Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems
Подождите немного. Документ загружается.
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
x Global Attractors of Non-autonomous Dissipative Dynamical Systems
autonomous systems satisfying the condition of uniform positive stability. There are
pointed conditions of keeping the property of dissipativity under homomorphisms.
It is selected a class of dynamical systems which allow full description of Levinson’s
center’s structure, called in this work systems with convergence. Some criteria of
convergence in terms of Lyapunov’s functions depending on two space variables are
given. It is shown that for non-autonomous dissipative dynamical systems with
finite-dimensional phase space all three types of dissipativity are equivalent. There
are given series of conditions that are equivalent to dissipativity in finite-dimensional
space. At last, it is proved that for linear systems dissipativity reduces to conver-
gence. Also there are given series of conditions equivalent to dissipativity of linear
systems.
The third chapter deals mostly with one special class of non-autonomous dissi-
pative dynamical systems called in the work C-analytic. It is proved that C-analytic
dissipative dynamical system has the property of uniform positive stability on com-
pact subsets. Full description of Levinson’s center of these systems is given. One
general construction allowing to connect with given non-autonomous dynamical
system an autonomous dynamical system in space of continuous sections is given.
With the help of these constructions are studied quasi-periodic solutions of analytic
systems with quasi-periodic coefficients. In conclusion conditions are given which
guarantee the dissipativity of weakly nonlinear systems of differential equations,
as is a condition which assure the existence of almost periodic solution of weakly
nonlinear system with almost periodic coefficients in Levinson’s center.
The fourth chapter is dedicated to a study of Levinson’s center’s structure
with condition of hyperbolicity on closure of recurrent motion’s set. There we
establish some topological properties of Levinson’s center of compact dissipative
dynamical system. In particular, it is shown that Levinson’s center is indecompos-
able if the phase space of dynamical systems is also decomposable. It is proved
that in connected and locally connected space Levinson’s center of compact dissi-
pative dynamical system both with continuous and discrete time is a connected set.
There we establish some properties of a set of chain recurrent motions of dissipative
system. A theorem is proved about the spectral decomposition of Levinson’s cen-
ter which is analogous to known Smale’s theorem. For one-dimensional dissipative
dynamical systems it was proved theorem precising theorem about spectral decom-
position of Levinson’s center and, particularly, it was shown that Levinson’s center
of such systems contains a local maximal hyperbolic Markov set. In the end of the
chapter an application of obtained results to periodic systems is given.
In the fifth chapter we develop the method of Lyapunov’s functions for research
of non-autonomous dissipative dynamical systems in finite-dimensional space. Cri-
teria of dissipativity in terms of Lyapunov’s functions, with the help of which we
can get sufficient tests for dissipativity suitable in applications, are discassed. With
the help of Lyapunov’s functions there were proved series of tests for dissipativity of
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Preface xi
multi-dimensional non-autonomous differential equations. On the basis developed
for research of non-autonomous dissipative systems methods a criterion of asymp-
totic stability of zero section of non-autonomous systems has been obtained. In
particular, it is proved the analog of known Barbashin–Krasovsky’s theorem for
non-autonomous dynamical systems. There are established some tests for conver-
gence of systems of differential equations with the help of Lyapunov’s functions
depending on two space variable. There were proved tests for dissipativity and
convergence of some systems of differential equations of the 2nd and 3rd order
appearing in applications.
The sixth chapter is dedicated to some applications of general results obtained
in previous chapters to difference equations, equations with impulses, functional-
differential equations and evolutionary equations x
0
+Ax = f with uniformly mono-
tone operator A. In particular, there are given tests for dissipativity and con-
vergence of weakly nonlinear systems of difference equations and equations with
impulses. It is proved the criterion of asymptotic stability of linear functional-
differential equations. It is established test for convergence of evolutionary equation
x
0
+ Ax = f with uniformly monotone operator A.
In the seventh chapter we systematically study of the problem of upper semi-
continuity of compact global attractors and compact pullback attractors of abstract
non-autonomous dynamical systems for small perturbations. Several applications
of our results are given for different classes of evolutionary equations.
The eighth chapter is devoted to the study of the relationship between the
global attractor of the skew–product system and the pullback and forward attractors
of the cocycle system. We also note that forward attractors are stronger than
global attractors if we suppose a compact set of non-autonomous perturbations.
An example is presented in which the cartesian product of the component subsets
of a pullback attractor is not a global attractor of the skew–product flow. This
set is, however, a maximal compact invariant subset of the skew–product flow. By
a generalization of some stability results of V. I. Zubov
[
336
]
it is asymptotically
stable. Thus a pullback attractor always generates a local attractor of the skew–
product system, but this need not be a global attractor. If, however, the pullback
attractor generates a global attractor in the skew–product flow and if, in addition, its
component subsets depend lower continuously on the parameter, then the pullback
attractor is also a forward attractor. Several examples illustrating these results are
presented in the final section.
In the ninth chapter we systematically study the global pullback attractors of
C−analytic cocycles. For the large class of C−analytic cocycles we give the de-
scription of the structure of their pullback attractors. Particularly we prove that
it is trivial, i.e. the fibers of these attractors contain only one point. Several ap-
plications of these results are given (ODEs, Caratheodory’s equations with almost
periodic coefficients, almost periodic ODEs with impulse).
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
xii Global Attractors of Non-autonomous Dissipative Dynamical Systems
The tenth chapter is dedicated to the investigation of the effect of time discretiza-
tion on the pullback attractor of a non-autonomous ordinary differential equation
for which the vector fields depend on a parameter that varies in time rather than
depending directly on time itself. The parameter space is assumed to be compact
so the skew product flow formalism as well as cocycle formalism also applies and
the vector fields have a strong dissipative structure that implies the existence of
a compact set that absorbs all compact sets under the resulting non-autonomous
dynamics. The numerical scheme considered is a general 1–step scheme such as the
Euler scheme with variable time-steps. Our main result is to show that the numeri-
cal scheme interpreted as a discrete time non-autonomous dynamical system, hence
discrete time cocycle mapping and skew product flow on an extended parameter
space, also possesses a cocycle attractor and that its component subsets converge
upper semi–continuously to those of the cocycle attractor of the original system
governed by the differential equation. We will also see that the corresponding skew
product flow systems have global attractors with the cocycle attractor component
sets as their cross-sectional sets in the original state space. Finally, we investigate
the periodicity and almost periodicity of the discretized pullback attractor when
the parameter dynamics in the ordinary differential equation is periodic or almost
periodic and the pullback attractor consists of singleton valued component sets, i.e.
the pullback attractor is a single trajectory.
In the eleventh chapter we study the non-autonomous Navier-Stokes equations.
It is proved that such systems admit compact global attractors. This problem is
formulated and solved in the terms of general non-autonomous dynamical systems.
We give conditions of convergence of non-autonomous Navier-Stokes equations . A
test of existence of almost periodic (quasi periodic, recurrent, pseudo recurrent)
solutions of non-autonomous Navier-Stokes equations is given. We prove the global
averaging principle for non-autonomous Navier-Stokes equations.
The twelfth chapter is devoted to the investigation of the global attractors of
general V - monotone non-autonomous dynamical systems and their applications to
different class of differential equations (ODEs, ODEs with impulse, some class of
evolution partial differential equations).
In the thirteenth chapter we study the linear almost periodic dynamical systems.
The bounded solutions, relation between different types of stability and uniform
exponential stability for those systems are studied. We give several applications the
obtained results for ODEs, PDEs and functional-differential equations.
Chapter 14 is devoted to the study of quasi-linear triangular maps: chaos, almost
periodic and recurrent solutions, integral manifolds, chaotic sets etc. This problem
is formulated and solved in the framework of non-autonomous dynamical systems
with discrete time. We prove that such systems admit an invariant continuous
section (an invariant manifold). Then, we obtain the conditions of the existence of
a compact global attractor and characterize its structure. We give a criterion for the
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Preface xiii
existence of almost periodic and recurrent solutions of the quasi-linear triangular
maps. Finally, we prove that quasi-linear maps with chaotic base admits a chaotic
compact invariant set.
The results given in this work belong mostly to the author. The results of
chapters eight, ten and twelve are jointly obtained by the author, Peter Kloeden and
Bjoern Schmalfuss
[
96, 97, 100, 106
]
. The results of chapter eleven and paragraph
six of chapter six are obtained jointly with Jinqiao Duan
[
104, 107
]
. The results of
chapter fourteen are obtained jointly with Cristiana Mammana
[
108
]
.
The reader needs no deep knowledge of special branches of mathematics, al-
though it should easier for readers who know the fundamentals of the qualitative
theory of differential equations.
The quality of English of this exposition is adversely affected by its not being
the native tongue of the author, for which fact the author asks for the reader’s kind
indulgence and understanding.
Acknowledgment: The research described in this publication, in part, was
possible due to Award No. MM1-3016 of the Moldovan Research and Development
Association (MRDA) and the U.S. Civilian Research & Development Foundation
for the Independent States of the Former Soviet Union (CRDF).
June 2003
D. N. Cheban
cheban@usm.md
http://www.usm.md/davidcheban
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
xiv Global Attractors of Non-autonomous Dissipative Dynamical Systems
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Contents
Preface vii
Notations xxi
1. Autonomous dynamical systems 1
1.1 Some notions, notations and facts from theory of dynamical systems 1
1.2 Limit properties of dynamical systems . . . . . . . . . . . . . . . . . 8
1.3 Center of Levinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Dissipative systems on the local compact spaces . . . . . . . . . . . . 16
1.5 Criterions of compact dissipativity . . . . . . . . . . . . . . . . . . . 18
1.6 Local dissipative systems . . . . . . . . . . . . . . . . . . . . . . . . 25
1.7 Global attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8 On a Problem of J. Hale . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.9 Connectedness of the Levinson’s center . . . . . . . . . . . . . . . . . 38
1.10 Weak attractors and center of Levinson . . . . . . . . . . . . . . . . 43
1.11 Asymptotic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2. Non-autonomous dissipative dynamical systems 53
2.1 On the stability of Levinson’s center . . . . . . . . . . . . . . . . . . 53
2.2 The positively stable systems . . . . . . . . . . . . . . . . . . . . . . 60
2.3 Behaviour of dissipative dynamical systems under homomorphisms . 64
2.4 Non-autonomous dynamical systems with convergence . . . . . . . . 68
2.5 Tests for convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.6 Global attractors of non-autonomous dynamical systems . . . . . . . 89
2.7 Global attractor of cocycles . . . . . . . . . . . . . . . . . . . . . . . 93
2.8 Global attractors of non-autonomous dynamical system with minimal
base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.9 Homogeneous dynamical systems . . . . . . . . . . . . . . . . . . . . 101
xv
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
xvi Global Attractors of Non-autonomous Dissipative Dynamical Systems
2.10 Power-law asymptotic of homogeneous systems . . . . . . . . . . . . 107
2.11 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3. Analytic dissipative systems 119
3.1 Skew-product dynamical systems and cocycles . . . . . . . . . . . . . 119
3.2 C-analytic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.3 Converse of Lyapunov’s theorem for C-analytic systems . . . . . . . 128
3.4 On the structure of compact attracting sets of C-analytic systems . . 132
3.5 Dynamical systems in spaces of sections . . . . . . . . . . . . . . . . 136
3.6 Quasi-periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . 139
3.7 The analogy of Cameron-Johnson’s theorem . . . . . . . . . . . . . . 142
3.8 Almost periodic solutions of the weak nonlinear dissipative systems . 145
4. The structure of the Levinson center of system with the condition
of the hyperbolicity 155
4.1 The chain recurrent motions . . . . . . . . . . . . . . . . . . . . . . . 155
4.2 The spectral decomposition of the Levinson’s center . . . . . . . . . 157
4.3 One-dimensional systems with hyperbolic center . . . . . . . . . . . 159
4.4 The dissipative cascades . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.5 The periodic dissipative systems . . . . . . . . . . . . . . . . . . . . 167
5. Method of Lyapunov functions 171
5.1 Criterions of dissipativity in term of Lyapunov functions . . . . . . . 171
5.2 Some criterions of dissipativity of differential equations . . . . . . . . 178
5.3 Theorem of Barbashin–Krasovskii for non-autonomous dynamical
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.4 Equations with convergence . . . . . . . . . . . . . . . . . . . . . . . 192
5.5 Dissipativity and convergence of some equations of 2nd and
3rd order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.6 Construction of Lyapunov function for homogeneous systems . . . . 208
5.7 Differentiable homogeneous systems . . . . . . . . . . . . . . . . . . 213
5.8 Global attractors of quasi-homogeneous systems . . . . . . . . . . . . 220
6. Dissipativity of some classes of equations 225
6.1 Difference equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.2 Equations with impulse . . . . . . . . . . . . . . . . . . . . . . . . . 228
6.3 Convergent periodic equations with impulse . . . . . . . . . . . . . . 232
6.4 Asymptotic stability of linear functional differential equations . . . . 236
6.5 Convergence of monotone evolutionary equations . . . . . . . . . . . 238
6.6 Global attractors of non-autonomous Lorenz systems . . . . . . . . . 244
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Contents xvii
6.6.1 Non-autonomous Lorenz systems . . . . . . . . . . . . . . . . 245
6.6.2 Non-autonomous dissipative dynamical systems and their
attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.6.3 Almost periodic and recurrent solutions of non-autonomous
Lorenz systems . . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.6.4 Uniform averaging principle . . . . . . . . . . . . . . . . . . . 253
6.6.5 Global averaging principle for the non-autonomous Lorenz
systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
7. Upper semi-continuity of attractors 263
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.2 Maximal compact invariant sets . . . . . . . . . . . . . . . . . . . . . 263
7.3 Upper semi-continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.4 Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
7.5.1 Quasi-homogeneous systems . . . . . . . . . . . . . . . . . . . 274
7.5.2 Monotone systems . . . . . . . . . . . . . . . . . . . . . . . . 276
7.5.3 Quasi-linear systems . . . . . . . . . . . . . . . . . . . . . . . 278
7.5.4 Non-autonomously perturbed systems . . . . . . . . . . . . . 279
7.5.5 Non-autonomous 2D Navier-Stokes equations . . . . . . . . . 280
7.5.6 Quasi-linear functional-differential equations . . . . . . . . . . 284
8. The relationship between pullback, forward and global attractors 287
8.1 Pullback, forward and global attractors . . . . . . . . . . . . . . . . 288
8.2 Asymptotic stability in α-condensing semi–dynamical systems . . . . 294
8.3 Uniform pullback attractors and global attractors . . . . . . . . . . . 299
8.4 Examples of uniform pullback attractors . . . . . . . . . . . . . . . . 300
8.4.1 Periodic driving systems . . . . . . . . . . . . . . . . . . . . . 300
8.4.2 Pullback attractors with singleton component sets . . . . . . 302
8.4.3 Distal dynamical systems . . . . . . . . . . . . . . . . . . . . 304
9. Pullback attractors of C-analytic systems 307
9.1 C-analytic cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
9.2 Some general facts about non-autonomous dynamical systems . . . . 310
9.3 Positively uniformly stable cocycles . . . . . . . . . . . . . . . . . . . 314
9.4 The compact global pullback attractors of C-analytic cocycles with
compact base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.5 The uniform dissipative cocycles with noncompact base . . . . . . . 319
9.6 The compact and local dissipative cocycles with noncompact base . 323
9.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
xviii Global Attractors of Non-autonomous Dissipative Dynamical Systems
9.7.1 ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
9.7.2 Caratheodory differential equations . . . . . . . . . . . . . . . 328
9.7.3 ODEs with impulses . . . . . . . . . . . . . . . . . . . . . . . 329
10. Pullback attractors under discretization 331
10.1 Non-autonomous dynamical systems and pullback attractors . . . . . 333
10.2 Non-autonomous quasi-linear differential equation . . . . . . . . . . 334
10.3 Cocycle property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
10.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
10.4.1 Existence of an absorbing set . . . . . . . . . . . . . . . . . . 341
10.4.2 Upper semi–continuity of the pullback attractor component
sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
10.4.3 Upper semi–continuous convergence of the discretized
pullback attractors . . . . . . . . . . . . . . . . . . . . . . . . 345
10.4.4 Upper semi–continuous convergence of the discretized
global attractors . . . . . . . . . . . . . . . . . . . . . . . . . 346
10.5 Singleton set-valued pullback attractor case . . . . . . . . . . . . . . 348
10.6 Appendix: Proof of Lemma 10.4 . . . . . . . . . . . . . . . . . . . . 353
11. Global attractors of non-autonomous Navier-Stokes equations 357
11.1 Non-autonomous Navier-Stokes equations . . . . . . . . . . . . . . . 358
11.2 Attractors of non-autonomous dynamical systems . . . . . . . . . . . 367
11.3 Almost periodic and recurrent solutions of non-autonomous Navier-
Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
11.4 Uniform averaging for a finite interval . . . . . . . . . . . . . . . . . 374
11.5 The global averaging principle for Navier-Stokes equations . . . . . . 379
12. Global attractors of V -monotone dynamical systems 385
12.1 Global attractors of V -monotone NDS . . . . . . . . . . . . . . . . . 385
12.2 On the structure of Levinson center of V -monotone NDS . . . . . . . 389
12.3 Almost periodic solutions of V -monotone systems . . . . . . . . . . . 391
12.4 Pullback attractors of V -monotone NDS . . . . . . . . . . . . . . . . 394
12.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
12.5.1 Finite-dimensional systems . . . . . . . . . . . . . . . . . . . . 395
12.5.2 Caratheodory’s differential equations . . . . . . . . . . . . . . 399
12.5.3 ODEs with impulse . . . . . . . . . . . . . . . . . . . . . . . . 400
12.5.4 Evolution equations with monotone operators . . . . . . . . . 402
13. Linear almost periodic dynamical systems 407
13.1 Bounded motions of linear systems . . . . . . . . . . . . . . . . . . . 407
September 27, 2004 16:46 WSPC/Book Trim Size for 9.75in x 6.5in GlobalAttractors/chapter 0
Contents xix
13.2 Bounded solutions of linear equations . . . . . . . . . . . . . . . . . 416
13.3 Finite-dimensional systems . . . . . . . . . . . . . . . . . . . . . . . 422
13.4 Relationship between different types of stability . . . . . . . . . . . . 426
13.5 Linear α-condensing systems . . . . . . . . . . . . . . . . . . . . . . 434
13.6 Exponential stable systems . . . . . . . . . . . . . . . . . . . . . . . 437
13.7 Linear system with a minimal base . . . . . . . . . . . . . . . . . . . 439
13.8 Some classes of uniformly exponentially stable equations . . . . . . . 441
13.9 Linear periodic systems . . . . . . . . . . . . . . . . . . . . . . . . . 450
13.9.1 Exponential stable linear periodic dynamical systems . . . . . 451
13.9.2 Some classes of linear uniformly exponentially stable
periodic differential equations . . . . . . . . . . . . . . . . . . 456
14. Triangular maps 461
14.1 Triangular maps and non-autonomous dynamical systems . . . . . . 462
14.2 Linear non-autonomous dynamical systems . . . . . . . . . . . . . . 463
14.3 Quasi-linear non-autonomous dynamical systems . . . . . . . . . . . 467
14.4 Global attractors of quasi-linear triangular systems . . . . . . . . . . 469
14.5 Almost periodic and recurrent solutions . . . . . . . . . . . . . . . . 475
14.6 Pseudo recurrent solutions . . . . . . . . . . . . . . . . . . . . . . . . 477
14.7 Chaos in triangular maps . . . . . . . . . . . . . . . . . . . . . . . . 478
Bibliography 481
Index 501