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Article Open Access October 26, 2022

Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map

O. Yu. Romanenko 1,* and O. M. Sharkovsky 1
1
Institute of Mathematics, National Academy of Sciences, Kyiv, Ukraine
Publihed in: Current Research in Public Health (Volume 1, Issue 2, 2022)
Page(s): 77-94
DOI: 10.31586/ijmebac.2022.468
Received
August 31, 2022
Revised
October 16, 2022
Accepted
October 24, 2022
Published
October 26, 2022
Keywords
One-Dimensional Map; Continuous-Time Difference Equation; Chaotic Dynamics
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright: Copyright © The Author(s), 2022. Published by Scientific Publications
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Cite This Article

APA Style
Romanenko, O. Y. , & Sharkovsky, O. M. (2022). Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map. Current Research in Public Health, 1(2), 77-94. https://doi.org/10.31586/ijmebac.2022.468
ACS Style
Romanenko, O. Y. ; Sharkovsky, O. M. Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map. Current Research in Public Health 2022 1(2), 77-94. https://doi.org/10.31586/ijmebac.2022.468
Chicago/Turabian Style
Romanenko, O. Yu., and O. M. Sharkovsky. 2022. "Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map". Current Research in Public Health 1, no. 2: 77-94. https://doi.org/10.31586/ijmebac.2022.468
AMA Style
Romanenko OY, Sharkovsky OM. Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map. Current Research in Public Health. 2022; 1(2):77-94. https://doi.org/10.31586/ijmebac.2022.468 
@Article{crph468,
AUTHOR = {Romanenko, O. Yu. and Sharkovsky, O. M.},
TITLE = {Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map},
JOURNAL = {Current Research in Public Health},
VOLUME = {1},
YEAR = {2022},
NUMBER = {2},
PAGES = {77-94},
URL = {https://www.scipublications.com/journal/index.php/IJMEBAC/article/view/468},
ISSN = {2831-5162},
DOI = {10.31586/ijmebac.2022.468},
ABSTRACT = {The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system  on some interval , so the equation  induces the infinite-dimensional dynamical system  on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence  (as in the discrete case) but also on the internal structure of  as . Assuming  to be continuous, we consider the iterations of  as the semigroup  generated by  on the space of continuous maps, and introduce the notion of a limit semigroup for  in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of.},
}
%0 Journal Article
%A Romanenko, O. Yu.
%A Sharkovsky, O. M.
%D 2022
%J Current Research in Public Health

%@ 2831-5162
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%N 2
%P 77-94

%T Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map
%M doi:10.31586/ijmebac.2022.468
%U https://www.scipublications.com/journal/index.php/IJMEBAC/article/view/468

TY  - JOUR
AU  - Romanenko, O. Yu.
AU  - Sharkovsky, O. M.
TI  - Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map
T2  - Current Research in Public Health
PY  - 2022
VL  - 1
IS  - 2
SN  - 2831-5162
SP  - 77
EP  - 94
UR  - https://www.scipublications.com/journal/index.php/IJMEBAC/article/view/468
AB  - The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system  on some interval , so the equation  induces the infinite-dimensional dynamical system  on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence  (as in the discrete case) but also on the internal structure of  as . Assuming  to be continuous, we consider the iterations of  as the semigroup  generated by  on the space of continuous maps, and introduce the notion of a limit semigroup for  in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of.
DO  - Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map
TI  - 10.31586/ijmebac.2022.468
ER  -