Time scale-induced asynchronous discrete dynamical systems

Stefan Siegmund; Petr Stehlík

doi:10.3934/dcdsb.2020151

Article Contents

2021, Volume 26, Issue 2: 1011-1029. Doi: 10.3934/dcdsb.2020151

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Time scale-induced asynchronous discrete dynamical systems

Stefan Siegmund^1, and
Petr Stehlík^2, ,

1.
Center for Dynamics & Institute for Analysis, Faculty of Mathematics, Technische Universität Dresden, 01062, Dresden, Germany
2.
Dept. of Mathematics and NTIS, University of West Bohemia, Univerzitní 8, 30614 Pilsen, Pilsen, Czech Republic

^* Corresponding author: Petr Stehlík
^* Corresponding author: Petr Stehlík

Received: July 2019

Revised: February 2020

Early access: May 2020

Published: February 2021

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Abstract

We study two coupled discrete-time equations with different (asynchronous) periodic time scales. The coupling is of the type sample and hold, i.e., the state of each equation is sampled at its update times and held until it is read as an input at the next update time for the other equation. We construct an interpolating two-dimensional complex-valued system on the union of the two time scales and an extrapolating four-dimensional system on the intersection of the two time scales. We discuss stability by several results, examples and counterexamples in various frameworks to show that the asynchronicity can have a significant impact on the dynamical properties.

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Mathematics Subject Classification: Primary: 37E99, 39A30; Secondary: 34N05, 91B64, 93C55.

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Figure 1. Time scales $ \mathbb{T}_3 $ and $ \mathbb{T}_5 $ of a $ (3,5) $-asynchronous discrete dynamical system (3)

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Figure 2. Time scales related to dynamically equivalent (2, 3)- and (6, 1)-asynchronous discrete dynamical systems from Example 7.3

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Table 1. 9 possible forms of the one-step evolution operator $ A(t) $, $ t,\sigma(t)\in \mathbb{T} $ associated with the system (8), see Corollary 2. The pictograms illustrate each quadruple $ (i,j,k,\ell) = \big(1_{ \mathbb{T}_{\mu}}(t), 1_{ \mathbb{T}_{\mu}}(\sigma(t)), 1_{ \mathbb{T}_{\nu}}(t), 1_{ \mathbb{T}_{\nu}}(\sigma(t))\big) $, squares correspond to $ \mathbb{T}_\mu $, circles to $ \mathbb{T}_\nu $, the left symbols to time $ t\in \mathbb{T} $ and the right symbols to $ \sigma(t)\in \mathbb{T} $

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