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Erschienen in:
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2020 | OriginalPaper | Buchkapitel

1. Introduction

verfasst von : Marat Akhmet

Erschienen in: Almost Periodicity, Chaos, and Asymptotic Equivalence

Verlag: Springer International Publishing

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Abstract

The introduction contains historical information which is most close to subjects of the book: almost periodicity, replication of chaos, generalized piecewise-constant argument, and asymptotical equivalence. It consists of four parts. In the first one, the origins of the almost periodic functions and almost periodic solutions of differential equations are shortly presented. Next, the process of development of the theory of discontinuous almost periodic functions as solutions for impulsive differential equations is described such that the reader will see the benefits of the reading of the book. It is important that influence of the notion on other types of differential equations and hybrid systems is mentioned. Dynamics on time scales and differential equations with generalized piecewise-constant argument are among them. That, the universal role of the concept considered in our research for development of many discontinuous dynamics is emphasized. The second part provides short and sufficiently complete description of main results on deterministic chaos. Then, the mechanism of replication of chaos is introduced with stressing that our proposals are a powerful instrument for shaping new irregular motions, while discovering a chaos is mainly under research in the theory. Applications of the method for neural networks, mechanics analysis as well as for Li-Yorke homoclinic chaos appearance are characterized. Preliminaries for generalized piecewise-constant argument are provided in the third section. Theoretical and application advantages of our suggestions for differential equations are pointed. Moreover, new results are described, which have not been considered in monographs before. The last section of this chapter gives historical description of the method of asymptotical equivalence. Cornerstone results are mentioned. An example that demonstrates that the results of the book are more widely applicable than other theorems in the field is provided. Extensions of the method for impulsive systems are described to give an argument for next theoretical applications.

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Nächstes Kapitel Generalities for Impulsive Systems
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Metadaten
Titel
Introduction
verfasst von
Marat Akhmet
Copyright-Jahr
2020
Buch
Almost Periodicity, Chaos, and Asymptotic Equivalence

Print ISBN: 978-3-030-19916-6

Electronic ISBN: 978-3-030-20572-0

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-030-20572-0_1

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