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In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
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Conley Index Approach to Sampled Dynamics.

Bogdan Batko1, Konstantin Mischaikow2, Marian Mrozek1

  • 1Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348 Kraków, Poland.

SIAM Journal on Applied Dynamical Systems
|August 2, 2021
PubMed
Summary
This summary is machine-generated.

This study enhances a topological method for reconstructing dynamics from time series data. It uses a multivalued map representation and Conley index theory for improved applicability with sparse data.

Keywords:
37B3037M0537M1054C6054H20Conley indexchaosdynamical systemnonlinear dynamicstopological data analysistopological semiconjugacy

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Area of Science:

  • Dynamical Systems
  • Topological Data Analysis
  • Computational Mathematics

Background:

  • The topological method for reconstructing dynamics from time series has limitations with sparse data.
  • Previous approaches required continuous selectors for data representation.
  • Stronger dynamical expansions pose challenges for existing methods.

Purpose of the Study:

  • To improve the applicability of topological dynamics reconstruction, especially for sparse data and strong expansions.
  • To adapt the Conley index theory for multivalued maps in time series analysis.
  • To develop a more robust method for inferring system dynamics from observational data.

Main Methods:

  • Reshaping the topological method using a multivalued map representation of time series data.
  • Employing a new version of Conley index theory for multivalued maps in computations.
  • Utilizing a multivalued map without requiring a continuous selector.

Main Results:

  • The enhanced method demonstrates improved applicability for reconstructing dynamics from sparse time series.
  • The existence of a continuous, single-valued generator of dynamics is guaranteed near the constructed multivalued map.
  • Numerical examples using Hénon-type maps validate the effectiveness of the approach.

Conclusions:

  • The modified topological method offers a more versatile approach to dynamical system reconstruction from time series.
  • The use of Conley index theory for multivalued maps provides a robust computational framework.
  • This advancement is particularly beneficial for analyzing complex systems with limited or noisy data.