Almost periodicity, chaos, and asymptotic equivalence 9783030199166, 9783030205720

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Nonlinear Systems and Complexity Series Editor: Albert C. J. Luo

Marat Akhmet

Almost Periodicity, Chaos, and Asymptotic Equivalence

Nonlinear Systems and Complexity Volume 27

Series editor Albert C. J. Luo Southern Illinois University Edwardsville, IL, USA

Nonlinear Systems and Complexity provides a place to systematically summarize recent developments, applications, and overall advance in all aspects of nonlinearity, chaos, and complexity as part of the established research literature, beyond the novel and recent findings published in primary journals. The aims of the book series are to publish theories and techniques in nonlinear systems and complexity; stimulate more research interest on nonlinearity, synchronization, and complexity in nonlinear science; and fast-scatter the new knowledge to scientists, engineers, and students in the corresponding fields. Books in this series will focus on the recent developments, findings and progress on theories, principles, methodology, computational techniques in nonlinear systems and mathematics with engineering applications. The Series establishes highly relevant monographs on wide ranging topics covering fundamental advances and new applications in the field. Topical areas include, but are not limited to: Nonlinear dynamics Complexity, nonlinearity, and chaos Computational methods for nonlinear systems Stability, bifurcation, chaos and fractals in engineering Nonlinear chemical and biological phenomena Fractional dynamics and applications Discontinuity, synchronization and control.

More information about this series at http://www.springer.com/series/11433

Marat Akhmet

Almost Periodicity, Chaos, and Asymptotic Equivalence

123

Marat Akhmet Department of Mathematics Middle East Technical University Ankara, Ankara, Turkey

ISSN 2195-9994 ISSN 2196-0003 (electronic) Nonlinear Systems and Complexity ISBN 978-3-030-19916-6 ISBN 978-3-030-20572-0 (eBook) https://doi.org/10.1007/978-3-030-20572-0 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To My Beloved Family

Preface

The main goal of this book is to show how to extend already existing concepts in mathematics to make our knowledge more adapted to the real world around us and secondly to provide new challenging theoretical problems and solve them. First of all, we will extend the concept of almost periodicity, which is well-studied in continuous dynamics, to other types of hybrid equations, when continuous motion is intermingled with different types of discontinuities. In addition, we suggest to consider chaos with ingredients more sophisticated than unstable periodical characteristics, namely, almost periodic ones. We thus demonstrate that complexity invasion in dynamics is unstoppable; of course, it should be accompanied by rigorous discussions. Finally, new equations are discovered, which require most subtle properties to be considered asymptotically equivalent. A distinguishing characteristic of the manuscript is that the author was involved in obtaining all of the results found in this book. Consequently, the book can be considered as a global product of personal research activity. Our world is a vast set of connected oscillators. That is, the motion of planets as well as molecules and atoms, the circadian rhythm of living organisms, and the dynamics of engines are all oscillators that may be connected. For instance, the rotation of the Earth affects the circadian rhythm. The most complex type of oscillations considered in applications are almost periodic motions. Nowadays, almost periodicity is observed not only in continuous dynamics but also for motions with different forms of discontinuity. It is important for researchers to find common characteristics in the discontinuous phenomena to provide more universal instruments for analysis. Our book is designed to be one to discuss these problems. The theory of differential equations focuses on equilibriums, periodic, and almost periodic oscillations. There is a large class of differential equations, which cannot be solved precisely for the oscillations. For this reason, asymptotic equivalence method is useful, when two systems, which are essentially equal ultimately, are considered, and one of them either admits exact solutions or we have more or less complete information about its solutions’ behavior. In theory of differential equations, oscillations meet the needs of many realworld problems related to mechanics, electronics, economics, biology, etc., if one vii

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searches for regular and stable motion. However, regular and isolated motions are not sufficient for modern and future demands of robotics, computer technologies, and the Internet, while chaotic dynamics comprise constructive properties for these applications. This is why it is important to join the power of deterministic chaos with the immensely rich source of methods for differential equations. In this book, we address the problems indicated above by considering the following: (1) almost periodic solutions with different types of discontinuity, discussed on the basis of the unique method developed in our papers and books; (2) new classes of asymptotic equivalent systems, which in some sense are the largest at the moment and are discovered in our research; and (3) a special way to generate chaotic attractors with almost periodic components that is created. The integrity of the book is achieved by research of almost periodic motions in chaotic dynamics and asymptotically equivalent systems. Our present study is a unique one, since it considers one of the most intricate subjects of the theory of differential equations, almost periodic solutions, and one of the most modern and complex dynamics, chaos, in the same book. Additionally, the powerful instrument for nonlinear systems study, asymptotic equivalence method, is delivered and applied for almost periodicity investigation. Thus, the manuscript serves to attract a specialist in differential equations to chaos investigation as well as invites to use results for differential equations in chaos analysis. Almost periodic solutions of differential equations are the most general oscillations in industrial applications and in interdisciplinary research in mathematics with biology, physics, mechanics, electronics, robotics, neural networks, etc. Modern development of industry and science demands discontinuous oscillations and differential equations with different types of discontinuity. The author is pioneering in the study of almost periodic motions in all types of models with discontinuities: impulsive differential equations, differential equations with discontinuous righthand side, differential equations with piecewise constant arguments, and differential equations on time scales. The book is unique as it contains analysis of all of these equations from the unified point of view. It will help readers to investigate almost periodic dynamics not only of all listed above models but also potential equations, where combination of the discontinuities with continuity can be involved. There are at least two more books on discontinuous almost periodicity, but they cannot be considered our competitors from the two points of view: we provide more precise historical background for the research, and, secondly, by considering deep analogies, the approach is extended for other types of differential equations with discontinuities. The asymptotic integration is a powerful instrument for solution and qualitative analysis of differential equations by comparison with more simple systems, if one considers the ultimate behavior of solutions. More complete investigations can be done, if the comparison is asymptotic equivalence. The focus of oscillation study will be essentially moved from regular and isolated motions to the chaotic dynamics in the very near future. This is why it is of

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indisputable importance to specify the structure of chaotic attractors observing their almost periodic oscillations, and not periodic motions as usually. On the basis of the theory developed in this book, the results are obtained, which increase the role of almost periodicity for chaos. The central subject of the book are oscillations. Almost periodic oscillations are chosen for our research, since they are most common in applications and most intricate for mathematical analysis. Moreover, they are proven in the book as a regular component of chaotic attractors. Thus, we pay attention to almost periodic functions (1) as stable (asymptotically) solutions of differential equations of different types, presumably discontinuous, and (2) as non-isolated oscillations in chaotic sets. Finally, we prove existence of almost periodic oscillations (asymptotic and biasymptotic) by asymptotic equivalence between systems. Thus, we are attracting attention of readers to the method, which is natural for oscillation discussion at the present and has a strong potential in the study of chaos, in the future. The book is useful for engineers and specialists in electronics, computer sciences, robotics, neural networks, artificial networks, and biology. Pragmatically, the book provides three powerful instruments for mathematical research of oscillations. The first one is the method of almost periodicity construction and recognition in hybrid systems, where discontinuous and continuous components are combined specifically. Another is the asymptotic equivalence method, which is useful to learn global portraits of dynamics in their ultimate form. In our book, it is utilized also to prove existence of asymptotic almost periodic solutions. The method can be applied for chaos analysis, in future research. And the third one is the replication of chaos method, which is useful for construction of chaotic sets with prescribed structures and properties. These three instruments promise to be helpful for mathematicians as well as researchers and specialists in many areas where dynamics are observable and applied. We are working how to extend the phenomenon of almost periodicity from continuous models to models with different types of discontinuity as well as to extend the role of differential equations in the research of chaos, and we hope that for this reason, the book will be of very great interest for mathematical society. Almost periodicity is a phenomenon, which is very often seen in applications and attractive for theoretical studies always. Discontinuous almost periodic functions are more difficult for analysis than continuous ones. We will present effective methods which were developed for almost periodic solutions’ investigation in differential equations with different types of discontinuities by using a unified approach. Asymptotic equivalence is a powerful method for qualitative analysis of differential equations which is underestimated in the theory. Recently, we suggested conditions for the equivalence which significantly extend the class of equivalent systems. In the book, we provide the full description of the method and results for equations with continuous and discontinuous dynamics of different types. The theoretical and practical roles of discontinuities are increasing in engineering, robotics, electronics, neural networks, computer sciences, and biological models. This is why the quick expansion of the content of the book is very desirable and inevitable.

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The chaos research is still centered on the recognition of chaos in the specific models. We suggest to replicate chaos by classes of differential equations such that chaos will become a routine object of investigation in the theory of differential and difference equations. It is known that periodicity is an ingredient of chaos in Devaney’s or Li-Yorke definitions, and there is an opinion that it can be replaced with almost periodic and even recurrent or Poisson stable motions. Using the replication of chaos method, we have solved the problem for almost periodic motions constructively. It is important for me to give the complete information about the results and describe future possible research in the direction. The reader of the book will find the rich diversity of differential equations which combine continuous and discontinuous dynamics in the models, differential equations with discontinuous right-hand side, differential equations with impulses, differential equations with piecewise constant argument, hybrid systems, and others. The investigation of the systems is intriguing and an attractive job. We must say that the number of mathematicians as well as physicists, engineers, biologists, and others working with the models is rapidly growing. The manuscript is a unique one, since it provides a uniform and effective approach to the research concerning the discussed problems. The book contains the most extended results which can be obtained with the asymptotic equivalence method. Theoretical aspects of the method can be found in their complete form in the manuscript. The reader will be able to apply the method for quasilinear systems in ordinary differential equations, partial differential equations, functional differential equations, hybrid systems, and various applications. One more interesting opportunity is to apply the method for chaos investigation. In this sense, we rely on our last results on unpredictable solutions [4, 5]. The unique feature of the book is that the mathematicians who are busy with qualitative theory of differential equations will learn the most convenient and short way for the chaos treatment. That is, they will find good samples how one can start the analysis of chaotic dynamics considering the material of the book as a basic one. Those who like to work on the interesting and fruitful application research of almost periodic dynamics will find nice opportunities related to differential equations with different types of discontinuity. The unified method is suggested in its complete form to consider almost periodicity for continuous and discontinuous components combined and arranged in convenient and constructive theoretical analysis way. These components are suitable for applications. We must say that the initial models for real-world problems were mostly considered as differential equations. The proper results about equilibriums, stability, periodic solution, etc., had been developed. Next, the models were extended by considering delays, impulses, non-smoothness, piecewise constant arguments, etc. Apparently, this type of extension of original models witnesses that their natural phenomena are confirmed by differential equations as models. That is equilibriums, periodic motions, and their stability can be kept if one changes the equations considering some kind of non-smoothness or deviation of arguments. Furthermore, definitely, these results, if not confirmed by immediate observations, give a strong belief to the development of experiments and research to satisfy

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obtained new mathematical observations, as it has been done already for other realworld problems. In this sense, our results will help to make some generalizations of previous well-known models and their investigations. The most likely target audience for this book would be those in engineering mathematics who have an interest in applying these systems to computational problems. Moreover, we are absolutely sure that enlarging the number of methods of analysis makes the general analysis deeper and recognition of this fact by the publication of our book will make the analysis faster. At the end of the preface, it is suitable to say that the line of periodic, almost periodic motions is prolonged in our research by unpredictable points and solutions [2–5]. Thus, we will continue in our next books serving for the chaos investigation as well as for introducing to the theory of differential equations the new type of solutions. Ultimately, we are aiming to the interesting problem, how to make what we call chaos the core for the modern theory of dynamical systems and how to shift the focus of the theory of differential equations to chaos analysis. Then, the modern dynamics theories will admit more theoretical challenges as well as become more attractive for applications. Next, we plan to consider more developed discontinuous almost periodicity. This will be done by removing certain restrictions on the connection between continuity and discontinuity. Concerning the asymptotic equivalence, it is difficult to give more strong push to the theory, than it is done in our papers [1, 6, 7]. In our research, we always try to keep a balance of complexity and applications. From this point of view, the results present in this book are sufficiently developed, and next extension will be done not to the depth of the theory but rather horizontally to different types of differential equations. The author would like to thank all those who helped him in his research. The following people were especially important in the writing of this book: Onur Fen, Michal Feckan, Madina Tleubergenova, Mokhtar Kirane, Roza Sejilova, Duygu Arugaslan, Ardak Kashkinbayev, Aysegul Kivilcim, and Mehmet Turan. Ankara, Turkey

Marat Akhmet

References 1. M.U. Akhmet, Asymptotic behavior of solutions of differential equations with piecewise constant arguments. Appl. Math. Lett. 21, 951–956 (2008) 2. M.U. Akhmet, M.O. Fen, Poincarè chaos and unpredictable functions. Commun. Nonlinear Sci. Numer. Simulat., 48, 85–94 (2016) 3. M.U. Akhmet, M.O. Fen, Unpredictable points and chaos. Commun. Nonlinear Sci. Numer. Simulat. 40, 1–5 (2016) 4. M.U. Akhmet, M.O. Fen, Existence of unpredictable solutions and chaos. Turk. J. Math. 41, 254-266 (2017)

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5. M.U. Akhmet, M.O. Fen, Non-autonomous equations with unpredictable solutions. Commun. Nonlinear Sci. Numer. Simulat. 59, 657–670 (2018) 6. M.U. Akhmet, M. Tleubergenova, On asymptotic equivalence of impulsive linear homogeneous differential systems. Math. J. 2(2), 15–18 (2002) 7. M.U. Akhmet, M. Tleubergenova, A. Zafer, Asymptotic equivalence of differential equations and asymptotically almost periodic solutions. Nonlinear Anal. Theory Methods Appl. 67, 1870–1877 (2007)

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Almost Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Piecewise Constant Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Asymptotic Behavior of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 5 13 22 33

2

Generalities for Impulsive Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Piecewise Continuous Functions and Their Points of Discontinuity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Description of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Equivalent Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Gronwall–Bellman Lemma for Piecewise Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Stability and Periodic Solutions of Systems with Fixed Moments of Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Basics of Periodic Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Basics of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Linear Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 The Adjoint System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Linear Exponentially Dichotomous Systems . . . . . . . . . . . . . . 2.5 Linear Non-homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 The Solution of Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 The Bounded Solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 43 44 44 44 46 49 49 49 50 52 52 54 58 59 61 61 62 64 67 xiii

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3

Discontinuous Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Unbounded Number Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Almost Periodic Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Derivative Almost Periodic Sequences . . . . . . . . . . . . . . . 3.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Discontinuous Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 The Basic Properties of Almost Periodic Functions . . . . . . 3.2.3 Invariance of Discontinuous Almost Periodicity . . . . . . . . . . 3.3 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 71 73 74 75 76 83 84

4

Discontinuous Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fundamental Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Linear Non-homogeneous Almost Periodic Systems . . . . . . . . . . . . . . . 4.3 Quasilinear Almost Periodic Impulsive Systems . . . . . . . . . . . . . . . . . . . 4.4 Systems with Periodic Homogeneous Parts . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 87 89 91 98 99

5

Bohr and Bochner Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bohr Discontinuous Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . 5.2 Bochner Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Amerio and Favard Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Mukhamadiev Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Recurrent and Limit Almost Periodic Solutions of Non-autonomous Impulsive Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103 104 109 112 114 118 120 120

Exponentially Dichotomous Linear Systems of Differential Equations with Piecewise Constant Argument . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Exponential Dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Non-homogeneous Linear System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 126 128 133 135 140

6

7

Differential Equations with Functional Response on Piecewise Constant Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Retarded Functional Differential Equations with Retarded Constancy of Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Retarded Functional Differential Equations with Alternate Constancy of Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Quasilinear Systems: Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 147 150 152

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7.5

Bounded Solutions of Quasilinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Further Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

156 160 162 168 171 174

Almost Periodic Solutions of Retarded SICNN with Functional Response on Piecewise Constant Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Existence and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Bounded Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

177 177 181 183 185 190 194 196 196

8

9

10

Differential Equations on Time Scales Through Impulsive Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Description of the DETC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 ψ-Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Reduction of the DETC (DETS) to Impulsive Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Homogeneous Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Non-homogeneous Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3 Linear Systems with Constant Coefficients . . . . . . . . . . . . . . . 9.6 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Description of Periodical DETC . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Floquet Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Massera Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Almost Periodic Solutions of Quasilinear Systems . . . . . . . . . . . . . . . . . 9.8 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost Periodicity in Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Chaos Through a Cascade of Almost Periodic Solutions . . . . . . . . . . 10.3.1 Extension of Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Cascade of Almost Periodic Solutions. . . . . . . . . . . . . . . . . . . . . 10.4 Li-Yorke Chaos with Infinitely Many Almost Periodic Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 204 205 207 207 209 210 212 212 214 216 217 220 221 223 223 226 226 228 228 229 231

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10.6 Control of Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 10.7 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 11

Homoclinic Chaos and Almost Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Bounded Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Li-Yorke Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

243 243 245 247 249 256 261 261

12

SICNN with Chaotic/Almost Periodic Postsynaptic Currents . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Exponential Decaying Input Currents (EDICs) . . . . . . . . . . . 12.2.2 Rectangular Input Currents (RICs) . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 The External Input, Lij (t), of the Cell Cij . . . . . . . . . . . . . . . 12.3 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1 Almost Periodicity of EDICs and RICs . . . . . . . . . . . . . . . . . . . 12.3.2 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Li-Yorke Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.1 The Li-Yorke Chaos in SICNNs . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 The Network with EDICs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 A Chaos Extension in SICNNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

265 265 269 269 270 271 273 273 275 279 280 286 287 289 295 296

13

Asymptotic Equivalence and Almost Periodic Solutions . . . . . . . . . . . . . . . 13.1 Introduction and Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Asymptotic Equivalence of Linear Systems and Asymptotically Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Asymptotic Equivalence of Linear and Quasilinear Systems . . . . . . 13.4 Bi-asymptotic Equivalence and Almost Periodic Solutions of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309 309

14

Asymptotic Equivalence of Hybrid Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 On Asymptotic Equivalence of Impulsive Linear Homogeneous Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.2 Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.3 Delay Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

312 315 318 319 320 323 323 324 325 326 328

Contents

Asymptotic Equivalence of the Quasilinear Impulsive Differential Equation and the Linear Ordinary Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Asymptotic Equivalence for EPCAG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Asymptotic Behavior of Linear Impulsive Integro-Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.1 Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Asymptotic Equilibriums. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.3 Asymptotic Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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329 329 330 332 333 334 340 340 342 349 353

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Chapter 1

Introduction

1.1 Almost Periodicity Our world is a collection of connected oscillators, and most developed in mathematics oscillations are almost periodic. The first functions which can be considered still as “periodic” and sufficiently determined for strict mathematical analysis were quasi-periodic functions introduced and investigated by Bohl [62, 63] and Esclangon [89] independently. Quasi-periodic motions encountered in celestial mechanics and radio techniques. As many of essential concepts of the dynamics, they were considered by Poincaré [155, 156]. In the manuscript [89] at the first time rather intuitive description of almost periodic functions, which cannot be covered by quasi-periodic ones, was mentioned. Later, the fundamental papers of Bohr [68– 71] provided the theory of almost periodic functions, which we call as Bohr almost periodic functions nowadays. Then different approaches to almost periodicity were found by Besicovitch [59], Bochner [66], Bogolyubov [61], Stepanov [188], and others. The book [97] is a good confirmation that almost periodic functions and their history are the most attractive objects of nonlinear dynamics for more than a hundred years. The almost periodic functions are of great importance for development of harmonic analysis on groups, Fourier series and integrals on groups. The paper [65] published by Bochner provided extension of the theory of almost periodic functions with values in a Banach space. The first paper on the existence of almost periodic solutions was written by Bohr and Neugebauer [71], and nowadays the theory of almost periodic equations has been developed in connection with problems of differential equations, stability theory, and dynamical systems. The list of the applications of the theory has been essentially extended, and includes not only ordinary differential equations and classical dynamical systems, but also wide classes of partial differential equations and equations in Banach spaces [51]. Thus, the concept of almost periodic functions proves its validity for the theory and applications of differential equations. This is why, it is important to give © Springer Nature Switzerland AG 2020 M. Akhmet, Almost Periodicity, Chaos, and Asymptotic Equivalence, Nonlinear Systems and Complexity 27, https://doi.org/10.1007/978-3-030-20572-0_1

1

2

1 Introduction

definitions of almost periodicity for hybrid systems, which combine continuous and discontinuous dynamics, and prove theorems on the existence of such kind of motions in differential equations with discontinuities of different types, impulses, piecewise constant arguments, discontinuous right-hand side, and dynamics on time scales. One of the main tasks of the present book is to provide comprehension of the universal role of almost periodicity and in this sense the hybrid systems are very useful. If one wants to avoid analysis of discontinuities in almost periodic functions, then can “hide” them through integrals, this can be done by applying either the Stepanov approach [108, 131] or the generalized functions [108, 183]. Another way to present non-continuous almost periodic functions is to describe oscillatory characteristics of moments of discontinuity. This approach has been started in papers by Wexler [195, 196] for linear equations and extended for the first time to nonlinear case in [38, 43, 48]. The results of Wexler were summarized in the book [108]. Let us start with the classical definition of the Bohr almost periodic function. Definition 1.1 ([93]) A continuous function F (t) is said to be almost periodic, if for any  > 0 the set T (F , ) = {ω : F (t + ω) − F (t) <  for all t ∈ R} is relatively dense, i.e., for any  > 0 there exists l > 0 such that for any interval with length l there exists a number ω in this interval satisfying F (t + ω) − F (t) <  for all t ∈ R. It is convenient to compare the last definition with another one for discontinuous functions. Let {ai }i∈Z be a sequence in Rn . An integer p is an -almost period of the sequence {ai }, if the inequality ||ai+p − ai || <  holds for all i ∈ Z. On the other hand, a set R ⊂ R is said to be relatively dense if there exists a number l > 0 such that [a, a + l] ∩ R = ∅ for all a ∈ R. Moreover, {ai } is almost periodic, if for any  > 0, there exists a relatively dense set of its -almost periods. Fix a sequence of real numbers ti , i ∈ Z, unbounded in both directions and j strictly increasing with respect to the index. Denote ti ≡ ti+j − ti , where i ∈ Z, j and obtain the sequence for each integer j. We will call ti , j ∈ Z, the derivative sequences of ti . We say that the sequence ti , i ∈ Z, admits θ -property if ti+1 − ti ≥ θ, i ∈ Z, for some positive θ. Consider a real-valued vector-function φ(t) = (φ1 (t), φ2 (t), . . . , φn (t)), which is determined on the real axis R. Assume that the function is left-continuous and the discontinuities at moments ti , i ∈ Z, are of the first kind. Definition 1.2 ([38, 43]) The function φ(t) is said to be discontinuous almost periodic (d.a.p.) if: j

(a) the sequences ti , j ∈ Z, are uniformly almost periodic; (b) for arbitrary positive  there exists a positive number δ such that |φ(t1 ) − φ(t2 )| <  if t1 and t2 are from the same interval of continuity and |t1 − t2 | < δ;

1.1 Almost Periodicity

3

(c) for arbitrary positive  there exists a respectively dense set T of -almost periods τ ∈ T such that φ(t + τ ) − φ(t)| <  if t ∈ R and |t − ti | >  for all integer i. We will call the property (b) in the last definition conditional uniform continuity of discontinuous functions or shortly, conditional uniform continuity. Apparently, the first results on the topic were published by Wexler [195, 196] and Halanay and Wexler [108]. Characteristics of the discontinuity points, which we call in this book “θ -property” and uniform almost periodicity of the “derivative” sequences, were introduced in the papers. Several interesting Lemmas 3.1–3.7 were proved for the sequences. Firstly, d.a.p. function was introduced in [108] as a solution of the impulsive system  dx = A(t)x + sk δ(t − tk ). dt

(1.1.1)

k

where the most important properties of d.a.p. functions were indicated, but not property (b). In this book we follow Definition 3.1 from [38], where the property (b) of conditional uniform continuity for d.a.p. functions was formulated for the first time. The definition was announced first time in [38, 43, 48]. The useful property of “diagonal” almost periodicity for the transition matrix of the homogeneous system, dx = A(t)x dt

(1.1.2)

where A(t) is a Bohr almost periodic matrix, was obtained in [108], if the system is uniformly asymptotically stable. In [38] the property was generalized (see Lemma 4.4) for the impulsive system dx = A(t)x, dt Δx|t=ti = Bi x,

(1.1.3)

with Bohr almost periodic matrix A(t) and almost periodic sequence of matrices Bi , and became the necessary instrument for almost periodicity investigation in impulsive systems. Another basics of the theory which were found in [38] are boundedness of discontinuous almost periodic solutions (Theorem 3.7), the lemma on common almost periods for d.a.p. functions (Lemma 3.8) and its application to prove invariance of the property with respect to summation, dot product, and division of functions (Theorems 3.10–3.12), the lemma on almost periodicity of the sequence of values of d.a.p. functions at discontinuity points (Lemma 3.9), the lemma on boundedness of function as criteria for its almost periodicity (Lemma 3.10, Corollary 3.2), the theorem on the average value (Theorem 3.13, Corollary 3.3).

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1 Introduction

Corollaries 3.4 and 3.5 are newly formulated in this book. They are not new ones, but for us it is important that the proof of them is much more shorter than in their original discussions, [168]. In [38] were proved and published in [43] the lemma on hybrid almost periods (Lemma 4.2), which became main instruments to investigate discontinuous almost periodic solutions for impulsive systems. Considering the list of the definitions, lemmas, and theorems mentioned above one can say that in the thesis [38] fundamentals of theory of discontinuous almost periodic functions and discontinuous almost periodic solutions of impulsive systems were laid. To ensure the reader in the truth of the assertion it is sufficient to say that more than 200 articles and books have been published on the subject according to Mathscinet. Among them [23, 24, 31, 72, 78, 135, 139, 154, 180, 184–187, 193, 203, 206] and many others. The results obtained in [38] were published in [43, 168] and [167]. The preprint [48] can be considered as an irregular publication of the results. It deserves to mention that the property of conditional uniform continuity is assumed in Definition 1.2 as the condition (b). In the classical H. Bohr’s theory the property of uniform continuity on the axis is not included in the definition, but it is a consequence of the definition and is formulated as a theorem. This is why, it is of big theoretical interest to provide theory of discontinuous almost periodic functions in the deep parallel form with the Bohr’s constructions. Initial steps, and it seems still last steps, in the direction were done in our paper [44], where on the basis of a special topology in the set of piecewise continuous functions a distance was introduced to determine definition of discontinuous almost periodic functions only by using the distance for shifts of functions along the axis. Thus, the properties of conditional uniform continuity and even uniform almost periodicity j for the derivative sequences ti became a subject of lemmas next to the definition. For the reader who are interested in this way for the development of the theory we recommend Chap. 5 of this book. In the same chapter we are concentrated on the Bochner definition of almost periodic function, [64–66], realized for the discontinuous case. First time this was done in our paper [41] and some elements can be found in [44]. The Bochner definition makes possible to look at almost periodic solutions of differential equations in the context of topological dynamics [145, 146, 175]. Initially, topological dynamics was applied either to autonomous equations or to non-autonomous periodic systems through the Poincaré map. It was the research by Miller [145] and Millionshchikov [146] which demonstrated how non-autonomous equations could be embedded in dynamical systems. The discontinuous almost periodic functions are object for intensive investigations considering them as solutions of impulsive differential equations as well as impulsive systems with piecewise constant arguments, time scales, deviated arguments, and others [6, 11, 14, 22, 24, 37–42, 46, 48, 76, 107, 117, 134, 139, 185]. We also want to attract attention of the readers to the functional differential equations with functional response on the piecewise constant argument [35]. Possibly, it is natural that the large number of papers on applications concerns neural networks [27, 35, 154, 185, 186], since the neural networks are most flexible models for real-world problems for the different types of motions as they

1.2 Chaos

5

are not restricted with fundamental lows in the modeling as, for example, the Newton’s second law of mechanics. Analogously, the almost periodic oscillations are proper for biology [53, 135, 201, 203, 204, 206]. Through introducing, in our papers [3–6, 11, 13] generalized piecewise constant argument it became possible to consider the continuous almost periodic solutions with discontinuous derivatives, such that the sequence of points of discontinuity admits almost periodic derivative sequences. This is why, the theory of discontinuous almost periodic functions is intensively applicable to the differential equations with piecewise constant argument [5, 6, 11, 14]. These results can be widely extended to differential equations and functional differential equations with functional response to the piecewise constant argument [15, 36, 45]. The first result concerning almost periodic solutions to dynamics on time scales was proved in paper [24], where the proof is based on the reduction of the original equation to an almost periodic impulsive differential equation. One can remark that many other results of the theory of impulsive systems can be analogously replicated for dynamics on time scales. The pioneering results in the direction were published in [24, 25]. Time scales in the papers are unions of isolated sections. This is why more general approach can be found in papers [132–134]. Nevertheless, if consider examples, the authors use time scales as unions of separated sections. This is the reason to say that in the future investigations, when analysis of applicable to real world problems models will be needed, researches again will pay attention to the uniformly almost periodic derivative sequences and correspondingly to the our approach to almost periodicity of time scales. Moreover, it seems that even in the theoretical sense the almost periodicity of time scales still needed to be analyzed more deeply. Then, one can consider the universal nature of the results connected to the sequences of discontinuity. We strongly believe that there are more relations than already exist between theory of differential equations and chaos investigation and realized this confidence in our papers. This time we discuss almost periodic motions present in chaotic attractors. This is a very interesting phenomenon, and we will describe its peculiarities in the next section. Another interesting subject to investigate is asymptotic presentation of almost periodic motions, and it will be presented in Sect. 1.3 of this Introduction.

1.2 Chaos The theory of dynamical systems starts with H. Poincaré, who studied nonlinear differential equations by introducing qualitative techniques to discuss the global properties of solutions [85]. His discovery of the homoclinic orbits figures prominently in the studies of chaotic dynamical systems. Poincaré first encountered the presence of homoclinic orbits in the three body problem of celestial mechanics [52]. A Poincaré homoclinic orbit is an orbit of intersection of the stable and unstable manifolds of a saddle periodic orbit. It is called structurally stable if the intersection

6

1 Introduction

is transverse, and structurally unstable or a homoclinic tangency if the invariant manifolds are tangent along the orbit [98]. In any neighborhood of a structurally stable Poincaré homoclinic orbit there exist nontrivial hyperbolic sets containing a countable number of saddle periodic orbits and continuum of non-periodic Poisson stable orbits [98, 177, 181]. For this reason, the presence of a structurally stable Poincaré homoclinic orbit can be considered as a criterion for the presence of complex dynamics [98]. The first mathematically rigorous definition of chaos is introduced by Li and Yorke [136] for one dimensional difference equations. According to [136], a continuous map F : J → J, where J ⊂ R is an interval, exhibits chaos if: (1) For every natural number p, there exists a p-periodic point of F in J ; (2) There is an uncountable set S ⊂ J containing no periodic points such  that for  every s1 , s2 ∈ S with s1 = s2 we have lim supk→∞ F k (s1 ) − F k (s2 ) > 0 and lim infk→∞ F k (s1 ) − F k (s2 ) = 0; (3)  For every s ∈ S and periodic point σ ∈ J we have lim supk→∞ F k (s) − F k (σ ) > 0. In the paper [136], it was proved that if a map on an interval has a point of period three, then it is chaotic. Generalizations of Li-Yorke chaos to high dimensional difference equations were provided in [49, 125, 143]. According to Marotto [143], if a repelling fixed point of a differentiable map has an associated homoclinic orbit that is transversal in some sense, then the map must exhibit chaotic behavior. More precisely, if a multidimensional differentiable map has a snap-back repeller, then it is chaotic. In the paper [137], Marotto’s theorem was used to prove rigorously the existence of Li-Yorke chaos in a spatiotemporal chaotic system. Furthermore, the notion of LiYorke sensitivity, which links the Li-Yorke chaos with the notion of sensitivity, was studied in [49], and generalizations of Li-Yorke chaos to mappings in Banach spaces and complete metric spaces were considered in [125]. Another mathematical definition of chaos for discrete-time dynamics was introduced by Devaney [85]. It is mentioned in [85] that a map F : J → J, where J ⊂ R is an interval, has sensitive dependence on initial conditions if there exists δ > 0 such that for any x ∈ J and any neighborhood N of x there exists y ∈ J and a  positive integer k such that F k (x) − F k (y) > δ. On the other hand, F is said to be topologically transitive if for any pair of open sets U, V ⊂ J there exists a positive integer k such that F k (U ) ∩ V = ∅. According to Devaney, a map F : J → J is chaotic on J if: (1) F has sensitive dependence on initial conditions; (2) F is topologically transitive; (3) Periodic points of F are dense in J. In other words, a chaotic map possesses three ingredients: unpredictability, indecomposability, and an element of regularity. Symbolic dynamics, whose earliest examples were constructed by Hadamard [105] and Morse [147], is one of the oldest techniques for the study of chaos. Symbolic dynamical systems are systems whose phase space consists of one-sided or two-sided infinite sequences of symbols chosen from a finite alphabet. Such dynamics arises in a variety of situations such as in horseshoe maps and the logistic map. The set of allowed sequences is invariant under the shift map, which is the most important ingredient in symbolic dynamics [85, 100, 110, 124, 198, 199].

1.2 Chaos

7

Moreover, it is known that the symbolic dynamics admits the chaos in the sense of both Devaney and Li-Yorke [9, 10, 20, 85, 163]. The Smale horseshoe map is first studied by Smale [182] and it is an example of a diffeomorphism which is structurally stable and possesses a chaotic invariant set [85, 124, 199]. The horseshoe arises whenever one has transverse homoclinic orbits, as in the case of the Duffing equation [101]. People used the symbolic dynamics to discover chaos, but we suppose that it can serve as an “embryo” for the morphogenesis of chaos. From the mathematical point of view, chaotic systems are characterized by local instability and uniform boundedness of the trajectories. Since local instability of a linear system implies unboundedness of its solutions, chaotic system should be necessarily nonlinear [94]. Chaos in dynamical systems is commonly associated with the notion of a strange attractor, which is an attractive limit set with a complicated structure of orbit behavior. This term was introduced by Ruelle and Takens [166] in the sense where the word strange means the limit set has a fractal structure [98]. The dynamics of chaotic systems are sensitive to small perturbations of initial conditions. This means that if we take two close but different points in the phase space and follow their evolution, then we see that the two phase trajectories starting from these points eventually diverge [85, 104]. The sensitive dependence on the initial condition is used both to stabilize the chaotic behavior in periodic orbits and to direct trajectories to a desired state [173]. It was Lorenz [141] who discovered that the dynamics of an infinite dimensional system being reduced to three dimensional equation can be next analyzed in its chaotic appearances by application of the simple unimodal one dimensional map. Smale [182] explained that the geometry of the horseshoe map is underneath of the Van der Pol equation’s complex dynamics which was investigated by Cartwright and Littlewood [75] and later by Levinson [130]. Nowadays, the Smale horseshoes with its chaotic dynamics are one of the basic instruments when one tries to recognize a chaos in a process. Guckenheimer and Williams [103] gave a geometric description of the flow of Lorenz attractor to show the structural stability of codimension 2. In addition to this, it was found out that the topology of the Lorenz attractor is considerably more complicated than the topology of the horseshoe [101]. It is natural to discover a chaos [79, 80, 116, 136, 141, 143, 157, 164–166, 170] and proceed by producing basic definitions and creating the theory. On the other hand, one can shape an irregular process by inserting chaotic elements in a system which has regular dynamics (let us say comprising an asymptotically stable equilibrium, a global attractor, etc.). This approach to the problem also deserves consideration as it may allow for a more rigorous treatment of the phenomenon, and helps to develop new methods of investigation. Our results are of this type. In this book, we use the idea that chaos can be utilized as input in systems of equations. To explain the input–output procedure realized in our book, let us introduce examples of systems called as the base-system, the replicator, and the generator, which will be intensively used in the manuscript. Consider the following system of differential equations,

8

1 Introduction

dz = B(z). dt

(1.2.4)

The system (1.2.4) is called the base-system. We assume that the system admits a regular property. For example, there is a globally asymptotically stable equilibrium of (1.2.4). Next we apply to the system a perturbation, I (t), which will be called an input and obtain the following system: dy = B(y) + I (t), dt

(1.2.5)

which will be called as the replicator. Suppose that the input I admits a certain property, let us say, it is a bounded function. We assume then that there exists a unique solution, y(t), of the last equation, the replicator, with the same property of boundedness. This solution is considered as an output. The process for obtaining the solution y(t) of the replicator system by applying perturbation I (t) to the base-system (1.2.4) is called the input– output mechanism, and sometimes we shall call it the machinery. It is known that for certain base-systems, if the input is periodic, almost periodic, bounded, then there exists an output, which is also periodic, almost periodic, bounded, respectively. In our book, we consider inputs of the new nature: chaotic sets and chaotic functions. The motions which are in the chaotic attractor of the Lorenz system considered altogether provide us an example of a chaotic set of functions. Any element of this set is considered as a chaotic function. Both of these types of inputs will be used effectively. For example, to prove rigorously, by verification of all ingredients, that there exists a certain type of chaos generated by the input–output procedure, we use the concept of the chaotic set. For simulations we shall use inputs in the form of chaotic functions. The main sources of chaos in theory are difference and differential equations. For this reason we consider inputs which are solutions of some systems of differential equations or discrete equations. These systems will be called generators in this book. Thus, we can consider the following system of differential equations: dx = G(x), dt

(1.2.6)

and it is assumed that this system possesses chaos. We shall call this system a generator. If x(t) is a solution of the system from the chaotic attractor, that is, it is a chaotic solution, then we notate I (t) = x(t) and use the function I (t) in Eq. (1.2.5). Types of Eqs. (1.2.4), (1.2.5), and (1.2.6) considered as a base-system, replicator system, and a generator system, respectively, can be varied in the future. For example, the systems may be non-autonomous and an input may be involved nonlinearly.

1.2 Chaos

9

In our theoretical results of chaos extension, we use coupled systems in which the generator influences the replicator in a unidirectional way, that is, the generator affects the behavior of the replicator, but not the converse. The possibility of making use of more than one replicators and nonidentical systems in the machinery is an advantage of the procedure. On the other hand, contrary to the method that we present, in the synchronization of chaotic systems, one does not consider the type of the chaos that the master and slave systems admit. The problem that whether the synchronization of systems implies the same type of chaos for both master and slave has not been taken into account yet. To have a comprehensible discussion in this introduction, let us give an outline of a consequence of our results for collectives of systems. Suppose that there is a system, S1 , which is autonomous and possesses chaos. That is, a chaotic attractor of the system exists and the presence of chaos is proved by applying one of the definitions of chaos: Li-Yorke chaos, Devaney chaos, chaos through period-doubling cascade and sensitivity, etc. We call S1 as the generator system (generator of chaos). Assume that there are other systems, S2 , S3 , . . . , Sn , which are all interconnected in the unidirectional fashion. In Fig. 1.2, an example of the connection for the case n = 16 is depicted. The connection’s nature is very simple. Solutions of system S1 are utilized as perturbations for systems S2 , S5 , S7 , and S11 . Next, solutions of the perturbed systems are utilized to perturb other adjoint systems and so on. That is, in their own turn systems S2 , S5 , S7 , and S11 become generator systems for all others, except S1 , etc. So, these connections will continue, while all the systems are connected in the net, which is seen in Fig. 1.2. Systems Si , i > 1, admit globally stable equilibriums, if they are isolated (unperturbed). In this unconnected state, they are called base-systems. It implies from our results that when the connections are valid, they are all chaotic under certain conditions. Thus, all of the cells shown in Fig. 1.1 are chaotic such that the whole system S, the union of subsystems, Si , i = 1, 2, . . . , 16, is chaotic with the same type of chaos as S1 . Thus, in what follows we shall refer to the collection S of the chaotified systems to illustrate concepts of the discussion. The idea of chaos control is based on the fact that chaotic attractors have a skeleton made of an infinite number of unstable periodic orbits [85, 99, 104, 123, 172]. Stability can be described as the ability of a system to keep itself working properly even when perturbations act on it, and this is the main goal to be achieved Fig. 1.1 Collection of chaotified systems

10

1 Introduction

Fig. 1.2 Chaos extension mechanism in the net

by the control strategy that is embedded in the system [172]. In other words, the aim of chaos control is to stabilize a previously chosen unstable periodic orbit by means of small perturbations applied to the system, so the chaotic dynamics is substituted by a periodic one chosen at will among the several available [99]. That is, when control is present, a chaotic trajectory transforms into a periodic one [94]. Experimental demonstrations of chaos control methods were presented in the papers [54, 56, 60, 87, 96, 114, 144, 171]. Small perturbations applied to control parameters can be used to stabilize chaos, keeping the parameters in the neighborhood of their nominal values, and this idea is first introduced by Ott et al. [152]. Experimental applications of the OGY control method require a permanent computer analysis of the state of the system. The method deals with a Poincaré map and therefore, the parameter changes are discrete in time. Using this method, one can stabilize only those periodic orbits whose maximal Lyapunov exponent is small compared to the reciprocal of the time interval between parameter changes [159]. Another control method has been developed by Pyragas [159] to stabilize unstable periodic orbits applying small time continuous control to a parameter of a system while it evolves in continuous time, instead of a discrete control at the crossing of a surface [99]. Pyragas control method uses a delayed feedback employing a suitably amplified difference of an output measurement of the chaotic system and the respectively delayed measurement for control. The control signal vanishes in the post-transient behavior for the stabilized

1.2 Chaos

11

orbit. For this reason, the delay time has to be the exact value of the period of the unstable periodic orbit that will be stabilized [119]. Both of the OGY and Pyragas control methods will be utilized throughout the book. Scientists are interested in the chaos theory because of the fact that it can offer new controlling strategies which have some particularly interesting insights for economic policies. There was opinion among economists that dynamics of chaos is neither predictable nor controllable due to sensitivity. Results of Ott et al. [152] showed that control of a chaos can be made by very small corrections of parameters [99, 120]. This achievement has been widely used in economics by Kaas et al. [50, 121, 122, 126] and many others. Our results demonstrate that the control may not be local (applied to an isolated model) but a global phenomenon with strong effectiveness. A control applied to a model, which is realizable easily (for example, the logistic map or Feichtinger’s generic model [91]), can be sufficient to rule the process in all models joined with the controlled one. Another benefit of our studies is that in the literature controls are applied to those systems which are simple and low-dimensional. It is worth noting that control of chaos (unstable periodic motions) becomes difficult if dimensions of systems increase and the construction of Poincarè sections is complexified. For this reason, the idea to control the generated chaos by controlling the exogenous shocks is useful for applications. In the present book, the control of an economic system through the application of the OGY control to the logistic map is demonstrated. A chaos control cannot be realized if we do not know the period of unstable the motion to be controlled. In our case, the control is applicable to models with arbitrary dimension if just the basic period of the generator is known. It is obvious that our methods provide us a scheme of investigations, which can be accompanied with detailed studies in the future. Control of chaos is a synonym to the suppression of chaos nowadays. Thus, our results give another way of suppression of chaos. If we find the controllable link (member) in a chain (collection) of connected chaotic systems, then we can suppress chaos in the whole chain. This is an effective consequence of our studies. The chaos phenomenon has been observed in the dynamics of neural networks [1, 2, 95, 102, 128, 148, 149, 158, 176, 179, 189, 191], and chaotic dynamics applying as external inputs are useful for separating image segments [176], information processing [148, 149], and synchronization of neural networks [138, 142, 205]. Aihara et al. [2] proposed a model of a single neuron with chaotic dynamics by considering graded responses, relative refractoriness, and spatiotemporal summation of inputs. Chaotic solutions of both the single chaotic neuron and the chaotic neural network composed of such neurons were demonstrated numerically in [2]. Focusing on the model proposed in [2], dynamical properties of a chaotic neural network in chaotic wandering state were studied concerning sensitivity to external inputs in [128]. On the other hand, in the paper [176], Aihara’s chaotic neuron model is used as the fundamental model of elements in a network, and the synchronization characteristics in response to external inputs in a coupled lattice based on a Newman–Watts model are investigated. Besides, in the studies [148, 149], a network consisting of binary neurons which do not display chaotic behavior is considered, and by means of the reduction of synaptic connectivities it is shown that the state

12

1 Introduction

of the network in which cycle memories are embedded reveals chaotic wandering among memory attractor basins. Moreover, it is mentioned that chaotic wandering among memories is considerably intermittent. Chaotic solutions to the Hodgkin– Huxley equations with periodic forcing have been discovered in [1]. The paper [102] indicates the existence of chaotic solutions in the Hodgkin–Huxley model with its original parameters. An analytical proof for the existence of chaos through period-doubling cascade in a discrete-time neural network is given in [191], and the problem of creating a robust chaotic neural network is handled in [158]. Generally speaking, it is recognized that chaos is a friend of mind. Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [85], Li and Yorke [136], and the one obtained through period-doubling cascade [92]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [98, 177, 181]. It was certified by Shilnikov [178] and Seifert [174] that it is possible to replace periodic solutions by Poisson stable or almost periodic motions in a chaotic attractor. Despite the fact that the idea of replacing periodic solutions by other types of regular motions is attractive, very few results have been obtained on the subject. The present study contributes to the chaos theory in that direction. In this book, we take into account chaos both through a cascade of almost periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke definition is modified by replacing infinitely many periodic motions with almost periodic ones, which are separated from the motions of the scrambled set. The theoretical results are valid for systems with arbitrary high dimensions. Formation of the chaos is exemplified by means of unidirectionally coupled Duffing oscillators. The controllability of the extended chaos is demonstrated numerically by means of the Ott et al. [152] control technique. In particular, the stabilization of tori is illustrated. In Chap. 11, we consider a system with a homoclinic solution under the influence of a chaotic forcing term. We have vigorously proved that a Li-Yorke chaotic perturbation of a system with a homoclinic orbit creates chaos along each periodic trajectory. To emphasize the role of the homoclinic solution in the book, we call the dynamics Li-Yorke homoclinic chaos. The structure of the chaos is investigated, and the existence of infinitely many almost periodic orbits out of the scrambled sets is revealed. Ott et al. and Pyragas control methods are utilized to stabilize almost periodic motions. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Results of Chap. 11 are of a significant interest due to the theoretical importance and perspectives for applications. This is the first time in the literature that chaos is obtained as a union of infinitely many sets of chaotic motions for a single equation by means of a perturbation.

1.3 Piecewise Constant Argument

13

1.3 Piecewise Constant Argument A new class of differential equations (EP CAG) was introduced in [3], and then developed in many papers and books [4–6, 8, 11–19, 26, 27, 30, 33, 34]. They contain as a subclass differential equations with piecewise constant argument (EPCA). For basics of the theory, one can find in the book [12]. In this Introduction we pay attention to the most fresh investigations in the theory. The first one is exponentially dichotomous systems [14] and another one are functional differential equations with functional response on the piecewise constant argument [15, 16]. The carefully described concepts provide a first glance at what we understand as differential equations with piecewise constant argument. More results as well as applications can be found in Chaps. 6–8, where we discuss mainly almost periodicity for different types of EPCAGs. The results of the chapters demonstrate one more time that differential equations with different types of discontinuity are naturally connected with each other, and the almost periodicity is one of the best concepts for this. Differential equations with piecewise constant argument (EPCA) were proposed for investigations by founders of theory and many others, the traditional method has been effectively applied to various interesting problems of differential equations and their applications. The traditional method means that the constant argument is assumed to be a multiple of the greatest integer function, and analysis is made on the basis of reduction to discrete equations. In fact, the simple type of the constancy and the method of investigation strongly relate to each other, since, if the argument is a multiple of the greatest integer function, then one can respectively easily reduce the linear system to a discrete equation. In paper [3], we not only generalized the functions of the piecewise constant argument, but, what is most important, proposed to investigate the newly introduced systems by reduction them to integral equations. This innovation became very effective, and there are two main reasons for that. At first, now, it is possible to investigate systems, which are essentially nonlinear, or more precisely, nonlinear with respect to values of solution at the discrete moments of time. While the main and unique method of analysis for EPCA is the reduction to discrete equation and, hence, only those equations are considered, where values of solutions at discrete moments appear linearly [197]. Moreover, one can, now, analyze existence and stability of solutions not only for initial moments, which coincide with discontinuity moments, but arbitrarily for chosen real numbers. Thus, we have deepened the analysis insight significantly. Further our proposals were used in papers [4–6, 8, 11– 16, 18, 19, 26, 30, 33, 34], and [153, 154], not only from theoretical point of view, but also from applications [32, 33, 57]. Despite our proposals were delivered in several papers, and even the books [12, 21, 27], the advantages, which can provide the integral presentations for solutions, have not been used properly yet. So, in the present book, we try to give more lights on the subject. Exponential dichotomy for EPCAG is considered at first time, if the associated linear homogeneous system is EPCAG. Existence of bounded solutions,

14

1 Introduction

Fig. 1.3 The graph of the argument

x x=t

x = β( t)

θ

i-2

θ

i-1

θ

i

θ

i+1

θ

t i+2

its stability, almost periodic, periodic solutions are under discussion. We do hope that our presentations are so clear that many questions, which relate to linear and quasilinear systems, will be solved later, by using results of Chaps. 6–8 of this book. We start with piecewise constant argument functions. Let θi , i ∈ Z be a strictly ordered sequence of real numbers such that |θi | → ∞ as |i| → ∞. We define the argument function as β(t) = θi if θi ≤ t < θi+1 , i ∈ Z. The greatest integer function [t], which is equal to the maximal among all integers less than t, is a β(t) function with θi = i, i ∈ Z. Similarly, β(t) = 2[t/2] if θi = 2i, i ∈ Z. One can see the graph of a β-type function in Fig. 1.3. Let two real-valued sequences θi , ζi , i ∈ Z, be defined such that θi < θi+1 , θi ≤ ζi ≤ θi+1 for all i ∈ Z, |θi | → ∞ as |i| → ∞. The argument function γ (t) is defined by γ (t) = ζi , if θi ≤ t < θi+1 , i ∈ Z. One can easily find, for example, that 2[ t+1 2 ] is γ (t) function with θi = 2i − 1, ζi = 2i. In Fig. 1.4 the typical graph of γ (t) function is seen. Finally, we say that a function is of χ -type, and denote it χ (t), if ζi = θi+1 , i ∈ Z. The function [t + 1] is a good example of χ (t) function with θi = i, i ∈ Z. We shall consider the following equation [45, 47]: z (t) = A0 (t)z(t) + A1 (t)z(γ (t)),

(1.3.7)

where z ∈ Rn , t ∈ R, γ (t) = ζi , if t ∈ [θi , θi+1 ), i ∈ Z. We assume that the coefficients A0 (t), A1 (t) are continuous on R, n × n, realvalued matrices. One can easily see that Eq. (1.3.7) has the form of functional differential equation:

1.3 Piecewise Constant Argument

15

Fig. 1.4 The graph of the argument γ (t)

x x=t

x = γ( t )

θ

i-2

θ

i-1

θ

i

θ

z (t) = A0 (t)z(t) + A1 (t)z(ζi ),

i+1

θ

t i+2

(1.3.8)

if t ∈ [θi , θi+1 ), i ∈ Z. That is, the system has the structure of a continuous dynamical system within the intervals [θi , θi+1 ), i ∈ Z. In our book we assume that the solutions of EP CAGs are continuous functions. But the deviating function γ (t) is discontinuous. Hence, in general, the right-hand side of (1.3.7) has discontinuities at moments θi , i ∈ Z. Summarizing, we consider the solutions of the equation as functions, which are continuous and continuously differentiable within intervals [θi , θi+1 ), i ∈ Z. We use the following definition, which is a version of a definition from [197], modified for our general case. Definition 1.3 ([8, 12]) A continuous function z(t) is a solution of (1.3.7) on R if: (i) the derivative z (t) exists at each point t ∈ R with the possible exception of the points θi , i ∈ Z, where the one-sided derivatives exist; (ii) the equation is satisfied for z(t) on each interval (θi , θi+1 ), i ∈ Z, and it holds for the right derivative of z(t) at the points θi , i ∈ Z. Let I be the n × n identity matrix. Denote by X(t, s), X(s, s) = I, s ∈ R, the fundamental matrix of solutions of the system x (t) = A0 (t)x(t),

(1.3.9)

16

1 Introduction

which is associated with system (1.3.7). We introduce a matrix function Mi (t), i ∈ Z, [45, 47],  Mi (t) = X(t, ζi ) +

t

X(t, s)A1 (s)ds, ζi

useful in what follows. From now on we make the assumption: • For every fixed i ∈ Z, det[Mi (t)] = 0, ∀t ∈ [θi , θi+1 ]. We shall call this property, the regularity condition. Theorem 1.1 ([12]) For every (t0 , z0 ) ∈ R × Rn there exists a unique solution z(t) = z(t, t0 , z0 ) of (1.3.7) in the sense of Definition 1.3 such that z(t0 ) = z0 if and only if the regularity condition is valid. The last theorem arranges the correspondence between points (t0 , z0 ) ∈ R × Rn and the solutions of (1.3.7) in the sense of Definition 1.3, and there exists no solution of the equation out of the correspondence. Using this assertion we can say that the definition of the IVP for EPCAG is similar to the problem for ordinary differential equations. Particularly, the dimension of the space of all solutions is n. Hence, the investigation of problems considered in our paper does not need to be supported by results of theory of functional differential equations [109], despite the fact EPCAG are equations with deviated arguments. System (1.3.7) is a differential equation with a delay argument. That is why it is reasonable to suppose that the initial “interval” must consist of more than one point. The following arguments show that in our case we need only one initial moment. Indeed, assume that (t0 , z0 ) is fixed, and θi ≤ t0 < θi+1 for a fixed i ∈ Z. We suppose that t0 = ζi . The solution satisfies, on the interval [θi , θi+1 ], the following functional differential equation z (t) = A0 (t)z + A1 (t)z(ζi ).

(1.3.10)

Formally we need the pair of initial points (t0 , z0 ) and (ζi , z(ζi )) to proceed with the solution, but since z0 = Mi (t0 )z(ζi ), where matrix Mi (t0 ) is nonsingular, we can say that the initial condition z(t0 ) = z0 is sufficient to define the solution. Theorem 1.1 implies that the set of the solutions of (1.3.7) is an n-dimensional linear space. Hence, for a fixed t0 ∈ R there exists a fundamental matrix of solutions of (13.1.1), Z(t) = Z(t, t0 ), Z(t0 , t0 ) = I such that dZ = A0 (t)Z(t) + A1 (t)Z(γ (t)). dt

1.3 Piecewise Constant Argument

17

Without loss of generality, assume that θi < t0 < ζi for a fixed i ∈ Z, and define the matrix for increasing t [12],

Z(t) = Ml (t)

i+1 

 Mk−1 (θk )Mk−1 (θk ) Mi−1 (t0 ),

(1.3.11)

k=l

if t ∈ [θl , θl+1 ], for arbitrary l > i. Similarly, if θj ≤ t ≤ θj +1 < . . . < θi ≤ t0 ≤ θi+1 , then Z(t) = Mj (t)

i−1 

 Mk−1 (θk+1 )Mk+1 (θk+1 ) Mi−1 (t0 ).

(1.3.12)

k=j

One can easily see that Z(t, s) = Z(t)Z −1 (s), t, s ∈ R,

(1.3.13)

and a solution z(t), z(t0 ) = z0 , (t0 , z0 ) ∈ R × Rn , of (13.1.1) is equal to z(t) = Z(t, t0 )z0 , t ∈ R.

(1.3.14)

The last formulas (1.3.11)–(1.3.14) have been obtained in [7], and are of exceptional importance for our theory of linear systems. It is well known that linear systems with constant coefficients are the source for all concepts of general linear systems theory. We cannot provide an analog for EPCAG with the same simplicity level, since linear EPCAG are not with constant coefficients. They are even not smooth. In these circumstances, we suppose that the best candidate to be the main and most simple one to illustrate concepts of our theory is a periodic linear system. So, we propose to consider a periodic system (13.1.1) as an analog of a linear system with constant matrix of coefficients for ordinary differential equations. Let us describe the system in detail in the next example. Example 1.1 ([14]) Assume that there are two numbers, ω ∈ R, p ∈ Z,such that p θk+p = θk + ω, ζk+p = ζk + ω, k ∈ Z. Then denote by Q the product k=1 Gk , −1 where matrices Gk are equal to Mk (θk )Mk−1 (θk ), k ∈ Z. We call the matrix Q, the monodromy matrix, and eigenvalues of the matrix, ρj , j = 1, 2, . . . , n, multipliers. Let us have some benefits from these definitions. From conditions stated above it follows that, if all multipliers are less than one in absolute value, then there exist positive numbers, R ≥ 1, α, such that Z(t, s) ≤ Re−α(t−s) , t ≥ s. More exactly, if β = ω1 maxi ln |ρi |, then for arbitrary  > 0, there exists a number R() ≥ 1, such that Z(t, s) ≤ R()e−(β+)(t−s) , t ≥ s. It is obvious that with |ρi | = 1, for all i, we have the so-called a hyperbolic homogeneous equation. It is recommended to

18

1 Introduction

the reader to discuss the problem of the exponential growth and decay for solutions of a hyperbolic system, depending on the multipliers. The last example shows that for (1.3.7) one can introduce the concept of exponential dichotomous system. We say that system (1.3.7) satisfies exponential dichotomy [14], if there exists a projection P and positive constants σ1 , σ2 , K1 , K2 , such that ||Z(t)P Z −1 (s)|| ≤ K1 exp(−σ1 (t − s)), t ≥ s, ||Z(t)(I − P )Z −1 (s)|| ≤ K2 exp(σ2 (t − s)), t ≤ s. It is not an easy task to provide examples of exponentially dichotomous systems, since there are no EPCAG systems with constant coefficients, generally. Of course, if one excepts the primitive case, when two subsystems are completely separated. Nevertheless, we can provide in our paper an advanced example, considering a periodic system. More precisely, one can say that periodic systems are analogs, in their simplicity, of the linear system with constant matrix of coefficients for ordinary differential equations. So, let us pay attention to the periodic system that has been discussed above. Assume that the monodromy matrix, Q, admits no multipliers on the unit circle. More exactly, suppose that k multipliers are inside and n − k of them outside of the unit disc. It is easy to see that there exists a k-dimensional subspace of solutions tending to zero uniformly and exponentially as t → ∞. As well as there exits an n − k-dimensional subspace of solutions tending to infinity uniformly and exponentially as t → ∞. On the basis of these observations we can repeat discussions of [83, pp. 10–12], which have been made for linear homogeneous systems with variable bounded matrices of coefficients, to prove that our periodic system is exponentially dichotomous. The simple case, when k = n, has been considered above in Example 1.1. Example 1.2 ([14]) Consider sequences of scalars bi , θi , i ∈ Z, which satisfy bi+p = bi , θi+p = θi + ω, i ∈ Z, for some positive ω ∈ R, p ∈ Z. Define the following EPCAG: x = bx(γ (t)), where γ (t) = θi if θ ≤ t < θi+1 . One can find that Q =

(1.3.15) 1 

[1+b(θi −θi−1 )]. Let us

i=p

give some analysis by using the last expression. It is seen that the zero solution of the equation is uniformly exponentially stable if, for example, −1 < 1 + b(θi − θi−1 ) < −2 1, i = 1, 2, . . . , p. That is, if θi −θ < b < 0, for all i. i−1

1.3 Piecewise Constant Argument

19

Similarly, if γ (t) = θi+1 , if θ ≤ t < θi+1 , then Q =

1 

[1 + b(θi+1 −

i=p

θi )]−1 . Then, the equation is uniformly exponentially stable, if b > 0 or b < −2 θi −θi−1 , for all i. Thus, we obtain that the simple Malthus model admits decaying solutions, even with positive coefficient, if the piecewise constant argument with anticipation is inserted. This fact certainly requests a biological conceive. Let us finalize the study with exact estimations. Assume that |Q| < 1, and denote −α = ω1 ln |Q|. Moreover, set R = maxi {max{maxt |Mi (t)|, maxt |Mi−1 (t)|}}. Then, |x(t, t0 , x0 )| ≤ Re−α(t−t) |x0 |, for arbitrary solution of the equation. Example 1.3 ([14]) Consider the following system:

z (t) =



01 00 z+ z([t]). 00 q0

(1.3.16)

We can see that θi = i, ζi = θi+1 = i + 1, i ∈ Z, for this system. One can find that

1 + q2 1 −1 Q = Gi ≡ Mi (θi )Mi (θi+1 ) = . q 1 Denote by ρj , j = 1, 2, eigenvalues of matrix Q. They are multipliers of system (1.3.16). From (1.3.11), it implies that the zero solution of (1.3.16) is exponentially stable, if and only if absolute values of both multiplies less than one. We find that  ρ1,2

q =1+ ± 4

1+

q q2 − , 16 2

and |ρ1 | > 1 for all q ∈ R. The second multiplier is less than one in absolute value, if − 43 < q < 4. Its absolute value is one if q = − 43 , 4. Otherwise the absolute value is larger than 1. Thus, issuing from (1.3.11), the regularity condition, one can say that conditions of our theorem are not valid for this example. Considering the general theory of linear system, we can say that the critical case is present, if q = − 43 , 4, otherwise we deal with the general hyperbolic case. If − 43 < q < 4, then the system is exponentially dichotomous. We must say, on the basis of the last examples, that novelty of our results, if one compares them with theory of EPCA [197], is not only in the fact that the argument functions are of generalized type, but also the stability is obtained is uniform. Our evaluations of the decay are the same for all initial moments in Example 1.2, for instance. In papers [192, 194] delay differential equations with piecewise constant argument have been investigated and interesting problems mainly related to the existence of periodic and almost periodic solutions were considered. Investigation in these

20

1 Introduction

papers continues to be through reduction to discrete equations and only linear equations have been discussed. Though, in our papers more general equations with piecewise constant argument were introduced and the research has been implemented for nonlinear systems. In Chaps. 7 and 8 we suggest to investigate more general functional differential equations with functional response on piecewise constant argument, and describe the class. We introduce the following differential equations: x = f (t, xt , xβ(t) ),

(1.3.17)

x = f (t, xt , xγ (t) ),

(1.3.18)

x = f (t, xt , xχ (t) ).

(1.3.19)

and

In these equations, xt must be understood as for functional differential equations (F DE) [74, 86, 118]. More precise description will be given in the next sections. We shall call systems (1.3.17) to (1.3.19) functional differential equations with piecewise constant deviation of argument, and abbreviate F DEP CA. This is a large class of equations, and has the following three subclasses: with retarded constancy of argument, RCA; with advanced constancy of argument, ACA; and with alternate (delay/advanced) constancy of argument, MCA. Examples of these equations are provided in Sect. 7.6. The models (1.3.17) to (1.3.19) are much more general than those investigated in [192, 194], where the delay is constant τ = 1 and it is equal to the step of the greatest integer function, [t]. Moreover, it is the first time, when we introduce the functional depending on xγ (t) . Thus, we can say that in this paper we begin to investigate functional differential equations with piecewise constant argument in the most general form. Let us consider some examples of F DEP CA, 1. x (t) =

m 

Aj (t)x(t − τj ) +

j =1

k 

Bi (t)x(γ (t) − ωj ),

i=1

where τj and ωi are fixed positive numbers. The linear equation is with constant delays and alternate constancy of argument; 2. x (t) =

m  j =1

Aj (t)x(t − τj (t)) +

k  i=1

Bi (t)x(γ (t) − ωj (t)),

1.3 Piecewise Constant Argument

21

where τj (t) and ωi (t) are fixed bounded positive functions. This linear system is with variable delays and alternate constancy of argument; 3. x (t) =



0 −τ

 K(s, x(γ (t) + s))ds +

0 −γ (t)

L(s, x(β(t) + s))ds.

The system is with bounded distributed delay and constancy of argument is of two types, alternate and retarded. 4.

x (t) =



0 −∞

 K(s, x(t + s))ds +

0 −∞

L(s, x(β(t) + s))ds.

The equation is with unbounded distributed delay and retarded constancy of argument. Let us clarify that γ (t) is a function of the alternate constancy. Fix k ∈ N and consider t ∈ [θk , θk+1 ). Then, γ (t) = ζk . If argument t satisfies θk ≤ t < ζk , then γ (t) > t and the piecewise constant argument is advanced. Similarly, if ζk < t < θk+1 , then γ (t) < t, and the constancy is retarded. Consequently, (1.3.17) is a retarded functional differential equation with alternate constancy of argument, RF DEMCA. Equation (1.3.17) is a retarded functional differential equation with retarded constancy of argument, RF DERCA, and (1.3.19) is a retarded functional differential equation with advanced constancy of argument, RF DEACA. Since there are three types of F DE: retarded, advanced, and neutral [109], we submit the following classes: RF DERCA, RF DEMCA, RF DEACA, retarded functional differential equations with piecewise constant argument; N F DERCA, N F DEMCA, NF DEACA, neutral functional differential equations with piecewise constant argument; and AF DERCA, AF DEMCA, AF DEACA, advanced functional differential equations with piecewise constant argument. In the present paper, we focus on retarded functional differential equations and neutral functional differential equations both with retarded and alternate constancy of argument. That is, RF DERCA, RF DEMCA, NF DERCA, and N F DEMCA are under discussion. It is obvious that they are functional differential equations. Existence and uniqueness of solutions, continuation, continuous dependence, periodic, almost periodic solutions, integral manifolds, asymptotic properties, that is, the standard list of problems for any theory of differential equations has to be under investigation for F DEP CA. Arguments, to ensure that newly introduced systems can play important role in applications, are the same as for F DE [127], since our systems are F DE in their generalized form. After all, they are differential equations with piecewise constant argument. Consequently, applications discussed in [12, 27, 84, 196] are also suitable. Moreover, providing F DEP CA as a new object for investigation, we are confident that it will provoke modeling activity, since this has been true for any type of differential equations.

22

1 Introduction

1.4 Asymptotic Behavior of Solutions In this book we suggest to consider conditions on asymptotic equivalence, which are most weak in the modern theory. Our results are compared with Wintner and Yakubovich theorems. The first paper on asymptotic equivalence of differential equations was published by Levinson in 1948 [129]. In his paper, the asymptotic nature of solutions of linear systems of differential equations was studied. The important asymptotic result was established (it is called the Levinson theorem nowadays, (see [129] or ([81] Theorem 8.1 in Chapter 3) or ([88] Theorem 1.3.1)). One must underline that the linear system as an etalon equation was considered. This was continued in the next paper [113], in 1955. The authors got another important result, called the Hartman– Wintner theorem. Later, the problem of asymptotic equivalence for systems of ordinary differential equations has been studied by many authors, and progress has been made in extending this works [73, 88, 111, 151], etc. One must say, that the most common property of these all papers is that all solutions of the non-perturbed linear system are bounded on the axis, with very particular exceptions. For example, when coefficient matrices are diagonal [162]. It is usual that results for differential equations can be extended in other types of dynamics. Thus, Coffman [82] considered asymptotic behavior of solutions of difference equations with almost constant coefficients, Benzaid and Lutz [58] obtained several asymptotic results, and a discrete analog of the Levinson theorem among them, Bohner and Lutz [67] investigated asymptotic behavior of dynamic equations on time scales. Let us formulate the result by N. Levinson. The paper is concerned with the linear system y = [A + B(t)]y,

(1.4.20)

which may be viewed as a perturbation of x = Ax,

(1.4.21)

where x, y ∈ Rn , A is a constant n × n real-valued matrix, B(t) is a continuous n × n real-valued matrix. Definition 1.4 ([77, 150]) A homeomorphism H between the sets of solutions x(t) and y(t) is called an asymptotic equivalence if y(t) = H (x(t)) implies that x(t) − y(t) → 0 as t → ∞. The classical theorem of Levinson [129] states that if the trivial solution of (1.4.21) is uniformly stable, and  0

∞

|| B(t) || dt < ∞,

(1.4.22)

1.4 Asymptotic Behavior of Solutions

23

then (1.4.20) and (1.4.21) are asymptotically equivalent. In the case when A is not a constant matrix, Wintner [200] proved that the above conclusion remains valid if all solutions of (1.4.21) are bounded, (1.4.22) is satisfied, and 

t

lim inf t→∞

Trace[B(s)] ds > −∞.

0

Let us describe the results of Chap. 12 of this book. Let X(t), X(0) = I , be a fundamental matrix solution of (1.4.21). Denote P (t) = X−1 (t)B(t)X(t). Assume that  ∞ ||P (t)||dt < ∞. (C1 ) 0

The following fundamental lemma has been obtained by Ráb [160, 161]. Lemma 1.1 ([160]) If (C1 ) is valid, then the matrix differential equation Ψ = P (t)(Ψ + I )

(1.4.23)

has a solution Ψ (t) which satisfies Ψ (t) → 0 as t → ∞. The main novelty of our research is the decision to apply the last lemma and it analogs for asymptotic equivalence of different types of differential equations. We have started with ordinary differential equations. Assume additionally to the Ráb conditions that (C2 ) limt→∞ X(t)Ψ (t) = 0. to get the following result. Theorem 1.2 ([29]) Suppose that conditions (C1 ) and (C2 ) hold. Then (1.4.20) and (1.4.21) are asymptotically equivalent. The last theorem firstly had been obtained in our paper [22] for impulsive linear systems, and in [29] for ordinary differential equations. One can easily see that we do not request boundedness of solutions for the unperturbed system. By the next example from [29] we demonstrate that there are systems, which are not asymptotically equivalent by our predecessors results, but they are asymptotically equivalent according to the last theorem. More precisely, the fundamental matrix of solutions is not bounded in the example. Example 1.4 ([29]) Let b(t) be a continuous function such that |b(t)| ≤ K1 e−αt for all t ∈ R+ for some α > 0, K1 > 0, and C ∈ R5×5 . Consider y = (A + B(t))y,

(1.4.24)

24

1 Introduction

where ⎛

0 ⎜ −1 ⎜ ⎜ A=⎜ 0 ⎜ ⎝ 0 0

1 0 0 0 0

0 0 0 −π 0

0 0 π 0 0

⎞ 0 0⎟ ⎟ ⎟ 0 ⎟, ⎟ 0⎠ β

B(t) = b(t)C, and β > 0 satisfies α − 2β > 0. The associated equation x = Ax has a fundamental matrix ⎛

⎞ cos t sin t 0 0 0 ⎜ − sin t cos t 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ X(t) = ⎜ 0 0 cos π t sin π t 0 ⎟ . ⎜ ⎟ ⎝ 0 0 sin π t cos π t 0 ⎠ 0 0 0 0 eβt The equality P (t) = X−1 (t)B(t)X(t) = b(t)X−1 (t)CX(t) implies that there exists a K > 0 such that ||P (t)|| ≤ Ke−(α−β)t for all t ∈ R+ . Therefore, (C1 ) is valid. We also see that ||Ψ (t)|| ≤

∞  (Ke−(α−β)t )k k=1

(α − β)k k!

= 1 − eK(α−β)

−1 e−(α−β)t

,

and hence X(t)Ψ (t) → 0 as t → ∞. Thus, Theorem 1.2 is verified for the couple. In the same time the result by Wintner in [200] is not applicable for the example, since not all solutions are bounded on the real axis (β > 0). The first paper, where we started to develop our method, concerns impulsive differential equations [22]. We considered the following systems of impulsive differential equations: dx = A(t)x, t = θi , dt Δx(θi ) = Bi x(θi )

(1.4.25)

dy = [A(t) + C(t)]y, t = θi , dt Δy(θi ) = [Bi + Di ]y(θi )

(1.4.26)

and

1.4 Asymptotic Behavior of Solutions

25

where x, y ∈ R n , (D1 ) A(t), C(t), t ∈ Bi , Di , i ∈ Z, are real-valued n × n-matrices, A(t), C(t) ∈ C(R+ ); (D2 ) {θi } ⊂ R, t0 < θ1 < θ2 < . . . θi → ∞ as i → ∞; (D3 ) matrices Bi , Di satisfy the inequalities det(I + Bi ) = 0, det(I + Bi + Di ) = 0, for all i ∈ Z.

(1.4.27)

Let X(t) be a fundamental matrix of (1.4.25). We start with the change of the dependent variable y(t) = X(t)u(t),

(1.4.28)

which transforms (1.4.26) to the system du = P (t)u, t = θi , dt Δu(ζi ) = Qi u(ζi ),

(1.4.29)

where P (t) = X−1 (t)B(t)X(t), Qi = X−1 (θi +)Di X(θi ).

(1.4.30)

Let us assume that 

∞

|| P (t) || dt +

t0



|| Qi ||< ∞

(1.4.31)

t0 0 or k < −2. Example 7.3 ([11]) √ Consider the following sequence, θi = i + ai , where ai = 1 | sin(i) − cos(i 2)|. By repeating proof in [21, 22] or Chap. 3, one can verify that 4 j θ satisfies the conditions of the last theorem. That is, sequences θi are uniformly almost periodic, and there are positive numbers θ¯ and θ such that θ < θi+1 − θi ≤

170

7 Functional Response on Piecewise Constant Argument

θ¯ , i ∈ Z. Now, introduce a sequence ζ such that ζi = θi+12+θi . One can easily verify that this sequence also satisfies all conditions of the theorem. Moreover, ζ¯ = θ¯ , ζ = θ . So, we fix the chosen sequences, and define on this basis the function γ (t). Introduce the following RF DEP CA: x (t) = αx(t) + βx(γ (t)) + f (t, xt , xγ (t) ),

(7.5.26)

in which α, β are fixed real constants, the identification function γ (t) is defined above, function f can be chosen, for example, as the following one:  f (t, xt , xγ (t) ) = L sin(γ (t))cos (π t) 2

0

−2

[x 2 (t + s) + x 4 (γ (t) + s)]ds.

It is obvious that it remains to check if the zero solution of the equation y (t) = αy(t) + βy(γ (t)),

(7.5.27)

is uniformly exponentially stable. We investigated the equation in [7] with simple case of γ (t). One can evaluate that  t $ β # α(t−ζi ) α(t−ζi ) e + eα(t−s) βds = eα(t−ζi ) + −1 . Mi (t) = e α ζi Then Mi (θi ) = e−α Mi−1 (θi ) = eα

θi+1 −θi 2

θi −θi−1 2

+

+

β −α θi+1 −θi 2 (e − 1), α

β α θi −θi−1 2 (e − 1), α

and

Mi−1 (θi )Mi−1 (θi ) =

eα

θi −θi−1 2

e−α

θi+1 −θi 2

Now, we assume that α < 0, β > 0, and

+

β α

+

β α

|β| |α|



θi −θi−1 eα 2 − 1

. θi+1 −θi e−α 2 − 1

< 1. Then one can find that

  θi −θi−1  α θi −θi−1  2  e + βα (eα 2 − 1)    ≤ q < 1, i ∈ Z, θi+1 −θi  −α θi+1 −θi  β −α e 2 2 + α (e − 1) 

7.6 Further Investigations

171

with some positive number q. The last inequality is sufficient for the zero solution of (7.5.27) to be uniformly exponentially stable, and then Eq. (7.5.26) admits a unique almost periodic solution, if the constant L is sufficiently small.

7.6 Further Investigations In this part of the chapter, we describe some of problems, which can be studied on the basis of our present results. Firstly, several models (population dynamics, optimal control, ship stabilization) will be considered. Secondly, we introduce differential equations, which are much more general, in some sense, than those we investigated in the paper, but they still are F DEP CA. A number of models with piecewise argument are considered in [7, 26]. In what follows we propose a very short list of models which are F DEP CA. One may investigate the following prey-predator Volterra system with piecewise constant argument: x (t) = [a1 − b1 y(t) −



−τ



y (t) = [a2 + b2 y(t) +

0

0

−τ

Q1 (s)y(β1 (t) + s)ds]x(t), Q2 (s)x(β2 (t) + s)ds]y(t).

(7.6.28)

In this equation we assume that the effect of the species accumulation is seen only near the moments of discontinuity, which can, for example, depend on seasonal behavior of animals. It is natural that two different constancy functions, β1 , β2 are considered for each of the two species. The model is RF DERCA. Similar arguments for investigation can be accepted for the following equation of an isolated population [28]:

x (t) = −α[



0 −τ

x(β(t) + s)dη(s)](1 + x(t)).

(7.6.29)

More biological arguments for equations with piecewise constant argument can be found in our book [7]. It is obvious that piecewise continuous control is easier to apply than continuous one. The following control problem for F DE with MCA can be investigated, x (t) = P (t)x(t) + B(t)u(γ (t)), y(t) = Q(t)x(t),  0 

u (t) = [ds η(t, s)]y(γ (t) + s) + −τ

0 −τ

[ds μ(t, s)]u(γ (t) + s). (7.6.30)

172

7 Functional Response on Piecewise Constant Argument

which is generalization of the optimal control problem in [34]. One has to emphasize that alternate constancy function γ does not make theoretical investigation more difficult as it is for advanced argument in F DE [32, 35]. Consider one technical problem, ship stabilization. The following retarded functional differential equation is discussed in [33]:





aφ (t) + bφ (t) + cφ (t − τ ) + dφ(t − τ ) = 0, In the last equation, φ is the ship deviation angle. The time lag τ is utilized, since it is impossible to measure ship deviation instantaneously. Technically it is easy to use the deviation function β(t) for control. So, we obtain the following F DEP CA:





aφ (t) + bφ (t) + cφ (t − τ ) + dφ(t − τ ) + eφ (β(t)) + f φ(β(t)) = 0. Finally, anticipated deviation one can be considered for control, and we obtain the following F DEP CA:





aφ (t) + bφ (t) + cφ (t − τ ) + dφ(t − τ ) + eφ (β(t)) +f φ(β(t)) + gφ (χ (t)) + hφ(χ (t)) = 0. Thus, F DE with RCA and MCA can be investigated for the ship stabilization. Next, we consider argument functions of more general type than those have been analyzed in the main body of the paper. Denote by θ = {θi }, i ∈ Z, θ ⊂ R, a strictly ordered sequence of real numbers such that |θi | → ∞ as |i| → ∞. Let, also, ζ = {ζi }, i ∈ Z, be another sequence of elements of R. This sequence may be strictly increasing or nondecreasing, and it is not necessary that ζi ∈ [θi , θi+1 ]. We say that a function, which is defined on R, is of the η-type, and denote it η(t), if it is equal to ζi , whenever θi ≤ t < θi+1 , i ∈ Z. This type of functions has been introduced in [7]. Now, we can suggest to consider the following functional differential equation with piecewise constant argument: x (t) = f (t, xt , xη(t) ).

(7.6.31)

What is the main difference between the last system and Eqs. (7.1.1)–(7.1.3)? Values of function η are not necessary in the interval of constancy, where t lies in. Consequently, one has more interesting opportunities in investigation. For example, one can request that η(t) = θi+1 for t ∈ [θi , θi+1 ). It is of great mathematical interest to investigate these type of equations, as well it provides more opportunities for applications. Since any differential equation with piecewise constant argument is a differential equation with deviated argument, one can suppose that they are functional differential equations. We, on the basis of our experience, propose to consider two classes of equations, which relate to each other similarly to ordinary differential

7.6 Further Investigations

173

equations and functional differential equations. That is, there have to be EP CAG and F DEP CA. If one considers Eqs. (7.1.1)–(7.1.3), there is no doubt that they are F DEP CA. Let us introduce another example, to illustrate our point of view, x = f (t, x(t), x(η1 (t)), x(η2 (t)), . . . , x(ηm (t))),

(7.6.32)

where t ∈ R, x ∈ Rn , function f is continuous in all arguments, ηj , j = 1, 2, . . . , m, are piecewise constant functions of η-type. We call (7.6.32) differential equation with piecewise constant argument (EPCAG, and not F DE), if the following condition holds, (A) For each j = 1, 2, . . . , m, and t ∈ J, the value of function ηj (t) and t are in the same interval of constancy of ηj (t). If there is a function ηj in (7.6.32) such that (A) does not hold for all t from the domain, then (7.6.32) is a functional differential equation with piecewise constant argument (F DEP CA). Assume that function f in (7.6.32) is defined not for all t ∈ R, but in some subset J ⊂ R. In this case one should request that ηj (t) ∈ J for all j and t ∈ J. Let us provide more concrete examples. Fix strictly increasing sequences of real j j j j numbers θi , i ∈ Z, j = 1, 2, . . . , m, and sequences ζi , θi ≤ ζi ≤ θi+1 , i ∈ j j j Z, j = 1, 2, . . . , m. Define functions ηj (t) = ζi , if t ∈ [θi , θi+1 ). One can easily see that (7.6.32) is EP CAG, and this system is not F DEP CA. Now, define j j j ηj (t) = ζi−1 , if t ∈ [θi , θi+1 ). In this case condition (A) is not valid, and (7.6.32) is F DEP CA. One can call it retarded functional differential equation with piecewise constant argument. Nevertheless, this equation is not of the same type as (7.1.2). Finally, we suggest a new name for functional differential equations with piecewise constant argument, functional differential equations with functional response on piecewise constant argument (F DEF RP CA). This class of systems may involve all equations considered in the present paper, and moreover, we suggest to investigate, and to apply in modeling of real-world problems, the following types of equations: x = f (t, xt , xβ(t−τ (t)) ),

(7.6.33)

x = f (t, xt , xγ (t−τ (t)) ),

(7.6.34)

x = f (t, xt , xχ (t−τ (t)) ),

(7.6.35)

and

where τ (t) is a deviation function, which is positive, negative, or alternate type.

174

7 Functional Response on Piecewise Constant Argument

Our proposals have been formulated just above show that there are various functional differential equations with piecewise constant argument, and they may provide more theoretical challenges and ways of solutions for real-world problems.

References 1. M.U. Akhmet, On the integral manifolds of the differential equations with piecewise constant argument of generalized type, in Proceedings of the Conference on Differential and Difference Equations at the Florida Institute of Technology, August 1–5, 2005, Melbourne, Florida, ed. by R.P. Agarval, K. Perera (Hindawi Publishing Corporation, London, 2006), pp. 11–20 2. M.U. Akhmet, Integral manifolds of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. 66, 367–383 (2007) 3. M.U. Akhmet, On the reduction principle for differential equations with piecewise constant argument of generalized type. J. Math. Anal. Appl. 336, 646–663 (2007) 4. M.U. Akhmet, Almost periodic solutions of differential equations with piecewise constant argument of generalized type. Nonlinear Anal. Hybrid Syst. 2, 456–467 (2008) 5. M.U. Akhmet, Asymptotic behavior of solutions of differential equations with piecewise constant arguments. Appl. Math. Lett. 21, 951–956 (2008) 6. M.U. Akhmet, Almost periodic solutions of the linear differential equation with piecewise constant argument. Discrete Impuls. Syst. Ser. A Math. Anal. 16, 743–753 (2009) 7. M.U. Akhmet, Nonlinear Hybrid Continuous/Discrete Time Models (Atlantis Press, Amsterdam, 2011) 8. M.U. Akhmet, Almost periodic solutions of second order neutral functional differential equations with piecewise constant argument. Discontinuity Nonlinearity Complexity 1, 1–6 (2012) 9. M.U. Akhmet, Exponentially dichotomous linear systems of differential equations with piecewise constant argument. Discontinuity Nonlinearity Complexity 1, 337–352 (2012) 10. M.U. Akhmet, Quasilinear retarded differential with functional dependence on piecewise constant argument. Commun. Pure Appl. Anal. 13, 929–947 (2014) 11. M.U. Akhmet, Functional differential equations with piecewise constant argument, in Regularity and Stochasticity of Nonlinear Dynamical Systems. Nonlinear Systems and Complexity, vol. 21 (Springer, Cham, 2018), pp. 79–109 12. M.U. Akhmet, D. Aru˘gaslan, Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete Contin. Dyn. Syst. 25, 457–466 (2009) 13. M.U. Akhmet, C. Buyukadali, Periodic solutions of the system with piecewise constant argument in the critical case. Comput. Math. Appl. 56, 2034–2042 (2008) 14. M.U. Akhmet, C. Buyukadali, Differential equations with a state-dependent piecewise constant argument. Nonlinear Anal. Theory Methods Appl. 72, 4200–4210 (2010) 15. M.U. Akhmet, E. Yılmaz, Impulsive Hopfield-type neural network system with piecewise constant argument. Nonlinear Anal. Real World Appl. 11, 2584–2593 (2010) 16. M. Akhmet, E. Yılmaz, Neural Networks with Discontinuous/Impact Activations (Springer, New York, 2014) 17. M.U. Akhmet, C. Buyukadali, T. Ergenc, Periodic solutions of the hybrid system with small parameter. Nonlinear Anal. Hybrid Syst. 2, 532–543 (2008) 18. M.U. Akhmet, D. Aru˘gaslan, E. Yılmaz, Stability analysis of recurrent neural networks with piecewise constant argument of generalized type. Neural Netw. 23, 805–811 (2010) 19. M.U. Akhmet, D. Aru˘gaslan, E. Yılmaz, Stability in cellular neural networks with piecewise constant argument. J. Comput. Appl. Math. 233, 2365–2373 (2010) 20. M.U. Akhmet, D. Aru˘gaslan, E. Yılmaz, Method of Lyapunov functions for differential equations with piecewise constant delay. J. Comput. Appl. Math. 235, 4554–4560 (2011)

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21. M.U. Akhmetov, Almost periodic solutions and stability of Lyapunov exponents of differential equations with impulse actions (in Russian). Ph.D. Thesis, Kiev State University, Kiev, 1984 22. M.U. Akhmetov, N.A. Perestyuk, Almost periodic solutions of a class of systems with impulse action (in Russian). Ukrain. Mat. Zh. 36, 486–490 (1984) 23. M.U. Akhmetov, R. Sejilova, The control of the boundary value problem for linear impulsive integro-differential systems. J. Math. Anal. Appl. 236, 312–326 (1999) 24. G. Bao, S. Wen, Z. Zeng, Robust stability analysis of interval fuzzy Cohen–Grossberg neural networks with piecewise constant argument of generalized type. Neural Netw. 33, 32–41 (2012) 25. T.A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations (Academic Press, Orlando, 1985) 26. L. Dai, Nonlinear Dynamics of Piecewise Constant Systems and Implementation of Piecewise Constant Arguments (World Scientific, Hackensack, 2008) 27. O. Diekmann, S.A. van Gils, L. Verduyn, M. Sjoerd, H.-O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis. Applied Mathematical Sciences (Springer, New York, 1995) 28. G. Dunkel, Single-species model for population growth depending on past history, in Seminar of Differential Equations and Dynamical Systems. Lecture Notes in Mathematics (Springer, Berlin, 1968) 29. A.M. Fink, Almost Periodic Differential Equations. Lecture Notes in Mathematics (Springer, Berlin, 1974) 30. A. Halanay, D. Wexler, Qualitative theory of impulsive systems (in Romanian). Edit. Acad. RPR, Bucuresti, 1968 31. J. Hale, Functional Differential Equations (Springer, New York, 1971) 32. M.E. Hernández, M.L. Pelicer, Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations. Appl. Math. Lett. 18, 1265–1272 (2005) 33. V.B. Kolmanovskii, Stability of Functional Differential Equations (Academic Press, Orlando, 1986) 34. M.A. Krasnosel’skii, V.S. Burd, Y.S. Kolesov, Nonlinear Almost Periodic Oscillations (Wiley, New York, 1973) 35. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993) 36. M. Pinto, Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Math. Comput. Model. 49, 1750–1758 (2009) 37. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations (World Scientific, Singapore, 1995) 38. G. Wang, Periodic solutions of a neutral differential equation with piecewise constant arguments. J. Math. Anal. Appl. 326, 736–747 (2007) 39. Y. Wang, J. Yan, Oscillation of a differential equation with fractional delay and piecewise constant argument. Comput. Math. Appl. 52, 1099–1106 (2006) 40. D. Wexler, Solutions périodiques et presque-périodiques des systémes d’équations différetielles linéaires en distributions. J. Differ. Equ. 2, 12–32 (1966)

Chapter 8

Almost Periodic Solutions of Retarded SICNN with Functional Response on Piecewise Constant Argument

We consider a new model for shunting inhibitory cellular neural networks, retarded functional differential equations with piecewise constant argument. The existence and exponential stability of almost periodic solutions are investigated. An illustrative example is provided.

8.1 Introduction Cellular neural networks (CN N s) have been paid much attention in the past two decades [35–38, 54–56, 65, 75, 93, 95, 97]. Exceptional role in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing is played by shunting inhibitory cellular neural networks (SI CN N s), which was introduced by Bouzerdoum and Pinter [28]. One of the most attractive subjects for this type of neural networks is the existence of almost periodic solutions. This problem has been investigated for models with different types of activation functions [32–34, 46, 69, 80, 84, 95, 99]. In the present study, we investigate a new model of SI CN Ns by considering deviated as well as piecewise constant time arguments, and prove the existence of exponentially stable almost periodic solutions. All the results are discussed for the general type of activation functions, but they can be easily specified for applications. Extended information about differential equations with generalized piecewise constant argument [3–7, 9–11] can be found in the book [8]. As a subclass, they contain differential equations with piecewise constant argument (EP CA) [1, 2, 20, 21, 31, 40–42, 44, 61, 64, 70, 74, 76–78, 80, 87, 88, 94, 98], where the piecewise constant argument is assumed to be a multiple of the greatest integer function.

© Springer Nature Switzerland AG 2020 M. Akhmet, Almost Periodicity, Chaos, and Asymptotic Equivalence, Nonlinear Systems and Complexity 27, https://doi.org/10.1007/978-3-030-20572-0_8

177

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8 SICNN with Functional Response on PCA

Differential equations with piecewise constant argument are very useful as models for neural networks. This was shown in the studies [61, 80, 100], where the authors utilized EP CA. We propose to involve a new type of systems, retarded functional differential equations with piecewise constant argument of generalized type, in the modeling. It will help to investigate a larger class of neural networks. In papers [3–6, 9], differential equations with piecewise constant argument of generalized type (EP CAG) were introduced. We not only maximally generalized the argument functions, but also proposed to reduce investigation of EP CAG to integral equations. Due to that innovation, it is now possible to analyze essentially nonlinear systems, that is, systems nonlinear with respect to values of solutions at discrete moments of time, where the argument changes its constancy. Previously, the main and unique method for EP CA was reduction to discrete equations and, hence, only equations in which values of solutions at the discrete moments appear linearly [1, 2, 20, 21, 31, 40–42, 44, 61, 64, 70, 74, 76–78, 80, 87, 88, 94, 98] have been considered. The crucial novelty of the present chapter is that the piecewise constant argument in the functional differential equations is of alternate (advanced-delayed) type. In the literature, biological reasons for the argument to be delayed were discussed [53, 68]. However, the role of advanced arguments has not been analyzed properly yet. Nevertheless, the importance of anticipation for biology was mentioned by some authors. For example, in the paper [29], it is supposed that synchronization of biological oscillators may request anticipation of counterparts behavior. Consequently, one can assume that equations for neural networks may also need anticipation, which is usually reflected in models by advanced argument. Therefore, the systems taken into account in the present study can be useful in the future analyses of SI CN N s. Furthermore, the idea of involving both advanced and delayed arguments in neural networks can be explained by the existence of retarded and advanced actions in a model of classical electrodynamics [47]. Moreover, mixed type deviation of the argument may depend on traveling waves emergence in CN N s [93]. Understanding the structure of such traveling waves is important due to their potential applications including image processing (see, for example, [35–38, 54–56, 65, 75, 93, 95, 97]). More detailed analysis of deviated arguments in neural networks can be found in [12–15]. Shunting inhibition is a phenomenon in which the cell is “clamped” to its resting potential when the reversal potential of Cl − channels is close to the membrane resting potential of the cell [28, 79]. It occurs through the opposition of an inward current, which would otherwise depolarize the membrane potential to threshold, by an inward flow of Cl − ions [79]. From the biological point of view, shunting inhibition has an important role in the dynamics of neurons [23, 67, 81]. According to the results of Vida et al. [81] networks with shunting inhibition are advantageous compared to the networks with hyperpolarizing inhibition such that in the former type networks oscillations are generated with smaller tonic excitatory drive, network frequencies are tuned to the γ band, and robustness against heterogeneity in the excitatory drive is markedly improved. It was demonstrated by Mitchell and Silver [67] that shunting inhibition can modulate the gain and offset

8.1 Introduction

179

of the relationship between output firing rate and input frequency in granule cells when excitation and/or inhibition are mediated by time-dependent synaptic input. Besides, Borg-Graham et al. [23] proposed that nonlinear shunting inhibition may act during the initial stage of visual cortical processing, setting the balance between opponent “On” and “Off” responses in different locations of the visual receptive field [23]. On the other hand, shunting neural networks are important for various engineering applications [22–28, 48, 67, 72, 73]. For example, in vision, shunting lateral inhibition enhances edges and contrast, mediates directional selectivity, and causes adaptation of the organization of the spatial receptive field and of the contrast sensitivity function [25–28, 58, 59]. Moreover, such networks are appropriate to be used in medical diagnosis [22]. Therefore, the investigation of the dynamics of SI CN Ns, which are biologically inspired networks designed upon the shunting inhibition concept [28], is important for the improvement of the techniques used in medical diagnosis, adaptive pattern recognition, image processing, etc. [22– 28, 48, 67, 72, 73] and may shed light on neuronal activities concerning shunting inhibition [23, 67, 81]. Exponential stability of neural networks has been widely studied in the literature (see, for example, [45, 51, 60, 66, 84, 90, 91, 95, 96]). According to Liao et al. [66], the exponential stability has importance in neural networks when the exponentially convergence rate is used to determine the speed of neural computations. The studies [66, 96] were concerned with the exponential stability and estimation of exponential convergence rates in neural networks. In the paper [66], Lyapunov–Krasovskii functionals and the linear matrix inequality (LMI) approaches were combined to investigate the problem, whereas the boundedness of the Dini derivative of the neuron input output activations was required in [96]. The exponential stabilization problem of memristive neural networks was considered in [91] by means of the Lyapunov–Krasovskii functional and free weighting matrix techniques. Additionally, the Lyapunov–Krasovskii functional method was considered by Wen et al. [90] to analyze the passivity of stochastic impulsive memristor-based piecewise linear systems, and the free weighting matrix approach was utilized in [51] to derive an LMI-based delay dependent exponential stability criterion for neural networks with a time varying delay. On the other hand, exponential stability criteria were derived by Dan et al. [45] for an error system in order to achieve lag synchronization of coupled delayed chaotic neural networks. The concept of lag synchronization was taken into account also within the scope of the papers [89] and [92] for memristive neural networks and for a class of switched neural networks with time-varying delays, respectively. Furthermore, the Banach fixed point theorem and the variant of a certain integral inequality with explicit estimate were used to investigate the global exponential stability of pseudo almost periodic solutions of SI CN N s with mixed delays in the study [34]. Almost periodic and in particular quasi-periodic motions are important for the theory of neural networks. According to Pasemann et al. [71], periodic and quasi-periodic solutions have many fundamental importance in biological and artificial systems, as they are associated with central pattern generators, establishing stability properties and bifurcations (leading to the discovery of periodic solutions).

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Besides, the sinusoidal shape of neural output signals is, in general, associated with appropriate quasi-periodic attractors for discrete-time dynamical systems. In the book [57], the dynamics of the brain activity is considered as a system of many coupled oscillators with different incommensurable periods. Signals from the neurons have a phase shift of π/2, and may be useful for various kinds of applications; for instance, controlling the gait of legged robots [62]. Furthermore, an alternative discrete-time model of coupled quasi-periodic and chaotic neural network oscillators were considered by Wang [82]. Let us describe the model of SI CN N s in its most original form [28]. Consider a two dimensional grid of processing cells arranged into m rows and n columns, and let Cij , i = 1, 2, . . . , m, j = 1, 2, . . . , n, denote the cell at the (i, j ) position of the lattice. In SI CN N s, neighboring cells exert mutual inhibitory interactions of the shunting type. The dynamics of a cell Cij are described by the following nonlinear ordinary differential equation: dxij = −aij xij − dt



Cijkl f (xkl (t))xij + Lij (t),

Ckl ∈Nr (i,j )

(8.1.1)

where xij is the activity of the cell Cij ; Lij (t) is the external input to the cell Cij ; the constant aij > 0 represents the passive decay rate of the cell activity; Cijkl ≥ 0 is the coupling strength of postsynaptic activity of the cell Ckl transmitted to the cell Cij ; the activation function f (xkl ) is a positive continuous function representing the output or firing rate of the cell Ckl ; and the r-neighborhood of the cell Cij is defined as Nr (i, j ) = {Ckl : max(|k − i|, |l − j |) ≤ r, 1 ≤ k ≤ m, 1 ≤ l ≤ n}. It is worth noting that even if the activation function is supposed to be globally bounded and Lipschitzian, these properties are not valid for the nonlinear terms in the right-hand sides of the differential equations describing the dynamics of SI CN Ns, and this is one of the reasons why a sophisticated mathematical analysis is required for SI CN N s in general. Another reason is that the connections between neurons in SI CN N s act locally only in r-neighborhoods. This causes special ways of evaluations different than those customized for earlier developed neural networks in the mathematical analyses of the models. It is reasonable to say that the usage of deviated arguments in neural networks makes the models much closer to applications. For example, in [46] the model was considered with variable delays, dxij = −aij xij − dt



Cijkl f (xkl (t − τ (t)))xij + Lij (t).

(8.1.2)

Ckl ∈Nr (i,j )

In the present study, we introduce and investigate more general neural networks. The model will be described in the next section.

8.2 Preliminaries

181

8.2 Preliminaries Let Z and R denote the sets of all integers  and  real numbers, / respectively. 0 Throughout the paper, the norm u = max uij  , where u = uij ∈ Rm×n , (i,j )

and u = (u11 , . . . , u1n , . . . , um1 . . . , umn ), will be used. Suppose that θ = {θp } and ζ = {ζp }, p ∈ Z, are sequences of real numbers such that the first one is strictly ordered, |θp | → ∞ as |p| → ∞, and the second one satisfies θp ≤ ζp ≤ θp+1 for all p ∈ Z. The sequence ζ is not necessarily strictly ordered. We say that a function is of γ -type, and denote it by γ(t), if γ(t) = ζp for t +1 is a γ -type θp ≤ t < θp+1 , p ∈ Z. One can affirm, for example, that 2 205 function with θp = 2p − 1, ζp = 2p. Fix a nonnegative number τ ∈ R and let C 0 be the set of all continuous functions mapping the interval [−τ, 0] into R, with the uniform norm φ0 = max |φ(t)| . t∈[−τ,0]

Moreover, we denote by C the set consisting of continuous functions mapping the interval [−τ, 0] into Rm×n , with the uniform norm φ0 = max φ(t) . t∈[−τ,0]

In the present study, we propose to investigate retarded SI CN N s with functional response on piecewise constant argument of the following form: dxij = −aij xij − dt



Cijkl f (xklt , xklγ (t) )xij + Lij (t),

(8.2.3)

Ckl ∈Nr (i,j )

where f : C 0 × C 0 → R is a continuous functional. In network (8.2.3), the terms xklt and xklγ (t) must be understood in the way used for functional differential equations [30, 50, 63]. That is, xklt (s) = xkl (t + s) and xklγ (t) (s) = xkl (γ (t) + s) for s ∈ [−τ, 0]. Let us clarify that the argument function γ (t) is of the alternate type. Fix an integer p and consider the function on the interval [θp , θp+1 ). Then, the function γ (t) is equal to ζp . If the argument t satisfies θp ≤ t < ζp , then γ (t) > t and it is of advanced type. Similarly, if ζp < t < θp+1 , then γ (t) < t and, hence, it is of delayed type. Consequently, it is worth noting that the SI CN N (8.2.3) is with alternate constancy of argument. It is known that γ (t) is the most general among piecewise constant argument functions [19]. Our model is much more general than the equations investigated in [76, 77, 83, 85, 86], where the delay is constant τ = 1 and it is equal to the step of the greatest integer function [t]. Differential equations with functional response on the piecewise constant argument were first introduced in the paper [11]. In the present study, we apply the theory to the analysis of neural networks. All previous authors were at most busy with terms of the form x(γ (t)). Thus, one can say that retarded functional differential equations with piecewise constant argument in the most general form are investigated in this paper. Since the model (8.2.3) is a new one, we have to investigate not only the existence of almost periodic solutions and their stability, but also common problems

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8 SICNN with Functional Response on PCA

of the existence and uniqueness of solutions, their continuation to infinity and boundedness. One can easily see that system (8.1.2) is a particular case of (8.2.3). Additionally, results of the present paper are true or can be easily adapted to the following systems: dxij = −aij xij − dt



Cijkl f (xkl (γ (t)))xij + Lij (t),

(8.2.4)

Ckl ∈Nr (i,j )

that is, differential equations with piecewise constant argument, EP CAG, dxij = −aij xij dt  −

Cijkl f (xkl (t − τ (t)), xkl (γ (t) − τ (t)))xij + Lij (t),

Ckl ∈Nr (i,j )

(8.2.5) differential equations with variable delay and piecewise constant argument, dxij = −aij xij − dt −





Cijkl f (xkl (t − τ (t)))xij

Ckl ∈Nr (i,j )

Dijkl g(xkl (γ (t)))xij + Lij (t).

(8.2.6)

Ckl ∈Nr (i,j )

In other words, what we have suggested are sufficiently general models, which can be easily specified for concrete applications. Let us introduce/ the0initial condition / 0 for SI CNN (8.2.3). Fix a number σ ∈ R and functions φ = φij , ψ = ψij ∈ C , i =/ 1, 2, .0. . , m, j = 1, 2, . . . , n. In the case γ (σ ) < σ, we say that a solution x(t) = xij (t) of (8.2.3) satisfies the initial condition and write x(t) = x(t, σ, φ, ψ), t ≥ σ, if xσ (s) = φ(s), xγ (σ ) (s) = ψ(s) for s ∈ [−τ, 0]. In what follows, we assume that if the set [γ (σ ) − τ, γ (σ )] ∪ [σ − τ, σ ] is connected, then the equation φ(s) = ψ(s + σ − γ (σ )) is true for all s ∈ [−τ, γ (σ ) − σ ]. If γ (σ ) ≥ σ, then we look for a solution x(t) = x(t, σ, φ), t ≥ σ, such that xσ (s) = φ(s), s ∈ [−τ, 0]. Thus, if θp ≤ σ < θp+1 for some p ∈ Z, then there are two cases of the initial condition: (IC1) xσ (s) = φ(s), φ ∈ C , s ∈ [−τ, 0] if θp ≤ σ ≤ ζp < θp+1 ; (IC2) xσ (s) = φ(s), xγ (σ ) (s) = ψ(s), φ, ψ ∈ C , s ∈ [−τ, 0], if θp ≤ ζp < σ < θp+1 . Considering SI CN N (8.2.3) with these conditions, we shall say about the initial value problem (I V P ) for (8.2.3). To be short, we shall say only about I V P in the form x(t, σ, φ, ψ), specifying x(t, σ, φ) for (I C1), if needed. Thus, we can provide the following definition now.

8.3 Existence and Uniqueness

183

/ 0 Definition 8.1 ([16]) A function x(t) = xij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , n, is a solution of (8.2.3) with (I C1) or (I C2) on an interval [σ, σ + a) if: (i) it satisfies the initial condition; (ii) x(t) is continuous on [σ, σ + a); (iii) the derivative x (t) exists for t ≥ σ with the possible exception of the points θp , where one-sided derivatives exist; (iv) Equation (8.2.3) is satisfied by x(t) for all t > σ except possibly at the points of θ, and it holds for the right derivative of x(t) at the points θp . / 0 Definition 8.2 ([16]) A function x(t) = xij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , n, is a solution of (8.2.3) on R if: (i) x(t) is continuous; (ii) the derivative x (t) exists for all t ∈ R with the possible exception of the points θp , p ∈ Z, where one-sided derivatives exist; (iv) Equation (8.2.3) is satisfied by x(t) for all t ∈ R except at the points of θ, and it holds for the right derivative of x(t) at the points θp , p ∈ Z. The existence and uniqueness of solutions of (8.2.3) will be investigated in the next section.

8.3 Existence and Uniqueness Throughout the paper we suppose in SI CN N (8.2.3) that γ0 = min aij > 0 and (i,j )

Cijkl are nonnegative numbers. The following assumptions are required. (C1) The functional f satisfies the Lipschitz condition f (φ1 , ψ1 ) − f (φ2 , ψ2 ) ≤ L(φ1 − φ2 0 + ψ1 − ψ2 0 ), for some positive constant L, where (φ1 , ψ1 ) and (φ2 , ψ2 ) are from C 0 × C 0 ; |f (φ, ψ)| ≤ M; (C2) There exists a positive number M such that sup (φ,ψ)∈C 0 ×C 0

(C3) There exists a positive number θ¯ such that θp+1 − θp ≤ θ¯ for all p ∈ Z; (C4) |Lij (t)| ≤ Lij for all i, j and t ∈ R, where Lij are nonnegative real constants.  Cijkl , c¯ = In the remaining parts of the paper, the notations μ = max 

kl Ckl ∈Nr (i,j ) Cij



(i,j )

kl Ckl ∈Nr (i,j ) Cij

Ckl ∈Nr (i,j )

Lij , d¯ = max , L¯ = max Lij and l¯ = max aij (i,j ) 2aij − γ0 (i,j ) (i,j ) aij will be used. We assume that μθ¯ M < 1 and M c¯ < 1. Let us denote CH0 = {φ ∈ C : φ0 ≤ H0 }, where H0 is a positive number. max (i,j )

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8 SICNN with Functional Response on PCA

Lemma 8.1 ([16]) Suppose that the conditions (C1) − (C4) hold and fix an integer  ¯  2L(H0 + θ¯ L) p. If H0 is a positive number such that μθ¯ M + < 1, then for 1 − μθ¯ M every (σ, φ, ψ) ∈ [θp , θp+1 ] × CH0 × CH0 there exists a unique solution x(t) = x(t, σ, φ, ψ) of (8.2.3) on [σ, θp+1 ]. Proof We assume without loss of generality that θp ≤ σ ≤ ζp < θp+1 . That is, we consider (I C1) and the solution x(t, σ, φ). Fix an arbitrary/function 0 φ ∈ CH0 . Let us denote by Λ the set of continuous functions u(t) = uij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , n, defined on [σ − τ, θp+1 ] such that uσ (t) = φ(t), t ∈ [−τ, 0], and u1 ≤ K0 , where u1 = H0 + θ¯ L¯ max u(t) and K0 = . t∈[σ,θp+1 ] 1 − μθ¯ M Define on Λ an operator Φ such that ⎧ φij (t − σ ), t ∈ [σ − τ, σ ], ⎪ ⎪ ⎪   t ⎪ ⎨ −aij (t−σ ) e φij (0) − e−aij (t−s) Cijkl (Φu(t))ij = σ ⎪ ⎪ Ckl ∈Nr (i,j ) ⎪ ⎪ ⎩ ×f (ukls , uklγ (s) )uij (s) − Lij (s) ds, t ∈ [σ, θp+1 ]. #   One can confirm that (Φu(t))ij  ≤ H0 + MK0

$ Cijkl + L¯ θ¯ , t ∈

 Ckl ∈Nr (i,j )

¯ θ¯ = K0 is [σ, θp+1 ]. Accordingly, the inequality Φu1 ≤ H0 + (μMK0 + L) valid. Therefore, Φ(Λ) ⊆ Λ. / 0 / 0 On the other hand, if u(t) = uij (t) and v(t) = vij (t) belong to Λ, then we have for t ∈ [σ, θp+1 ] that   (Φu(t))ij − (Φv(t))ij  ≤



t

e σ

−aij (t−s)



Cijkl

Ckl ∈Nr (i,j )

  t  e−aij (t−s) − vij (s)ds + σ

   f (ukls , uklγ (s) ) uij (s)   

 Ckl ∈Nr (i,j )

Cijkl f (ukls , uklγ (s) )

   − f (vkls , vklγ (s) ) vij (s) ds ≤ θ¯ (M + 2K0 L) u − v1



Cijkl .

Ckl ∈Nr (i,j )

Hence, the inequality Φu − Φv1 ≤ μθ¯ (M + 2K0 L) u − v1 holds. Because  ¯  2L(H0 + θ¯ L) < 1, the operator Φ is a contraction. μθ¯ (M + 2K0 L) = μθ¯ M + 1 − μθ¯ M   Consequently, there exists a unique solution of (8.2.3) on [σ, θp+1 ].

8.4 Bounded Solutions

185

Lemma 8.2 ([16]) Suppose that the conditions (C1) − (C4) hold and fix an integer  ¯  2L(H0 + θ¯ L) p. If H0 is a positive number such that μθ¯ M + < 1, then for 1 − μθ¯ M every (σ, φ, ψ) ∈ [θp , θp+1 ] × CH0 × CH0 there exists a unique solution x(t) = x(t, σ, φ, ψ), t ≥ σ, of (8.2.3), and it satisfies the integral equation 

xij (t) = e−aij (t−σ ) φij (0) −

t

e−aij (t−s)



σ



Cijkl f (xkls , xklγ (s) )xij (s)

Ckl ∈Nr (i,j )

 − Lij (s) ds.

(8.3.7)

8.4 Bounded Solutions In this section, we will investigate the existence of a unique bounded solution of SI CN N (8.2.3). Moreover, the exponential stability of the bounded solution will be considered. An auxiliary result is presented in the following lemma. Lemma 8.3 ([16]) Assume that the conditions (C1)–(C4) are fulfilled. If H0 is a  ¯  2L(H0 + θ¯ L) positive number such that μθ¯ M + < 1, then a function x(t) = 1 − μθ¯ M / 0 xij , i = 1, 2, . . . , m, j = 1, 2, . . . , n, satisfying sup x(t) ≤ H0 is a solution t∈R

of (8.2.3) if and only if it satisfies the following integral equation:  xij (t) = −

t −∞

e−aij (t−s)





 Cijkl f (xkls , xklγ (s) )xij (s) − Lij (s) ds.

Ckl ∈Nr (i,j )

(8.4.8) Proof We consider only sufficiency. The necessity can be proved by using (8.3.7) in a very similar way to the ordinary differential equations case. One can obtain that   

t −∞

e−aij (t−s)

1 # ≤ MH0 aij





   Cijkl f (xkls , xklγ (s) )xij (s) − Lij (s) ds 

Ckl ∈Nr (i,j )



$ Cijkl + Lij .

Ckl ∈Nr (i,j )

Therefore, the integral in (8.4.8) is convergent. Differentiate (8.4.8) to verify that it is a solution of (8.2.3).  

186

8 SICNN with Functional Response on PCA

The following conditions are needed:  ¯  2L(H + θ¯ L) l¯ ; (C5) μθ¯ M + < 1, where H = 1 − M c¯ 1 − μθ¯ M (C6) (M+ 2LH )c¯ < 1; # $ ¯ (C7) 2d¯ M + LH eγ0 τ/2 1 + eγ0 θ/2 < 1. The main result concerning the existence and exponential stability of bounded solutions of (8.2.3) is mentioned in the next theorem. Theorem 8.1 ([16]) Suppose that conditions (C1)–(C6) hold. Then, (8.2.3) admits a unique bounded on R solution, which satisfies (8.4.8). If, additionally, the condition (C7) is valid, then the solution is exponentially stable with exponential convergence rate γ0 /2. Proof Let C0 (R) be the set of uniformly continuous functions defined on R such that if u(t) ∈ C0 (R), then u∞ ≤ H, where u∞ = sup u(t). Define on t∈R

C0 (R) the operator Π as  (Π u(t))ij ≡ −

t −∞

e−aij (t−s)





Cijkl f (ukls , uklγ (s) )uij (s)

Ckl ∈Nr (i,j )

 − Lij (s) ds.

(8.4.9)

/ 0 If u(t) = uij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , n, belongs to C0 (R), then we have that  t #  $   (Π u(t))ij  ≤ e−aij (t−s) Cijkl MH + Lij ds −∞

1 # = aij

Ckl ∈Nr (i,j )

$ Cijkl MH + Lij .

 Ckl ∈Nr (i,j )

Utilizing the last inequality one can show that (Π u)∞ ≤ cMH ¯ + l¯ = H. Therefore, Π u(t) ∈ C0 (R). / 0 Let us / verify 0 that this operator is contractive. Indeed, if u(t) = uij (t) and v(t) = vij (t) belong to C0 (R), then  t       (Π u(t))ij − (Π v(t))ij  ≤ e−aij (t−s) Cijkl f (ukls , uklγ (s) ) uij (s) −∞

Ckl ∈Nr (i,j )

   − vij (s)ds + ×

 Ckl ∈Nr (i,j )

t −∞



e−aij (t−s)

 Cijkl f (ukls , uklγ (s) )

8.4 Bounded Solutions

187

   − f (vkls , vklγ (s) ) vij (s) ds  t    ≤ e−aij (t−s) Cijkl M uij (s) − vij (s) ds −∞

 +

Ckl ∈Nr (i,j ) t

−∞

e−aij (t−s)

# Cijkl H L ukls − vkls 0

 Ckl ∈Nr (i,j )

. . $ + .uklγ (s) − vklγ (s) .0 ds  kl Ckl ∈Nr (i,j ) Cij u − v∞ . ≤ (M + 2LH ) aij Hence, the inequality Π u − Π v∞ ≤ (M + 2LH )c¯ u − v∞ is valid. In accordance with condition (C6), the operator /Π is contractive. Consequently, 0 SI CN N (8.2.3) admits a unique solution 1 v (t) = 1 vij (t) that belongs to C0 (R). We will continue with the investigation of the exponential stability. Fix an arbitrary number  > 0 and let δ be a sufficiently small positive number such that α1 < 1, α2 < 1, α3 < 1, and K (δ) < , where K (δ) =  ¯  δ 2L(H + δ + θ¯ L) ¯ M + , α2 = = μ θ , α 1 ¯ ¯ 1 − μθ¯ M 1 − 2d[M + LH eγ0 τ/2 (1 + eγ0 θ/2 )] ¯ (δ), and α3 = μθ¯ (M + 2LH ) + 2μθ¯ LK (δ). (M + 2LH )c¯ + 4LdK / 0 Suppose that 1 vσ (s) = η(s), s ∈ [−τ, 0]. Let u(t) = uij (t) be a solution of the network (8.2.3) with uσ (s) = φ(s), s ∈ [−τ, 0], where the function φ satisfies the inequality φ − η0 < δ. Without loss of generality we assume that γ (σ ) ≥ σ. Utilizing Lemma 8.2 one can verify for t ≥ σ that ( ) uij (t) − 1 vij (t) = e−aij (t−σ ) φij (0) − ηij (0)  t   − e−aij (t−s) Cijkl f (ukls , uklγ (s) )uij (s) σ

Ckl ∈Nr (i,j )

 vklγ (s) )1 vij (s) ds. − f (1 vkls ,1 0 / v (t). Then, w(t) satisfies the Denote by w(t) = wij (t) , the difference u(t) − 1 relation  t   ( ) vkls e−aij (t−s) Cijkl f (1 wij (t) = e−aij (t−σ ) φij (0) − ηij (0) − σ

Ckl ∈Nr (i,j )

 + wkls ,1 vklγ (s) + wklγ (s) )(1 vij (s) + wij (s)) − f (1 vkls ,1 vklγ (s) )1 vij (s) ds. (8.4.10)

188

8 SICNN with Functional Response on PCA

We will / solve0 Eq. (8.4.10) for σ = 0. Let Ψδ be the set of all continuous functions w(t) = wij (t) which are defined on [−τ, ∞) such that: (i) w(t) = φ(t) − η(t), t ∈ [−τ, 0]; (ii) w(t) is uniformly continuous on [0, +∞); (iii) ||w(t)|| ≤ K (δ)e−γ0 t/2 for t ≥ 0. Define on Ψδ an operator Π˜ such that ⎧ ⎪ ⎪ φij (t) − ηij (t), t ∈ [−τ, 0],  t ⎪  ⎪  ⎨ −aij t vkls + (φ (0) − η (0)) − e−aij (t−s) Cijkl f (1 e ij ij (Π˜ w(t))ij = 0 ⎪ Ckl ∈Nr (i,j ) ⎪  ⎪ ⎪ ⎩ w ,1 vij (s) + wij (s)) − f (1 vkls ,1 vklγ (s) )1 vij (s) ds, t > 0. kls vklγ (s) + wklγ (s) )(1

We shall show that Π˜ : Ψδ → Ψδ . Indeed, it is true for t ≥ 0 that −aij t ˜ |(Πw(t)) δ+ ij | ≤ e



t

e−aij (t−s)

0



Cijkl

Ckl ∈Nr (i,j )

   × f (1 vkls + wkls ,1 vklγ (s) + wklγ (s) ) − f (1 vkls ,1 vklγ (s) ) 1 vij (s) ds  t     vkls + wkls ,1 + e−aij (t−s) Cijkl f (1 vklγ (s) + wklγ (s) ) wij (s) ds 0

≤ e−aij t δ +

Ckl ∈Nr (i,j )



t 0



t

+

Ckl ∈Nr (i,j )



e−aij (t−s)

0

≤ e−aij t δ + t

+

t

= e−aij t δ +

Cijkl MK (δ)e−γ0 s/2 ds $ # ¯ Cijkl H LK (δ) eγ0 τ/2 + eγ0 (θ +τ )/2 e−γ0 s/2 ds



e−aij (t−s)

Ckl ∈Nr (i,j )



e−aij (t−s)

0

. ) . ( Cijkl H L wkls 0 + .wklγ (s) .0 ds

Ckl ∈Nr (i,j )

 0





e−aij (t−s)

Cijkl MK (δ)e−γ0 s/2 ds

Ckl ∈Nr (i,j )

'  * 2 Ckl ∈Nr (i,j ) Cijkl 2aij − γ0

¯

K (δ)[M + LH eγ0 τ/2 (1 + eγ0 θ /2 )]e−γ0 t/2 .

Thus, the inequality . . ¯ . .˜ ¯ (δ)[M + LH eγ0 τ/2 (1 + eγ0 θ/2 )]e−γ0 t/2 ≤ K (δ)e−γ0 t/2 .Π w(t). ≤ e−γ0 t δ + 2dK is valid for t ≥ 0.

8.4 Bounded Solutions

189

    1 (t) , w 2 (t) = w 2 (t) be elements of Ψ . One can Now, let w 1 (t) = wij δ ij confirm for t ≥ 0 that . .  t . ˜ 1 . 2 ˜ e−aij (t−s) .(Π w (t))ij − (Πw (t))ij . ≤ 0

#   2 $ vij (s) + wij Cijkl 1 (s)

 Ckl ∈Nr (i,j )

 # $ # $   1 1 2 2 vkls + wkls × f 1 − f 1 v ,1 vklγ (s) + wklγ + w ,1 v + w kls kls klγ (s) (s) klγ (s)  ds  t  # $    1 1 + vkls + wkls e−aij (t−s) Cijkl f 1 ,1 vklγ (s) + wklγ (s)  0

Ckl ∈Nr (i,j )

   1  2 × wij (s) − wij (s) ds  t  ≤ e−aij (t−s) 0

$ # Cijkl L H + K (δ)e−γ0 s/2

Ckl ∈Nr (i,j )

. . $ . #. . 1 . . 1 2 . 2 ds × .wkls − wkls − w . + .wklγ (s) klγ (s) . 0 0  t     1  2 + e−aij (t−s) Cijkl M wij (s) − wij (s) ds 0

Ckl ∈Nr (i,j )

kl . . . 1 . Ckl ∈Nr (i,j ) Cij 2 ≤ (M + 2LH ) sup .w (t) − w (t). (1 − e−aij t ) aij t≥0 kl # . . $ . Ckl ∈Nr (i,j ) Cij −γ0 t/2 . 1 2 +4LK (δ) sup .w (t) − w (t). e − e−aij t 2aij − γ0 t≥0

˜ 1 (t) − Πw ˜ 2 (t)|| ≤ α2 sup ||w 1 (t) − w 2 (t)||. Since Therefore, we have that sup ||Πw t≥0

t≥0

a contraction mapping argument that there exists α2 < 1, one can conclude by using / 0 a unique fixed point w 1(t) = w 1ij (t) of the operator Π˜ : Ψδ → Ψδ , which is a solution of (8.4.10). To complete the proof, we need to show that there does not exist a solution of (8.4.10) with σ = 0 different from w 1(t). Suppose that θ/p ≤ 0 0< θp+1 for w(t) = w ij (t) of (8.4.10) some p ∈ Z. Assume that there exists / a solution 0 different from w 1(t). Denote by z(t) = zij (t) the difference w(t) − w 1(t), and let max ||z(t)|| = m. ¯ It can be verified for t ∈ [0, θp+1 ] that

t∈[0,θp+1 ]



t

|zij (t)| ≤ 0

e−aij (t−s)

 Ckl ∈Nr (i,j )

  vij (s) + w Cijkl 1 1ij (s)

  vkls + wkls ,1 × f (1 vklγ (s) + w klγ (s) ) − f (1 vkls + w 1kls ,1 vklγ (s) + w 1klγ (s) ) ds

190

8 SICNN with Functional Response on PCA



t

+

e−aij (t−s)

0 t

≤

e−aij (t−s)

0 t

+ 0

   vkls + w kls ,1 Cijkl f (1 vklγ (s) + w klγ (s) ) zij (s) ds

Ckl ∈Nr (i,j )

 

  Ckl ∈Nr (i,j )

e−aij (t−s)



. . ) ( Cijkl L (H + K (δ)) zkls 0 + .zklγ (s) .0 ds   Cijkl M zij (s) ds

Ckl ∈Nr (i,j )

≤ θ¯ m ¯ [M + 2L (H + K (δ))]



Cijkl .

Ckl ∈Nr (i,j )

¯ Because α3 < 1 we obtain a contradiction. The last inequality yields z(t) ≤ α3 m. Therefore, w(t) = w 1(t) for t ∈ [0, θp+1 ]. Utilizing induction one can easily prove the uniqueness for all t ≥ 0.   Remark 8.1 In the proof of Theorem 8.1, we make use of the contraction mapping principle to prove the exponential stability. In the literature, Lyapunov–Krasovskii functionals, LMI technique, free weighting matrix method, and differential inequality technique were utilized to investigate the exponential stability in neural networks [33, 51, 66]. They may also be considered in the future to prove the exponential stability in networks of the form (8.2.3). The next section is devoted to the existence as well as the exponential stability of almost periodic solutions of (8.2.3).

8.5 Almost Periodic Solutions Let us denote by B0 (R) the set of all bounded and continuous functions defined on R. For g ∈ B0 (R) and α ∈ R, a translation of g by α is a function Qα g(t) = g(t + α), t ∈ R. A number α ∈ R is called an -translation number of a functional g ∈ B0 (R) if ||Qα g(t) − g(t)|| <  for every t ∈ R. Besides, a set S ⊂ R is said to be relatively dense if there exists a number h > 0 such that [ϑ, ϑ + h] ∩ S = ∅ for all ϑ ∈ R. A function g ∈ B0 (R) is said to be almost periodic, if for every positive number , there exists a relatively dense set of -translation numbers of g [43]. / 0 On the other hand, an integer  k0 is called  an -almost period of a sequence aqp , p ∈ Z, of real numbers if ap+k0 − ap  <  for any p ∈ Z [18, 49]. Let ζp = q ζ/p+q0 − ζp and θp = θp+q − θp for all p and q. We call the family of sequences q ζp p , q ∈ Z, uniformly almost periodic [18, 49] if for an arbitrary positive number / there exists a relatively dense set of -almost periods, common for all sequences q0 ζp p , q ∈ Z.

8.5 Almost Periodic Solutions

191

The following conditions are required: / q0 / q0 (C8) The sequences ζp , q ∈ Z, as well as the sequences θp , q ∈ Z, are uniformly almost periodic; (C9) There exist positive numbers θ and ζ such that θp+1 −θp ≥ θ and ζp+1 −ζp ≥ ζ for all p ∈ Z. It follows from condition (C8) that there exists a positive number θ¯ such that condition (C3) is valid, and |θp |, |ζp | → ∞ as |p| → ∞ [18, 49]. The next assertion is an analog of the first lemma on hybrid almost periods in Chap. 3. / 0 Lemma 8.4 ([16]) Assume that L(t) = Lij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , n is almost periodic in t and the conditions (C8), (C9) are valid. Then, for arbitrary η > 0, 0 < ν < η, there exist a respectively dense set of real numbers Ω and integers Q such that for α ∈ Ω and q ∈ Q, it is true that (i) L(t + α) − L(t) < η, t ∈ R; q (ii) |ζp − α| < ν, p ∈ Z; q (iii) |θp − α| < ν, p ∈ Z. The existence and exponential stability of the almost periodic solution of the network (8.2.3) is mentioned in the following theorem. Theorem 8.2 ([16]) Assume that the conditions (C1), (C2), (C4)–(C6), (C8), and (C9) are fulfilled. Then, the SI CNN (8.2.3) admits a unique almost periodic solution. If, additionally, the condition (C7) is valid, then the solution is exponentially stable with exponential convergence rate γ0 /2. Proof It follows/ from 0Theorem 8.1 that (8.2.3) admits a unique bounded on R solution u(t) = uij (t) , i = 1, 2, . . . , m, j = 1, 2, . . . , m, which is exponentially stable provided that the condition (C7) is valid. We will show that it is an almost periodic function. Consider the operator Π defined by Eq. (8.4.9) again. It is sufficient to verify that Π u(t) is almost periodic, if u(t) is. %  kl 1 Ckl ∈Nr (i,j ) Cij Let us denote β = max + (M + 3LH ) (i,j ) aij aij   4LH 2 C ∈Nr (i,j ) Cijkl kl . Fix an arbitrary positive number . Because Π u is + −aij θ 1−e

θ uniformly continuous, there exists a positive number η satisfying η < and 5    η≤ such that if t − t

 < 4η, then 3β . . .Π u(t ) − Π u(t

). <  . 305

(8.5.11)

192

8 SICNN with Functional Response on PCA

. . Next, we take a number ν with 0 < ν < η such that .u(t ) − u(t

). <  into

account  η whenever t − t  < ν, and let α and q satisfy the conditions of Lemma 8.4 such that α is an η-translation number for u(t). Assume that t ∈ (θp + η, θp+1 − η) for some p ∈ Z. Making use of the equation  (Π u(t + α))ij − (Π u(t))ij = −

t −∞

e−aij (t−s)



Cijkl

Ckl ∈Nr (i,j )

  × f (ukl(s+α) , uklγ (s+α) )uij (s + α) − f (ukls , uklγ (s) )uij (s) ds  t , + + e−aij (t−s) Lij (s + α) − Lij (s) ds, −∞

we obtain that   (Π u(t + α))ij − (Π u(t))ij   t    ≤ e−aij (t−s) Cijkl M uij (s + α) − uij (s) ds −∞

 +

t

−∞

 +

t

−∞

 +

t

−∞

Ckl ∈Nr (i,j )

e−aij (t−s)

 Ckl ∈Nr (i,j )

e−aij (t−s)

 Ckl ∈Nr (i,j )

. . Cijkl LH .ukl(s+α) − ukls .0 ds . . Cijkl LH .uklγ (s+α) − uklγ (s) .0 ds

  e−aij (t−s) Lij (s + α) − Lij (s) ds.

(8.5.12)

According to Lemma 8.4, (i), the inequality 

  η e−aij (t−s) Lij (s + α) − Lij (s) ds < a ij −∞ t

is valid. Moreover, since α is an η-translation number for u(t), one can confirm that 

t

−∞

and  t −∞

e−aij (t−s)

e−aij (t−s)

 Ckl ∈Nr (i,j )

 Ckl ∈Nr (i,j )

  Mη Cijkl M uij (s + α) − uij (s) ds < aij

. . LH η Cijkl LH .ukl(s+α) − ukls .0 ds < aij



Cijkl

Ckl ∈Nr (i,j )

 Ckl ∈Nr (i,j )

Cijkl .

8.5 Almost Periodic Solutions

193

On the other hand, we have 

t

−∞



Ckl ∈Nr (i,j )

t

θp +η ∞  

e−aij (t−s) θp−λ −η

θp−λ +η

λ=0 θp−λ −η



t

−∞

. . Cijkl LH .uklγ (s+α) − uklγ (s) .0 ds ≤

 Ckl ∈Nr (i,j )

λ=0 θp−λ−1 +η ∞  



e−aij (t−s)

. . Cijkl LH .uklγ (s+α) − ukl(γ (s)+α) .0 ds + 

e−aij (t−s)

Ckl ∈Nr (i,j )



e−aij (t−s)

Ckl ∈Nr (i,j )



e−aij (t−s)

Ckl ∈Nr (i,j )

(8.5.13)

. . Cijkl LH .uklγ (s+α) − ukl(γ (s)+α) .0 ds +

. . Cijkl LH .uklγ (s+α) − ukl(γ (s)+α) .0 ds +

. . Cijkl LH .ukl(γ (s)+α) − uklγ (s) .0 ds.

For any p ∈ Z, if s ∈ (θp + η, θp+1 − η), then one can show by using Lemma 8.4, (iii) that the number s + α belongs to the interval (θp+q , θp+q+1 ) so that  . .  .uγ (s+α) − uγ (s)+α . = max u(κ + ζp+q ) − u(κ + ζp + α) < η, 0 κ∈[−τ,0]

   q  since (κ + ζp+q ) − (κ + ζp + α) = ζp − α  < ν by Lemma 8.4, (ii). Besides, the inequality ∞  

θp−λ +η

λ=0 θp−λ −η

e−aij (t−s) ds ≤ 2η

∞ 

e−aij θλ =

λ=0

2η 1 − e−aij θ

is valid. Therefore, (8.5.13) yields 

t −∞

e−aij (t−s)

< LH η

 Ckl ∈Nr (i,j )

. . Cijkl LH .uklγ (s+α) − uklγ (s) .0 ds

'  2 Ckl ∈Nr (i,j ) Cijkl aij

+

4H



kl Ckl ∈Nr (i,j ) Cij 1 − e−aij θ

* .

It can be verified by means of (8.5.12) that  kl 1 Ckl ∈Nr (i,j ) Cij + (M + 3LH ) aij aij   4LH 2 Ckl ∈Nr (i,j ) Cijkl . + 1 − e−aij θ

  (Π u(t + α))ij − (Π u(t))ij  < η



194

8 SICNN with Functional Response on PCA

Hence, Π u(t + α) − Π u(t) < βη ≤ /3

(8.5.14)

for each t that belongs to the intervals (θp + η, θp+1 − η), p ∈ Z.   The inequality η < θ /5 ensures that t + 3η ∈ (θp + η, θp+1 − η) if t − θp  ≤ η. Now, utilizing the inequalities (8.5.11) and (8.5.14) we attain for t − θp  ≤ η, p ∈ Z, that Π u(t + α) − Π u(t) ≤ Π u(t + α) − Π u(t + α + 3η) + Π u(t + α + 3η) − Π u(t + 3η) + Π u(t + 3η) − Π u(t) < . The last inequality implies that α is an -translation number of Π u(t). Consequently, the SI CN N (8.2.3) admits a unique almost periodic solution.   Remark 8.2 The Bohr definition of almost periodicity is also suitable for the application of Lyapunov functional method and the technique of Young inequality [60] to show the existence, uniqueness, and exponential stability of almost periodic solutions in CN Ns.

8.6 An Example √ / 0 1 Consider the sequence θ = θp defined as θp = p+ | sin(p)−cos(p 2)|, p ∈ Z. 4 / q0 Utilizing the technique provided in [17, 18], one can verify that the sequences θp , q ∈ Z, are uniformly almost periodic. We take the function γ (t) with ζp = θp . One can confirm that the conditions (C3) and (C9) hold with θ¯ = 3/2 and θ = 1/2, respectively. Let us take into account the SI CNN  dxij = −aij xij − Cijkl f (xkl (γ (t) − τ ))xij + Lij (t), (8.6.15) dt Ckl ∈N1 (i,j )

s2 if |s| ≤ 0.1, f (s) = 0.005 if |s| > 0.1, in which i, j = 1, 2, 3, f (s) = 205 τ = 0.3, ⎛ ⎞ ⎛ ⎞ a11 a12 a13 9 3 5 ⎝ a21 a22 a23 ⎠ = ⎝ 6 5 4 ⎠ , 3 12 9 a31 a32 a33 ⎞ ⎛ ⎞ 0.08 0.01 0.02 C11 C12 C13 ⎝ C21 C22 C23 ⎠ = ⎝ 0.05 0.03 0.06 ⎠ , 0.04 0.07 0.02 C31 C32 C33 ⎛

8.6 An Example

195

⎛

⎞ L11 (t) L12 (t) L13 (t) ⎜ ⎟ ⎝ L21 (t) L22 (t) L23 (t) ⎠ L31 (t) L32 (t) L33 (t) ⎛ ⎞ √ √ 0.1 cos(t) + 0.2 sin( 205t) 0.2 cos(π t) + 0.1 sin( 2t) 0.15 cos(2t) − 0.12 cos(π t) √ √ ⎜ ⎟ = ⎝ 0.15 cos(3t) − 0.1 sin(π t) 0.2 cos(t) − 0.15 sin( 2t) 0.1 sin(t) + 0.2 cos( 305t) ⎠ . √ √ √ 0.2 cos( 2t) + 0.14 sin(π t) 0.2 cos( 2t) + 0.1 sin(t) 0.15 cos( 2t) − 0.13 cos(4t)

0

50

t

100

13

x 0

50

t

100

x23

0

50

t

100

0

50

t

33

0

50

t

100

100

0

50

100

0

50

100

t

0

t

0.02 0 −0.02

−0.02 0

50

0.05

100

0.02

0

0

−0.05

−0.05 0

0 −0.05

0.05

x32

x31

0 −0.05

0.1

−0.1

0.05

0.05

x

0.04 0.02 0 −0.02 −0.04

x12

0.04 0.02 0 −0.02 −0.04

x22

x

21

x

11

 kl = 0.17,  One can calculate that Ckl ∈N1 (1,1) C11 C kl = 0.25,  Ckl ∈N1 (1,2) 12  kl kl kl = 0.38, C = 0.12, Ckl ∈N1 (2,1) C21 = 0.28, Ckl ∈N1 (2,2) C22   Ckl ∈N1 (1,3) 13 kl = 0.19, kl C kl = 0.21, Ckl ∈N1 (3,1) C31 Ckl ∈N1 (3,2) C32 = 0.27, Ckl ∈N1 (2,3) 23 kl Ckl ∈N1 (3,3) C33 = 0.18. The conditions (C5) − (C7) are valid for (8.6.15) with γ0 = 3, μ = 0.38, c¯ = d¯ = 0.25/3, M = 0.005, L = 0.1, L¯ = 0.35, l¯ = 0.34/3. According to Theorem 8.2, the network (8.6.15) has a unique almost periodic solution, which is exponentially stable / with0 the rate of convergence 3/2. Consider the constant function φ(t) = φij (t) such that φ11 (t) = −0.025, φ12 (t) = 0.036, φ13 (t) = −0.014, φ21 (t) = 0.012, φ22 (t) = −0.021, φ23 (t) = 0.042, φ31 (t) = 0.023, φ32 (t) = −0.015, φ33 (t) = 0.012. We depict in Fig. 8.1 the / 0 1 solution x(t) = xij (t) of (8.6.15) with x(t) = φ(t), t ≤ σ = θ0 = . Figure 8.1 4 supports the result of Theorem 8.2 such that the represented solution converges to the unique almost periodic solution of SI CNN (8.6.15).

0

50

t

100

Fig. 8.1 The unique almost periodic solution of SI CN N (8.6.15)

t

196

8 SICNN with Functional Response on PCA

8.7 Note In this chapter, we investigate the existence as well as the exponential stability of almost periodic solutions in a new model of SI CN N s. The chapter contains results of the paper [16]. The usage of the functional response on alternate (advanceddelayed) type of piecewise constant arguments is the main novelty of our study, and it is useful for the investigation of a large class of neural networks. An illustrative example is provided to show the effectiveness of the theoretical results. Our approach concerning exponential stability may be used in the future to investigate synchronization of chaos in coupled neural networks and control of chaos in large communities of neural networks with piecewise constant argument. Differential equations with functional response on piecewise constant argument can be applied for the development of other kinds of recurrent networks such as Hopfield and Cohen–Grossberg neural networks [39, 52] and others. This will provide new opportunities for the analysis and applications of neural networks.

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Chapter 9

Differential Equations on Time Scales Through Impulsive Differential Equations

In this chapter we investigate differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (IDE). DETC are in some sense more general than dynamic equations on time scales [10, 18]. Basic properties of linear systems, existence and stability of periodic solutions and almost periodic solutions are considered. Appropriate examples are given to illustrate the theory.

9.1 Introduction The theory of dynamic equations on time scales (DETS) has been developed in the last several decades [1, 10, 18]. After a literature survey about DETS one can conclude that there are not so many results of the theory on the existence of periodic solutions and almost periodic solutions. To this moment, the investigations concerning linear DETS, integral manifolds, and the stability of equations have not been fully developed. Certainly, these results should be obtained to be able to benefit from the applications of the theory. In our paper, we make an attempt to expand our knowledge of these aspects of the theory. We also propose a way to obtain these theoretical results. Moreover, we investigate differential equations on certain time scales with transition conditions (DETC), which are in some sense more general than DETS. At the same time, we should recognize that significant theoretical results concerning oscillations, boundary value problems, positive solutions, hybrid systems, etc. have been achieved [1, 2, 7–14, 16, 18, 21]. We assume that our proposals may stimulate new ideas by which the theory can also be developed adding to the previous significant achievements in the direction. The main idea of the paper is to apply the results of the theory of impulsive differential equations (IDE) the investigated of which started in the late 1960s of the last century [15, 17, 20]. We note that certain classes of DETC, particular with time scales, can be reduced © Springer Nature Switzerland AG 2020 M. Akhmet, Almost Periodicity, Chaos, and Asymptotic Equivalence, Nonlinear Systems and Complexity 27, https://doi.org/10.1007/978-3-030-20572-0_9

201

202

9 Differential Equations on Time Scales

to IDE, if we apply a special transformation [3] of the independent argument (the time variable). This transformation allows the reduced IDE to inherit all similar properties of the corresponding DETC. Then the investigation of the IDE can proceed using the known results. Finally, by taking the properties of the independent argument transformation into account, we can make an interpretation of the obtained results for DETC.

9.2 Description of the DETC Throughout the chapter we consider a specific time scale of the following type. Fix a sequence {ti } ∈ R such that ti < ti+1 for all i ∈ Z, and |ti | → ∞ as |i| → ∞. Denote δi = t2i+1 − t2i , κi = t2i − t2i−1 and assume that:  ∞ (C0) κi = ∞. −∞ κi = ∞, 2∞ The time scale T0 = i=−∞ [t2i−1 , t2i ], is going to be considered throughout the paper. Consider the following system of differential equations: dy = f (t, y), t ∈ T0 , dt y(t2i+1 ) = Ji (y(t2i )) + y(t2i ),

(9.2.1)

where the derivative is one sided at the boundary points of T0 , f (t, y) : T0 × Rn → Rn , Ji (y) : Rn → Rn , for all i ∈ Z. We assume that functions f and J are continuous on their domains. More detailed characteristics of the functions will be given below when we consider specific problems. Let us introduce the following transition operator, Πi : {t2i } × Rn → {t2i+1 } × Rn , i ∈ Z, such that Πi (t2i , y) = (t2i+1 , Ji (y) + y). Thus the evolution of the process is described by: 1. the system of differential equations dy = f (t, y), dt

t ∈ T0 ;

(9.2.2)

2. the transition operator Πi , i ∈ Z; 3. the set T0 × Rn . We shall call Eq. (9.2.1) differential equation on time scales with transition condition (DETC). Let us show how to construct a solution of (9.2.1). Denote, by φ(t, κ, z), a solution of system (9.2.2) with an initial condition y(κ) = z, κ ∈ T0 , z ∈ Rn , and, by y(t), a solution of system (9.2.1) with an initial condition y(t 0 ) = y0 . Fix t 0 ∈ T0 such that t2k−1 < t 0 < t2k for some k ∈ Z. If t 0 ≤ t < t2k the solution is equal to φ(t, t 0 , y0 ), and y(t2k ) = φ(t2k −, t 0 , y0 ), where the left limit is assumed to exist. Now, applying the transition operator, we

9.2 Description of the DETC

203

obtain that y(t2k+1 ) = Jk (y(t2k )) + y(t2k ). Thus the solution is not defined in the interval (t2k , t2k+1 ). Next, on the interval [t2k+1 , t2(k+1) ) the solution is equal to φ(t, t2k+1 , y(t2k+1 )), and y(t2(k+1) ) = φ(t2(k+1) −, t2k+1 , y(t2k+1 )), and so on. If solution y(t) is defined on a set I ⊂ T0 , then the set {(t, y) : y = y(t), t ∈ I } is called an integral curve of the solution. Let us start with the general information about differential equations on time scales. We provide only those facts of the theory which directly concern the needs of our paper. More detailed description can be found in [1, 10, 18]. Any non-empty closed subset, T, of R is called a time scale. On a time scale the functions σ (t) := inf{s ∈ T : s > t} and ρ(t) := sup{s ∈ T : s < t} are called the forward and backward jump operators, respectively. The point t ∈ T is called right-scattered if σ (t) > t, and right-dense if σ (t) = t. Similarly, it is called left-scattered if ρ(t) < t, and left-dense if ρ(t) = t. Note that on time scale T0 , the points t2i−1 , i ∈ Z, are left-scattered and right-dense, and the points t2i , i ∈ Z, are right-scattered and left dense. Moreover, it is worth to mention that σ (t2i ) = t2i+1 , ρ(t2i+1 ) = t2i , i ∈ Z, and σ (t) = ρ(t) = t for any other t ∈ T0 . The Δ-derivative of a continuous function f, at a right-scattered point is defined as f (σ (t)) − f (t) , σ (t) − t

f Δ (t) :=

and at a right-dense point it is defined as f Δ (t) := lim s→t

f (t) − f (s) , t −s

if the limit exists. Let T be an arbitrary time scale. A function ϕ : T → R is called rd-continuous if: (i) it is continuous at each right-dense or maximal t ∈ T; (ii) the left sided limit ϕ(t−) = lim ϕ(ξ ) exists at each left-dense t. ξ →t −

Similarly, a function ϕ : T → R is called ld-continuous if: (i) it is continuous at each left-dense or minimal t ∈ T; (ii) the right sided limit ϕ(t+) = lim ϕ(ξ ) exists at each right-dense t. ξ →t +

A differential equation y Δ (t) = f (t, y),

t ∈T

(9.2.3)

is said to be a differential equation on time scale [18], where function f (t, y) : T × Rn → Rn in (9.2.3) is assumed to be rd-continuous on T × Rn .

204

9 Differential Equations on Time Scales

In our specific case we denote, by T0 , the set of all functions which are rdcontinuous on T0 . Moreover, we define a set of functions T01 ⊂ T which are continuously differentiable on T0 , assuming that the functions have a one-sided derivative at the boundary points of T0 , that is if φ ∈ T 10 , then φ ∈ T 0 . In general, by the derivative at boundary point, we mean a one-sided derivative.

9.3 ψ-Substitution It is common to simplify a given equation by a proper transformation in every theory of differential equations. Likewise, in this section, we introduce a transformation which plays the role of a bridge in the passage from DETC, as in (9.2.1), to an IDE. Without loss of2 generality, we assume that t−1 < 0 < t0 . The ψ-substitution, on the set T 0 = T0 \ ∞ i=−∞ {t2i−1 }, is defined as  ⎧ ⎪ t− δk , t ≥ 0 ⎪ ⎨ 0 0, there exists a relatively dense set of -translation numbers of f . The set of all almost periodic functions is denoted by A P(R). A function f ∈ C(R, Rn ) is called asymptotically almost periodic if there is a function g ∈ A P(R) and a function φ ∈ C(R, Rn ) with limt→∞ φ(t) = 0 such that f (t) = g(t) + φ(t). The basic definition of an almost periodic function given by H. Bohr has been modified by several authors [4, 6, 7, 10, 20, 25]. Below we introduce a new notion with regard to almost periodic functions. Definition 13.1 ([3]) A function f ∈ C(R, Rn ) is called bi-asymptotically almost periodic if f (t) = g(t) + φ(t) for some g ∈ A P(R) and φ ∈ C(R, Rn ) with limt→±∞ φ(t) = 0. Note that every bi-asymptotically almost periodic function is a pseudo almost periodic function [25], but not conversely. In this paper we are concerned with the linear system y = [A(t) + B(t)]y,

(13.1.1)

which may be viewed as a perturbation of x = A(t)x,

(13.1.2)

where x, y ∈ Rn , and A, B ∈ C(R, Rn×n ). Moreover we consider the quasilinear systems of the form y = Cy + f (t, y)

(13.1.3)

and the corresponding homogeneous linear system x = Cx,

(13.1.4)

where x, y ∈ Rn , C ∈ Rn×n , and f (t, x) ∈ C(R × Rn , Rn ) such that f (t, 0) = 0

for all t ∈ R.

Definition 13.2 ([5, 15]) A homeomorphism H between the sets of solutions x(t) and y(t) is called an asymptotic equivalence if y(t) = H (x(t)) implies that x(t)− y(t) → 0 as t → ∞.

13.1 Introduction and Preliminaries

311

Definition 13.3 A homeomorphism H between the sets of solutions x(t) and y(t) is called a bi-asymptotic equivalence if y(t) = H (x(t)) implies that x(t)−y(t) → 0 as t → ±∞. Our main objective is to investigate the problem of asymptotic equivalence of systems and to prove the existence of asymptotically and bi-asymptotically almost periodic solutions of (13.1.1) and (13.1.3). The classical theorem of Levinson [13] states that if the trivial solution of (13.1.2) is uniformly stable, A(t) ≡ A, and 

∞

|| B(t) || dt < ∞,

(13.1.5)

0

then (13.1.1) and (13.1.2) are asymptotically equivalent. In the case when A is not a constant matrix, Wintner [22] proved that the above conclusion remains valid if all solutions of (13.1.2) are bounded, (13.1.5) is satisfied, and  lim inf t→∞

t

Trace[B(s)] ds > −∞.

0

Later, Yakuboviˇc [23] considered (13.1.3) and obtained asymptotic equivalence of (13.1.4) and (13.1.3), see [15, 23] for details. After the pioneering works of Levinson, Wintner, and Yakuboviˇc, the problem of asymptotic equivalence of differential systems including linear, nonlinear, and functional equations has been investigated by many authors; see, e.g., [5, 11–19], and the references cited therein. Two interesting articles in this direction which also motivate our study here in this paper were written by Ráb [17, 18]. In fact, the main result in [18] is an improvement of the earlier one in [17], which we have employed in our work. Asymptotically almost periodic functions were introduced by Fréchet [8, 9]. The existence of this type of solutions was investigated by Fink [7] (Theorem 9.5) for the first time. For more results on the existence of asymptotically almost periodic solutions of different types of equations we refer to [11, 12, 14, 16, 21, 24] and the references cited therein. In this work we exploit the idea of A.M. Fink to obtain the existence of asymptotically almost periodic solutions of linear and quasilinear systems. Moreover, we prove a theorem about bi-asymptotic equivalence of linear systems and a theorem on the existence of bi-asymptotically almost periodic solutions. Apparently the notions of bi-asymptotic equivalence and a bi-asymptotically almost periodic function are introduced for the first time in the paper [3]. The chapter is organized as follows. In the next section, we prove a main lemma of Ráb and obtain sufficient conditions concerning the asymptotic equivalence of (13.1.1) and (13.1.2), and the existence of a family of asymptotically almost periodic solutions of the system (13.1.1). The third section is devoted to the problem of the asymptotic equivalence of systems (13.1.4) and (13.1.3) and the problem of existence of asymptotically almost periodic solutions of the system (13.1.3). The last section concerns with bi-asymptotic equivalence problem and the existence of

312

13 Asymptotic Equivalence and Almost Periodic Solutions

bi-asymptotically almost periodic solutions of (13.1.1). In addition, examples are given to illustrate the results.

13.2 Asymptotic Equivalence of Linear Systems and Asymptotically Almost Periodic Solutions Let X(t), X(0) = I , be a fundamental matrix solution of (13.1.2). Setting y = X(t)u, we easily see from (13.1.1) that u = P (t)u,

(13.2.6)

where P (t) = X−1 (t)B(t)X(t). Assume that  ∞ ||P (t)||dt < ∞. (C1 ) 0

The following lemma has been obtained Ráb [17, 18] for which we include a proof for convenience. Lemma 13.1 ([17]) If (C1 ) is valid, then the matrix differential equation Ψ = P (t)(Ψ + I )

(13.2.7)

has a solution Ψ (t) which satisfies Ψ (t) → 0 as t → ∞. Proof Construct a sequence of n × n matrices {Ψk } defined on R+ = [0, ∞) as follows:  ∞ Ψ0 (t) = I, Ψk (t) = − P (s)Ψk−1 (s) ds for k = 1, 2, . . . . t

Fix , 0 <  < 1. In view of (C1 ) there exists a t1 > 0 such that 

∞

||P (s)||ds < 

for all t > t1 .

t

 It follows that ||Ψk (t)|| <  k , k ∈ N, and consequently the series ∞ k=1 Ψk (t) is ∞ convergent uniformly for t ∈ [t1 , ∞). Letting Ψ (t) = k=1 Ψk (t), one can easily check that Ψ satisfies  ∞ Ψ (t) = − P (s)[I + Ψ (s)]ds (13.2.8) t

13.2 Asymptotically Almost Periodic Solutions of Linear Systems

313

and hence it is a solution (13.2.7). From (13.2.8) it also follows that Ψ → 0 as t → ∞, which completes that proof.   We may assume that (C2 ) limt→∞ X(t)Ψ (t) = 0. Theorem 13.1 ([3]) Suppose that conditions (C1 ) and (C2 ) hold. Then (13.1.1) and (13.1.2) are asymptotically equivalent. Proof Let t be sufficiently large, t ≥ t1 say. In view of (13.2.8) we see that the function u(t) = [I + Ψ (t)]c, c ∈ Rn , is a solution of (13.2.6) defined on [t1 , ∞) and hence y(t) = X(t)[I + Ψ (t)]c

(13.2.9)

is a solution of (13.1.1). Since Ψ (t) → 0 as t → ∞, there exists a t2 > t1 such that I + Ψ (t2 ) is nonsingular. Let x 0 = X(t2 )c and y 0 = X(t2 )(I + Ψ (t2 ))c. Denote by y(t, c) = y(t, t2 , x 0 ) and x(t, c) = x(t, t2 , y 0 ) the solutions of (13.1.1) and (13.1.2) satisfying x(t2 ) = x 0 and y(t2 ) = y 0 , respectively. Now, because of the existence and uniqueness of solutions of linear differential equations and the fact that I + Ψ (t2 ) is nonsingular, the relation y 0 = X(t2 )[I + Ψ (t2 )]X−1 (t2 )x 0 defines an isomorphism between solutions x(t) of (13.1.2) and y(t) of (13.1.1) such that y(t) = x(t) + X(t)Ψ (t)c for t > t1 . The last equality, in view of (C2 ), completes the proof.

 

Corollary 13.1 ([3]) Suppose that the system (13.1.2) has a k-parameter (k ≤ n) family σ2 of almost periodic solutions, and that the conditions (C1 ), (C2 ) are satisfied. Then there exists a k-parameter family σ1 of asymptotic almost periodic solutions of (13.1.1), and σ1 is isomorphic σ2 . Example 13.1 Consider the systems x

− 2(t + 1)−2 x = 0

(13.2.10)

y

− [2(t + 1)−2 + b(t)]y = 0,

(13.2.11)

and

314

13 Asymptotic Equivalence and Almost Periodic Solutions

where b(t) is a continuous function defined on R+ . We assume that there exist real numbers K1 > 0 and α > 0 such that | b(t) |< K1 e−α t

for all t ∈ R+ .

(13.2.12)

Notice that (13.2.10) has solutions x1 (t) = (t + 1)2 and x2 (t) = (t + 1)−1 . If we transform the above second order equations into systems of the form (13.1.1) and (13.1.2) we identify that  A=

0 1 2/(t + 1)2 0



 and B =

 0 0 . b(t) 0

It is easy to see that for a given  > 0 there exists K > 0 such that || P (t) ||≤ Ke(−α+)t

for all t ∈ R+ .

(13.2.13)

Fix  so that β := α +  < 0. Then (13.2.6) is satisfied, i.e., 

∞

||P (t)|| dt < ∞,

0

and || Ψ (t) ||≤ eK|β|

−1 eβt

− 1.

(13.2.14)

Moreover, using (13.2.13) and (13.2.14) one can show that X(t)Ψ (t) → 0 as t → ∞, i.e., (C2 ) holds. Since the conditions of Theorem 13.1 are fulfilled, we may conclude that (13.2.10) and (13.2.11) are asymptotically equivalent whenever (13.2.12) holds. Example 13.2 Let b(t) be a continuous function such that |b(t)| ≤ K1 e−αt for all t ∈ R+ for some α > 0, K1 > 0, and C ∈ R5×5 . Consider y = (A + B(t))y, where ⎛

0 ⎜ −1 ⎜ ⎜ A=⎜ 0 ⎜ ⎝ 0 0

1 0 0 0 0

0 0 0 −π 0

0 0 π 0 0

B(t) = b(t)C, and β > 0 satisfies α − 2β > 0.

⎞ 0 0⎟ ⎟ ⎟ 0 ⎟, ⎟ 0⎠ β

(13.2.15)

13.3 Asymptotic Equivalence of Linear and Quasilinear Systems

315

The associated equation x = Ax has a fundamental matrix ⎛

⎞ cos t sin t 0 0 0 ⎜ − sin t cos t 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎟ X(t) = ⎜ 0 0 cos π t sin π t 0 ⎟ . ⎜ ⎟ ⎝ 0 0 sin π t cos π t 0 ⎠ 0 0 0 0 eβt The equality P (t) = X−1 (t)B(t)X(t) = b(t)X−1 (t)CX(t) implies that there exists a K > 0 such that ||P (t)|| ≤ Ke−(α−β)t for all t ∈ R+ . Therefore, (C1 ) is valid. We also see that ||Ψ (t)|| ≤

∞  (Ke−(α−β)t )k k=1

(α

− β)k k!

= 1 − eK(α−β)

−1 e−(α−β)t

,

and hence X(t)Ψ (t) → 0 as t → ∞. In view of Corollary 13.1 we conclude that system (13.2.15) has a 4-parameter family of asymptotically almost periodic solutions. More precisely they are asymptotically “quasi-periodic” solutions and every such solution has a torus as the ω-limit set.

13.3 Asymptotic Equivalence of Linear and Quasilinear Systems Let α = minj λj and β = maxj λj , where λj denotes the real part of the eigenvalue λj of the matrix C. Let mα and mβ be the maximum of degrees of elementary divisors of C corresponding to eigenvalues with real part equal to α and β, respectively. Clearly, there exist constants κ1 , κ2 such that ||eCt || ≤ κ1 t mβ −1 eβt

and ||e−Ct || ≤ κ2 t mα −1 e−αt

for all t ∈ R+ = [0, ∞). The following conditions are to be assumed: (C3 ) ||f (t, x1 ) − f (t, x2 )|| ≤ η(t)||x1 − x2 || for all (t, x1 ), (t, x2 ) ∈ R+ × R n , and for some  nonnegative function η(t) defined on R+ ; ∞

(C4 ) L :=

t mβ +mα −2 e(β−α)t η(t) dt < ∞.

0

Lemma 13.2 ([3]) If (C3 ) and (C4 ) are valid, then every solution of u = e−Ct f (t, eCt u)

(13.3.16)

316

13 Asymptotic Equivalence and Almost Periodic Solutions

is bounded on R+ and for each solution u of (13.3.16) there exists a constant vector cu ∈ Rn such that u(t) → cu as t → ∞. Proof Let u(t) = u(t, t0 , u0 ) denote the solution of (13.3.16) satisfying u(t0 ) = u0 , t0 ≥ 0. It is clear that 

t

u(t) = u0 +

e−Cs f (s, eCs u(s)) ds,

t ≥ t0 .

t0

By using (C3 ) and f (t, 0) = 0, we see that  |u(t)| ≤ |u0 | + k1

t

s mβ +mα −2 e(β−α)s η(s)|u(s)| ds,

t ≥ t0

t0

for some k1 > 0. In view of (C4 ) and Gronwall’s inequality, we have  |u(t)| ≤ |u0 | e

t

k1

s mβ +mα −2 e(β−α)s η(s) ds

t0

≤ |u0 | ek1 L < ∞,

t ≥ t0 .

Let M0 = max{|u(t)| : t ∈ [0, t0 ]} and M = max{M0 , |u0 | ek1 L }. Then we have |u(t)| ≤ M for all t ∈ R+ . To prove the second part of the theorem, we first note that  t      e−Cs f (s, eCs u(s)) ds  ≤ Mk1   t0

∞

t mβ +mα −2 e(β−α)t η(t)dt < ∞.

0

So we may define  cu = u0 +

∞

e−Cs f (s, eCs u(s)) ds.

t0

It follows that 

∞

u(t) = cu −

e−Cs f (s, eCs u(s))ds,

t

 

which completes the proof. The following lemma can be easily justified by a direct substitution.

Lemma 13.3 ([3]) If y(t) is a solution of (13.1.3), then there is a solution u(t) of (13.3.16) such that y(t) = eCt u(t).

(13.3.17)

13.3 Asymptotic Equivalence of Linear and Quasilinear Systems

317

Conversely, if u(t) is a solution of (13.3.16), then y(t) in (13.3.17) is a solution of (13.1.3). Theorem 13.2 ([3]) If conditions (C3 ) and (C4 ) are satisfied, then every solution y(t) of (13.1.3) possesses an asymptotic representation of the form y(t) = eC t [c + o(1)], where c ∈ Rn is a constant vector and for a solution u(t) of (13.3.16), 

∞

o(1) = −

e−Cs f (s, eCs u(s)) ds.

t

Proof The proof follows from Lemma 13.2 and 13.3.

 

Theorem 13.3 ([3]) Assume that (C3 ) and (C4 ) are fulfilled, and  ∞ (C5 ) lim (s − t)mα −1 s mβ −1 eα(t−s) eβs η(s)ds = 0. t→∞ t

Then (13.1.3) and (13.1.4) are asymptotically equivalent. Proof In view of Lemma 13.2 we see that  y(t) = e [cu − Ct

t ∞

 = x(t) −

∞

e−Cs f (s, eCs u(s))ds]

eC(t−s) f (s, eCs u(s))ds,

t

where x(t) = eCt cu is a solution of (13.1.4) and u(t) = u(t, t0 , u0 ) is a solution of (13.3.16). It is clear that a given u0 results in a homeomorphism between x(t) and y(t). In view of (C5 ), we also see that x(t) − y(t) → 0 as t → ∞, which completes the proof of the theorem.   In [23], Yakubovich proved that if  lim

t→∞ t

∞

s mβ +mα −2 eβs η(s)ds = 0

(13.3.18)

then (13.1.3) and (13.1.4) are asymptotically equivalent. It is clear that if α > 0 then condition (C5 ) is weaker than (13.3.18). The following assertion is a simple corollary of Theorem 13.3 Corollary 13.2 ([3]) Suppose that conditions (C3 ), (C4 ), (C5 ) hold, and that system (13.1.4) has a k-parameter (k ≤ n) family γ1 of almost periodic solutions. Then (13.1.3) admits a k-parameter family γ2 of asymptotically almost periodic solutions, and γ1 is homeomorphic γ2 .

318

13 Asymptotic Equivalence and Almost Periodic Solutions

13.4 Bi-asymptotic Equivalence and Almost Periodic Solutions of Linear Systems With regard to systems (13.1.1) and (13.1.2) we shall make use of the following conditions: (C6 ) A(−t) = −A(t) for all t ∈ R. (C7 ) B(−t) = B(t) for all t ∈ R. We will rely on the following two lemmas. The first lemma is almost trivial. Lemma 13.4 ([3]) If (C6 ) is satisfied, then X(−t) = X(t) for all t ∈ R, and if in addition (C7 ) holds, then P (−t) = −P (t) for all t ∈ R. Lemma 13.5 ([3]) Assume that conditions (C1 ), (C6 ), (C7 ) are valid. Then (13.2.7) has a solution Ψ (t) which satisfies Ψ (−t) = Ψ (t) for all t ∈ R and Ψ → 0 as t → ∞. Proof By Lemma 13.1 there exists a solution Ψ+ (t) of (13.2.7) which is defined for t ≥ t1 and satisfies Ψ+ → 0 as t → ∞. Using P (−t) = −P (t) we see that 

−t1 −∞

 ||P (s)|| ds =

∞

||P (s)|| ds < .

t1

We may define a sequence of n × n matrices {Ψ˜ k } for t ∈ (−∞, 0] as follows: Ψ˜ k =

Ψ˜ 0 (t) = I,



t

−∞

P (s)Ψ˜ k−1 (s) ds for k = 1, 2, . . . .

As in the proof of Lemma 13.1, the matrix function Ψ− (t) =  Ψ (t) =

t

−∞

∞

˜

k=1 Ψk (t)

P (s)(I + Ψ (s))ds

and hence becomes a solution of (13.2.7) for t ≤ −t1 . On the other hand, since Ψ0 (−t) = Ψ˜ 0 (t) = I we have  Ψ1 (−t) = −  =

∞ −t

t −∞

 P (s)Ψ0 (s)ds =

−∞

P (−s)Ψ0 (−s)ds t

P (s)Ψ˜ 0 (s)ds = Ψ˜ 1 (t).

satisfies

13.5 Note

319

It follows by induction that Ψk (−t) = Ψ˜ k (t) for all k = 0, 1, 2, . . . and for all t ≤ −t1 . Hence Ψ+ (−t) = Ψ− (t) for all t ≤ −t1 . Continuing Ψ+ and Ψ− as solutions of (13.2.7), one can obtain that Ψ+ (−t) = Ψ− (t) for all t ≤ 0. Define " Ψ+ if t ≥ 0, Ψ (t) = Ψ− if t < 0. Clearly Ψ (t) is a solution of (13.2.7) satisfying Ψ (−t) = Ψ (t) for all t ∈ R and Ψ → 0 as t → ∞. This completes the proof.   The following results are analogous to Theorem 13.1 and Corollary 13.1. Theorem 13.4 ([3]) Suppose that (C1 ), (C2 ), (C6 ), (C7 ) are valid. Then (13.1.1) and (13.1.2) are bi-asymptotically equivalent. Corollary 13.3 ([3]) Suppose that (C1 ), (C2 ), (C6 ), (C7 ) are valid, and that (13.1.2) has a k-parameter (k ≤ n) family ν1 of almost periodic solutions. Then (13.1.1) admits a k-parameter family ν2 of bi-asymptotically almost periodic solutions, and ν1 is isomorphic ν2 . Example 13.3 Consider the system y = (A(t) + B(t))y,

(13.4.19)

where A(t) =

sin π t 0√ 0 sin 5t

,

B(t) = cos t e−α|t| C with α > 0 a real number and C ∈ R2×2 . Applying Corollary 13.3 one can conclude that every solution of (13.4.19) is bi-asymptotically quasi-periodic.

13.5 Note The problem of asymptotic equivalence of systems of differential equations is one of the most important in the qualitative analysis of the systems. In this chapter we ´ applied the Rab’s lemma [17, 18] for asymptotics of solutions of linear systems. New sufficient conditions for the asymptotic equivalence not only linear, but also quasilinear systems of ordinary differential equations were obtained. Thus, we developed the Yakubovich’s result [15, 23] on the asymptotic equivalence of a linear and a quasilinear system such that in this chapter most general results are obtained for asymptotic equivalence of differential equations. We proved the equivalence and

320

13 Asymptotic Equivalence and Almost Periodic Solutions

the existence of asymptotically almost periodic solutions of the systems under more weak conditions, and they are for the moment most developed ones. Moreover, in this chapter the definitions of bi-asymptotic equivalence for the equations and biasymptotically almost periodic solutions are introduced. Theorems on the sufficient conditions for the systems to be bi-asymptotically equivalent and for the existence of bi-asymptotically almost periodic solutions are obtained. Appropriate examples are constructed. The content of the chapter is in accordance with our papers [1–3].

References 1. M.U. Akhmet, M. Tleubergenova, On asymptotic equivalence of impulsive linear homogeneous differential systems. Math. J. 2(2), 15–18 (2002) 2. M.U. Akhmet, M. Tleubergenova, Asymptotic equivalence of the quasi-linear impulsive differential equation and the linear ordinary differential equation. Miskolc Math. Notes 8, 117– 121 (2007) 3. M.U. Akhmet, M. Tleubergenova, A. Zafer, Asymptotic equivalence of differential equations and asymptotically almost periodic solutions. Nonlinear Anal. Theory Methods Appl. 67, 1870–1877 (2007) 4. L. Amerio, G. Prouse, Almost Periodic Functions and Functional Equations (Van Nostrand Reinhold Company, New York, 1961) 5. L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations (Springer, New York, 1963) 6. C. Corduneanu, Almost Periodic Functions (Interscience, New York, 1961) 7. A.M. Fink, Almost periodic differential equations, in Lecture Notes in Mathematics (Springer, New York, 1974) 8. M. Fréchet, Sur le théoréme ergodique de Birkhoff (French). C. R. Acad. Sci. Paris 213, 607– 609 (1941) 9. M. Fréchet, Les fonctions asymptotiquement presque-periodiques continues (French). C. R. Acad. Sci. Paris 213, 520–522 (1941) 10. A. Halanay, D. Wexler, Qualitative Theory of Impulsive Systems (Edit. Acad. RPR, Bucuresti, 1968) (Romanian) 11. M.E. Hernández, M.L. Pelicer, Asymptotically almost periodic and almost periodic solutions for partial neutral differential equations. Appl. Math. Lett. 18, 1265–1272 (2005) 12. E.M. Hernández, M.L. Pelicer, J.P.C. Dos Santos, Asymptotically almost periodic and almost periodic solutions for a class of evolution equations. Electron. J. Diff. Equ. 61, 15 (2004) 13. N. Levinson, The asymptotic nature of solutions of linear systems of differential equations. Duke Math. J. 15, 111–126 (1948) 14. T. Morozan, Asymptotic almost periodic solutions for Riccati equations of stochastic control. Stud. Cerc. Mat. 46, 603–612 (1994) 15. V.V. Nemytskii, V.V. Stepanov, Qualitative theory of Differential Equations (Princeton University, Princeton, 1966) 16. V.F. Puljaev, Z. B. Caljuk, Asymptotically almost periodic solutions of a Volterra integral equation (Russian). Math. Anal. Kuban. Gos. Univ. Naucn. Trudy 180, 127–131 (1974) 17. M. Ráb, Über lineare perturbationen eines systems von linearen differentialgleichungen. Czech. Math. J. 83, 222–229 (1958) 18. M. Ráb, Note sur les formules asymptotiques pour les solutions d’un systéme d’équations différentielles linéaires. Czech. Math. J. 91, 127–129 (1966) 19. S. Saito, Asymptotic equivalence of quasilinear ordinary differential systems. Math. Jpn. 37, 503–513 (1992)

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20. A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations (World Scientific, Singapore, 1995) 21. C. Varsan, Stochastic differential equations and asymptotic almost periodic solutions. Rev. Roumaine Math. Pures Appl. 35, 485–493 (1990) 22. A. Wintner, Linear variations of constants. Am. J. Math. 68, 185–213 (1946) 23. V.A. Yakubovich, On the asymptotic behavior of systems of differential equations. Math. Sbornik 28, 217–240 (1951) 24. R. Yuan, Existence of almost periodic solutions of functional-differential equations of neutral type. J. Math. Anal. Appl. 165, 524–538 (1992) 25. C. Zhang, Pseudo almost periodic solutions of some differential equations. J. Math. Anal. Appl. 181, 62–76 (1994)

Chapter 14

Asymptotic Equivalence of Hybrid Systems

In the present chapter we extend the method of the asymptotic equivalence to hybrid equations. In Sect. 14.1 impulsive linear homogeneous differential equations are under discussion. A partial result is also proved when one of the equations is of delay type. Next, in Sect. 14.2 we obtain new sufficient conditions for the asymptotic equivalence of the quasilinear impulsive system of differential equations and the linear system of ordinary differential equations. It is easy to see that the results are generalizations of Chap. 12 such that if one removes the impulsive parts in equations of this chapter then the results of the last chapter will be obtained. The main goal of this section is to obtain sufficient conditions for the asymptotic equivalence of a linear system of ordinary differential equations and a quasilinear system of differential equations with piecewise constant argument. The results of the section were published in the paper [1]. They are one more confirmation of the universality of the approach which have been developed in papers [3, 4, 6]. Asymptotic equilibriums of linear integro-differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and solutions of linear integro-differential equations are established. A new Gronwall–Bellman type lemma for integro-differential inequalities is proved. An example is constructed to demonstrate the validity of one of the results.

14.1 On Asymptotic Equivalence of Impulsive Linear Homogeneous Differential Systems Sufficient conditions of asymptotic equivalence of impulsive linear homogeneous differential equations are obtained. A partial result is also proved when one of the equations is of delay type.

© Springer Nature Switzerland AG 2020 M. Akhmet, Almost Periodicity, Chaos, and Asymptotic Equivalence, Nonlinear Systems and Complexity 27, https://doi.org/10.1007/978-3-030-20572-0_14

323

324

14 Asymptotic Equivalence of Hybrid Systems

14.1.1 Introduction The problem of asymptotic equivalence for linear and nonlinear ordinary differential equations has been considered by many authors [10, 16, 18–20, 24, 27, 28] (see also bibliography of [10, 18]). And recently, there has been a lot of activity with solutions which undertake discontinuities or jumps at some definite instants [21]. It generates needs for investigation of the asymptotic behavior of impulsive systems [23]. Interesting article which motivates our study here in this article was written by Ráb [19]. Following the results of M. Ráb without requiring any special form and any boundedness condition for the solutions new results for asymptotic equivalence of impulsive systems are obtained. We should also note that Ráb did not consider asymptotic equivalence but rather gave asymptotic representation of solutions. Let Z, R be sets of all integers and real numbers, respectively, R+ = [t0 , ∞) for some t0 ∈ R, || · || be the Euclidean norm in R n , n ∈ N. We consider the following systems of impulsive differential equations: dx = A(t)x, t = θi , dt Δx(θi ) = Bi x(θi )

(14.1.1)

dy = [A(t) + C(t)]y, t = θi , dt Δy(θi ) = [Bi + Di ]y(θi )

(14.1.2)

and

where x, y ∈ R n , (C1 ) A(t), C(t), t ∈ Bi , Di , i ∈ Z, are real-valued n × n-matrices, A(t), C(t) ∈ C(R+ ); (C2 ) {θi } ⊂ R, t0 < θ1 < θ2 < . . . θi → ∞ as i → ∞; (C3 ) matrices Bi , Di satisfy the inequalities det(I + Bi ) = 0, det(I + Bi + Di ) = 0, for alli ∈ Z.

(14.1.3)

Remark 14.1 The conditions (C1 )–(C3 ) imply [21, p. 44] that solutions x(t, t0 , x0 ) and y(t, t0 , x0 ) of Cauchy’s problem for systems (14.1.1) and (14.1.2) with any x0 , y0 ∈ R n , t0 ∈ R exist and unique on R+ . The technique used in the present paper is also applied in a certain sense when (14.1.2) is replaced by an impulsive delay differential equation of the form dy = A(t)y(t) + C(t)y(t − τ )], t = θi , dt Δy(θi ) = Bi y(θi ) + Di y(θi−p ), where τ, p are positive real and integer numbers, respectively.

(14.1.4)

14.1 On Asymptotic Equivalence of Impulsive Linear Homogeneous. . .

325

14.1.2 Ordinary Differential Equations Let X(t) be a fundamental matrix of (14.1.1). We start with the change of dependent variable y(t) = X(t)u(t)

(14.1.5)

which transforms (14.1.2) into the system du = P (t)u, t = θi , dt Δu(ζi ) = Qi u(ζi )

(14.1.6)

where P (t) = X−1 (t)B(t)X(t), Qi = X−1 (θi +)Di X(θi ).

(14.1.7)

The substitution appearing in (14.1.5), which was also employed in [19, 27], is very crucial in the proof of our results. In fact, our method is based on a complete characterization of the function u(t) for t sufficiently large. Let us assume that  ∞  || P (t) || dt + || Qi ||< ∞ (14.1.8) t0

t0