Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems
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GLOBAL ATTRACTORS OF NON-AUTONOMOUS DISSIPATIVE
DYNAMICAL SYSTEMS
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Dedicated to my wife Ivanna
and my children Olga and Anatoli
v
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vi
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Preface
In the qualitative theory of differential equations non-local problems play an im-
portant role, especially in regard to questions of boundedness, periodicity, almost
periodicity, Poisson stability, asymptotic behaviour, dissipativity, etc.
The present work takes a similar approach and is dedicated to the study of
abstract non-autonomous dissipative dynamical systems and their application to
differential equations.
In applications there often occur systems
u
0
= f(t, u), (0.1)
which have every one of their solutions driven into fixed bounded domain and kept
there under further increase of time, becouse of natural dissipation. Such systems
are called dissipative ones
[
135
]
-
[
137
]
,
[
270
]
-
[
272
]
,
[
325
]
,
[
326
]
. Solutions of dissipative
systems are called limit (finally) bounded
[
325, 326
]
.
Dynamical systems occur in hydrodynamics studying turbulent phenomena,
meteorology, oceanography, theory of oscillations, biology, radio engineering and
other domains of sciences and engineering technics related to the study of asymp-
totic behaviour. Lately interest in dissipative systems increased even more
because of intensive elaboration of strange attractors (see, e.g.,
[
143, 239, 296,
306
]
).
The study of the dissipative systems there are dedicated plenty of works, begin-
ning from the classical works of N. Levinson. Among works on dissipative systems
of ordinary differential equations two directions can be made out.
To the first belong works which contain some conditions assuring the dissipativ-
ity of system (0.1), some class or concrete system, representing theoretical or applied
interest. Examples are works of P. V. Atrashenok
[
9
]
, B. P. Demidovich
[
135
]
-
[
137
]
,
V. I. Zubov
[
336
]
-
[
338
]
, V. M. Matrosov
[
251
]
, V. V. Nemytski
[
259
]
-
[
260
]
, V. A.
Pliss
[
270
]
, V. N. Schennikov
[
298, 299
]
, C. Corduneanu
[
125
]
, N. Levinson
[
237
]
, N.
Pavel
[
264
]
-
[
266
]
, R. Reissig
[
272
]
, T. Talpalaru
[
309
]
and a lot of other authors.
To the second direction belong works in which are studied inner conditions
vii
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viii Global Attractors of Non-autonomous Dissipative Dynamical Systems
of dissipative systems, that is, conditions relating to the character of behaviour
of solution of the system when assuming its dissipativity, for different classes of
differential equations. Among them are the works of V. M. Gershtein
[
159
]
-
[
163
]
,
V. V. Zhikov
[
331
]
-
[
332
]
, I. L. Zinchenko
[
334
]
, M. A. Krasnoselsky
[
159
]
, S. Yu.
Pilyugin
[
269
]
, V. A. Pliss
[
270
]
-
[
271
]
, M. L. Cartwright and J. E. Littlewood
[
44
]
-
[
46
]
, N. Levinson
[
237
]
, G. Fusco and M. Oliva
[
157
]
, J. Skowronski and S. Ziemba
[
307
]
and other authors.
For a compact map on a Banach space, V. M. Gerstein
[
162
]
, V. M. Ger-
stein and M. A. Krasnoselskii
[
159
]
investigated the existence of maximal com-
pact invariant set and studied some of its properties. In works of G. Billoti,
G. Cooperman, J. LaSalle, O. Lopes, P. Massat, M. Slemrod, J. Hale
[
175, 26,
124
]
,
[
171
]
-
[
176
]
,
[
235
]
,
[
246
]
-
[
248
]
and a lot of other authors many important results
obtained for ordinary differential equations
[
270
]
-
[
271
]
are generalized to functional-
differential equations.
Of late in the theory of partial differential equations there appeared works of
A. V. Babin and M. I. Vishik
[
12
]
-
[
16
]
, Ju. S. Ilyashenko
[
194
]
-
[
197
]
, O. A. La-
dyzhenskaya
[
230
]
, A. N. Sharkovsky
[
296
]
, R. Temam
[
314
]
and other authors,
in which, are studied evolutionary equations with maximal attractors (dissipative
evolutionary equations).
We note that all works mentioned above (with rare exceptions) studied periodical
or autonomous systems. These results are presented in monographs
[
175
]
,
[
270
]
-
[
272
]
,
[
296
]
.
If the right hand side f of the equation (0.1) is non-periodic, e.g. quasi-periodic
(almost periodic by Bohr, recurrent in sense the of Birkhoff, almost periodic by
Levitan, stable by Poisson) or depending on time in more complicated way, then
the situation essentially complicates already in the class of almost periodic systems.
It is caused at least by two reasons.
First, the definition of dissipativity in the non-autonomous case needs to be made
more precise because Levinson’s definition in the class of non-periodical systems
divides on some non equivalent notions and we need to choose one which allows
us develop a general theory which would contain as particular case most essential
results obtained for periodical dissipative systems.
Second, in the study of periodic dissipative systems an important role is played
by discrete dynamical system (cascade) generated by degrees of Poincar´e’s trans-
formation (mapping). For non-periodic systems there is no Poincar´e’s transforma-
tion and, consequently, the approach created for research on periodical dissipative
systems is not useful in the more general case. That is why to study non-periodical
dissipative systems we need new ideas; that is, making a theory of non-autonomous
dissipative dynamical systems demands making corresponding methods of research.
Our approach to the study of dissipative systems of differential equations consists
of drawing to the study of non-autonomous dissipative systems ideas and methods
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Preface ix
developed in the theory of abstract dynamical systems. We select one class of
dynamical systems (called in this work, dissipative) modelling the properties of
dissipative differential equations. The selected class is systematically researched
and then the general results obtained are applied to the study of dissipative systems
of differential and some other classes of equations.
The idea of applying methods of the theory of dynamical systems to the study
of non-autonomous differential equations is not new. It has been successfully
applied to the resolution of different problems in the theory of linear and non-
linear non-autonomous differential equations for more than thirty years. First
this approach to non-autonomous differential equations was introduced in works
of V. M. Millionshchikov
[
253
]
-
[
255
]
, B. A. Shcherbakov
[
300, 302
]
, L. G. Dey-
seach and G. R. Sell
[
139
]
, R. K. Miller
[
252
]
, G. Seifert
[
289
]
, G. R. Sell
[
290,
291
]
, later in works of V. V. Zhikov
[
331
]
, I. U. Bronshtein
[
32
]
, R. A. Johnson
[
203
]
-
[
204
]
and many other authors. This approach consists of naturally linking
with equation (0.1) a pair of dynamical systems and a homomorphism of the first
onto the second. In one dynamical system is put the information about right hand
side of equation (0.1) and in the other about the solutions of equation (0.1).
We note that there exists the another approach offered in the works of V. I.
Zubov
[
336
]
and then developed in works of C. M. Dafermos
[
128
]
-
[
131
]
, J. K. Hale
[
175
]
, I. Hitoshi
[
189
]
and many other authors (for details, see survey
[
303
]
). It con-
sists of linking with every non-autonomous differential equation a two-parametric
family of mappings (by terms of some authors – process).
In the beginning a these two approaches developed independently but later there
was found a relation between them (see, e.g.
[
130
]
,
[
187
]
).
This author adheres to the first approach because in his opinion it is better
adapted for resolving those problems which are studied in this work.
The proposed work consists of fourteen chapters.
In the first chapter for autonomous dynamical systems there are introduced and
studied different kinds of dissipativity: point, compact, local, bounded and weak
one. Criteria of point, compact and local dissipativity are given. It is shown that
for dynamical systems in locally compact spaces all three types of dissipativity
are equivalent. Examples are given showing that in the general case the notions of
point, compact and local dissipativity are different. The notion of Levinson’s center,
which is an important characteristic of compact dissipative system, is introduced.
The solution of one J. K. Hale’s problem for locally bounded dynamical systems is
given.
The second chapter is dedicated to non-autonomous dissipative dynamical
systems. It is noted that in the general case Levinson’s center of non-autonomous
dissipative dynamical system is not orbitally stable. The question of stability of
Levinson’s center of non-autonomous system is studied. A simple geometric descrip-
tion ensuring its stability is given, as is a description of Levinson’s center of non-