinity of the graph of the multivalued map constructed from data. Some numerical examples based on time series derived from the iteration of Hénon-type maps are presented.

Original languageEnglish (US)
Pages (from-to)665-704
Number of pages40
JournalSIAM Journal on Applied Dynamical Systems
Volume19
Issue number1
DOIs
StatePublished - 2020

All Science Journal Classification (ASJC) codes

  • Analysis
  • Modeling and Simulation

Keywords

  • Chaos
  • Conley index
  • Dynamical system
  • Fixed point
  • Index pair
  • Invariant set
  • Isolating neighborhood
  • Nonlinear dynamics
  • Periodic orbit
  • Topological data analysis
  • Topological semiconjugacy
  • Weak index pair

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