Table Of ContentSpringer Monographs in Mathematics
David N. Cheban
Nonautonomous
Dynamics
Nonlinear Oscillations and Global
Attractors
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David N. Cheban
Nonautonomous Dynamics
Nonlinear Oscillations and Global Attractors
123
DavidN.Cheban
Faculty of Mathematics
State University of Moldova
Chisinau, Moldova
ISSN 1439-7382 ISSN 2196-9922 (electronic)
SpringerMonographs inMathematics
ISBN978-3-030-34291-3 ISBN978-3-030-34292-0 (eBook)
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Mathematics Subject Classification (2010): Primary: 34C12, 34C27, 34D25, 37B55, 37B35, 37B20,
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To my grandchildren
Olivia and Nicholas
Preface
The present monograph is dedicated to the abstract theory of nonau-
tonomous dynamical systems, which is a new branch of the theory of
dynamical systems.
In this monograph, I present the developments of the basic ideas and
methods for nonautonomous dynamical systems and their applications over
the past ten years.
Ourmainapplicationsarenonautonomousordinarydifferential/difference
equations, functional differential/difference equations and some classes of
partial differential equations.
In recent years, there seems to be a growing interest in nonautonomous
differential/difference equations, both finite-dimensional (ordinary
differential/difference equations) and infinite-dimensional (functional
differential/differenceequationsandpartialdifferentialequations).
Nonlocal problems concerning the conditions of existence of different
classes of solutions play an important role in the qualitative theory of dif-
ferential equations. Here we include the problems of boundedness, periodic-
ity,Bohr/Levitanalmostperiodicity,almostautomorphy,almostrecurrence
in the sense of Bebutov, recurrence in the sense of Birkhoff, stability in the
sense of Poisson, problems of existence of limit regimes of different types,
convergence, dissipativity, etc.
The present work belongs to this direction, and it is devoted to the
mathematical theory of nonautonomous dynamical systems and applica
tions. The main goal of this book is to study Bohr/Levitan almost periodic
and almost automorphic systems,different classes ofPoisson stable motions,
and global attractors of Bohr/Levitan almost periodic systems with contin-
uous and discrete time.
Thus,therearetwokeyobjectsthatarethesubjectsofstudyinthisbook.
These are various oscillatory regimes (Bohr/Levitan almost periodic and
Poisson stable movements) and global attractors and application of the
obtained general results (related to abstract nonautonomous dynamical
vii
viii Preface
systems) to different classes of nonautonomous differential and difference
equations.
The problems that we consider in this book are mainly motivated by
nonautonomous differential/difference equations.
The monograph presents ideas and methods, developed by the author, to
solve the problem of existence of Bohr/Levitan almost periodic (respectively,
almostrecurrentinthesenseofBebutov,almostauthomorphic,Poissonstable)
solutions, and global attractors of nonautonomous differential/difference equa-
tions. Namely, the text provides answers to the following problems:
1. ProblemofexistenceofatleastoneBohr/Levitanalmostperiodicsolution
for linear almost periodic differential/difference equations without
Favard’s separation condition (Favard theory);
2. Problem of existence of Bohr/Levitan almost periodic solution for
monotone differential/difference equations;
3. Problem of existence of at least one Bohr/Levitan almost periodic
solution for uniformly stable and dissipative differential equations
(I. U. Bronshtein’s conjecture, 1975);
4. Problem of description of the structure of the global attractor for holo-
morphic and gradient-like nonautonomous dynamical systems.
Chapters I and IV–VI are devoted to nonlinear oscillations, and global
attractors are studied in Chaps. II, III, and VI.
One fundamental question of the qualitative theory of nonautonomous
differential/differenceequationsistheproblemofalmostperiodicity,ormore
generally Poisson stability (in particular, Levitan almost periodicity,
Bochner almost automorphy, almost recurrence in the sense of Bebutov,
recurrence in the sense of Birkhoff, and so on) of solutions.
Thetheoryofalmostperiodicfunctionswasmainlycreatedandpublished
byBohr[42–45](inthisrelationseealsotheimportantresultsofBohl[40,41]
and Esclangon [145–147]). Bohr’s theory was substantially extended by
Bochner[35],Weyl[321],Besicovitch[22],Favard[151],vonNeumann[235],
Stepanoff [308], Bogolyubov [37–39] and others.
Levitan [212] introduced a new class offunctions (the so-called N-almost
periodic or Levitan almost periodic functions) that includes all Bohr almost
periodic functions, but does not coincide with the latter. The foundation of
this type of function was created in the works of Levitan [214], Levin [210,
211], and Marchenko [220, 221].
A notion of almost automorphic function was introduced by Bochner
[34–36] which also is an extension of Bohr almost periodicity. Substantial
resultsaboutalmostautomorphicfunctionswereobtainedbyVeech[314–316].
ThedifferentclassesofPoissonstablefunctions(inparticular,recurrentin
the sense of Birkhoff [25], almost recurrent in the sense of Bebutov [13],
pseudo recurrent [283, 284, 302], and so on) were introduced and studied by
Shcherbakov [283–296].
Preface ix
The theory of Bohr/Levitan almost periodic, almost automorphic, and
PoissonstablefunctionsiswidelypresentedinthemonographsofAmerioand
Prouse [3], Bochner [35], Corduneanu [129, 130], Toka Diagana [140],
Fink [155],Favard[151],Hinoetal.[179],Levitan[214],LevitanandZhikov
[215], Pankov [244], N’Guerekata [236, 237], Shcherbakov [290, 294],
Shen and Yi [297], Yoshizawa [324], Zaidman [325] and others (see also the
references therein).
Inthelast25–30years,thetheoryofBohr/Levitanalmostperiodic,almost
automorphic and Poisson stable differential/difference equations has been
developed in connection with problems of differential/difference equations,
stability theory, dynamical systems, and so on. The main achievements are
related to the application of ideas and methods of dynamic systems in the
study of the above problems.
Globalattractorsplayaveryimportantroleinthestudyoftheasymptotic
behavior of dynamical systems (both autonomous and nonautonomous). In
thelast20–25yearsmanyworksdedicatedtothestudyofglobalattractorsof
dynamical systems (including the infinite-dimensional systems) have been
published.See,forexample,BabinandVishik[9],Chueshov[109],Hale[171],
Ladyzhenskaya [208], Robinson [255], Temam [310, 311] (for autonomous
systems), Carvalho et al. [67], Cheban [84, 91], Chepyzhov and Vishik [106],
Haraux [174], Kloeden and Rasmussen [197] (for nonautonomous systems),
and references therein.
In this book we study global attractors for a special class of nonau-
tonomous dynamical systems, namely for the Bohr/Levitan almost periodic
systems. We establish the structure of global attractors for this class of
systems and the existence at least one almost periodic motion belonging to
the Levinson center (maximal compact global invariant attractor).
Our approach to the study of the problem of Bohr/Levitan almost peri-
odicity of solutions of almost periodic differential equations and their com-
pact global attractors consists in applying to the study of nonautonomous
systemstheideasandmethodsdevelopedinthetheoryofabstractdynamical
systems.Theideaofapplyingmethodsofthetheoryofdynamicalsystemsto
the study of nonautonomous differential equations is not new. It has been
successfully applied to the resolution of different problems in the theory of
linear and nonlinear nonautonomous differential equations for more than 50
years. This approach to nonautonomous differential equations was first
introduced in the works of Millionshchikov [226–228], Shcherbakov [290,
294], Deyseach and Sell [139], Miller [225], Seifert [273], Sell [275, 276], later
inworksofZhikov[327],Bronshtein[48],Johnson[188,189],andmanyother
authors. This approach consists in naturally associating with the equation
x0 ¼fðt;xÞ ð1Þ
x Preface
apairofdynamicalsystemsandahomomorphismofthefirstontothesecond.
One assigns the information about the right-hand side of Eq. (1) to one
dynamicalsystem,andtheinformationaboutthesolutionsof(1)totheother.
PlentyofworksarededicatedtothestudyoftheproblemofBohr/Levitan
almost periodicity, almost automorphy, and different classes of Poisson sta-
bility of solutions for differential/difference equations. We survey briefly
some of these works in our book.
Note that a bibliography of papers on Bohr/Levitan almost periodic,
almost automorphic, and Poisson stable solutions and compact global
attractors of almost periodic differential/difference equations contains over
300 items, i.e., it is still a very active area of research.
The body of the book consists of six chapters.
In the first chapter, on semigroup dynamical systems, different kinds of
Poisson stability of motions and their comparability by character of recur-
rence are introduced and studied: Bohr/Levitan almost periodicity, almost
automorphy, Bebutov almost recurrence, Birkhoff recurrence, pseudorecur-
rence, and other types of Poisson stability.
Thesecondchapter isdedicatedtothestudyofcompactglobalattractors
of dynamical systems (both autonomous and nonautonomous). For autono-
mous systems, we study different kinds of dissipativity for autonomous
dynamical systems: point, compact, local, and bounded. Criteria of point,
compact, and local dissipativity are given. We show that for dynamical
systems in locally compact spaces, any three types of dissipativity are
equivalent.Wegiveexamplesshowingthatinthegeneralcase,thenotionsof
point, compact, and local dissipativity are distinct. The notion of the
Levinson center, which is an important characteristic of compact dissipative
systems, is introduced. We study the dissipative nonautonomous dissipative
dynamical systems with minimal (in particular, with Bohr almost periodic,
almost automorphic, or recurrent in the sense of Birkhoff) base. We give a
description of the Levinson center of nonautonomous systems satisfying the
condition of uniform positive stability. We give series of conditions that are
equivalent to dissipativity in finite-dimensional space, and we prove that for
linear systems, dissipativity reduces to convergence. Also we give series of
conditions equivalent to dissipativity of linear systems.
The third chapter is dedicated to the study of one special class of
nonautonomous dissipative dynamical systems that we call C-analytic. We
prove that a C-analytic dissipative dynamical system has the property of
uniform positive stability on compact subsets. A full description of the
Levinson center of these systems is given. Finally, we study C-analytic dis-
cretedynamicalsystemsoninfinite-dimensionalspaces.Apositiveanswerto
the Belitskii-Lyubich conjecture (for C-analytic discrete dynamical systems
and flows) is given.
In the fourth chapter we present some new results about Bohr/Levitan
almost periodic, almost automorphic, and Poisson stable solutions of linear
differential equations that complement the classical theory of Favard. In
