Global Attractors of Non-autonomous Dissipative Dynamical Systems

Preface......Page 8
Contents......Page 16
Notations......Page 22
1.1 Some notions, notations and facts from theory of dynamical systems......Page 25
1.2 Limit properties of dynamical systems......Page 32
1.3 Center of Levinson......Page 34
1.4 Dissipative systems on the local compact spaces......Page 40
1.5 Criterions of compact dissipativity......Page 42
1.6 Local dissipative systems......Page 49
1.7 Global attractors......Page 53
1.8 On a Problem of J. Hale......Page 57
1.9 Connectedness of the Levinson's center......Page 62
1.10 Weak attractors and center of Levinson......Page 67
1.11 Asymptotic stability......Page 71
2.1 On the stability of Levinson's center......Page 77
2.2 The positively stable systems......Page 84
2.3 Behaviour of dissipative dynamical systems under homomorphisms......Page 88
2.4 Non-autonomous dynamical systems with convergence......Page 92
2.5 Tests for convergence......Page 103
2.6 Global attractors of non-autonomous dynamical systems......Page 113
2.7 Global attractor of cocycles......Page 117
2.8 Global attractors of non-autonomous dynamical system with minimal base......Page 122
2.9 Homogeneous dynamical systems......Page 125
2.10 Power-law asymptotic of homogeneous systems......Page 131
2.11 Linear systems......Page 137
3.1 Skew-product dynamical systems and cocycles......Page 143
3.2 C-analytic systems......Page 147
3.3 Converse of Lyapunov's theorem for C-analytic systems......Page 152
3.4 On the structure of compact attracting sets of C-analytic systems......Page 156
3.5 Dynamical systems in spaces of sections......Page 160
3.6 Quasi-periodic solutions......Page 163
3.7 The analogy of Cameron-Johnson's theorem......Page 166
3.8 Almost periodic solutions of the weak nonlinear dissipative systems......Page 169
4.1 The chain recurrent motions......Page 179
4.2 The spectral decomposition of the Levinson's center......Page 181
4.3 One-dimensional systems with hyperbolic center......Page 183
4.4 The dissipative cascades......Page 188
4.5 The periodic dissipative systems......Page 191
5.1 Criterions of dissipativity in term of Lyapunov functions......Page 195
5.2 Some criterions of dissipativity of differential equations......Page 202
5.3 Theorem of Barbashin{Krasovskii for non-autonomous dynamical systems......Page 211
5.4 Equations with convergence......Page 216
5.5 Dissipativity and convergence of some equations of 2nd and 3rd order......Page 228
5.6 Construction of Lyapunov function for homogeneous systems......Page 232
5.7 Differentiable homogeneous systems......Page 237
5.8 Global attractors of quasi-homogeneous systems......Page 244
6.1 Difference equations......Page 249
6.2 Equations with impulse......Page 252
6.3 Convergent periodic equations with impulse......Page 256
6.4 Asymptotic stability of linear functional differential equations......Page 260
6.5 Convergence of monotone evolutionary equations......Page 262
6.6 Global attractors of non-autonomous Lorenz systems......Page 268
6.6.1 Non-autonomous Lorenz systems......Page 269
6.6.2 Non-autonomous dissipative dynamical systems and their attractors......Page 272
6.6.3 Almost periodic and recurrent solutions of non-autonomous Lorenz systems......Page 275
6.6.4 Uniform averaging principle......Page 277
6.6.5 Global averaging principle for the non-autonomous Lorenz 6.6.5 Global averaging principle for the non-autonomous Lorenz......Page 283
7.2 Maximal compact invariant sets......Page 287
7.3 Upper semi-continuity......Page 290
7.4 Connectedness......Page 296
7.5.1 Quasi-homogeneous systems......Page 298
7.5.2 Monotone systems......Page 300
7.5.3 Quasi-linear systems......Page 302
7.5.4 Non-autonomously perturbed systems......Page 303
7.5.5 Non-autonomous 2D Navier-Stokes equations......Page 304
7.5.6 Quasi-linear functional differential equations......Page 308
The relationship between pullback, forward and global attractors......Page 311
8.1 Pullback, forward and global attractors......Page 312
8.2 Asymptotic stability in α-condensing semi-dynamical systems......Page 318
8.3 Uniform pullback attractors and global attractors......Page 323
8.4.1 Periodic driving systems......Page 324
8.4.2 Pullback attractors with singleton component sets......Page 326
8.4.3 Distal dynamical systems......Page 328
Pullback attractors of C-analytic systems......Page 331
9.1 C-analytic cocycles......Page 333
9.2 Some general facts about non-autonomous dynamical systems......Page 334
9.3 Positively uniformly stable cocycles......Page 338
9.4 The compact global pullback attractors of C-analytic cocycles with 9.4 The compact global pullback attractors of C-analytic cocycles with......Page 340
9.5 The uniform dissipative cocycles with noncompact base......Page 343
9.6 The compact and local dissipative cocycles with noncompact base......Page 347
9.7.1 ODEs......Page 350
9.7.2 Caratheodory differential equations......Page 352
9.7.3 ODEs with impulses......Page 353
Pullback attractors under discretization......Page 355
10.1 Non-autonomous dynamical systems and pullback attractors......Page 357
10.2 Non-autonomous quasi-linear differential equation......Page 358
10.3 Cocycle property......Page 362
10.4.1 Existence of an absorbing set......Page 365
10.4.2 Upper semi-continuity of the pullback attractor component sets......Page 368
10.4.3 Upper semi-continuous convergence of the discretized pullback attractors......Page 369
10.4.4 Upper semi-continuous convergence of the discretized global attractors......Page 370
10.5 Singleton set-valued pullback attractor case......Page 372
10.6 Appendix: Proof of Lemma 10.4......Page 377
Global attractors of non-autonomous Global attractors of non-autonomous......Page 381
11.1 Non-autonomous Navier-Stokes equations......Page 382
11.2 Attractors of non-autonomous dynamical systems......Page 391
11.3 Almost periodic and recurrent solutions of non-autonomous Navier-Stokes equations......Page 393
11.4 Uniform averaging for a......Page 398
11.5 The global averaging principle for Navier-Stokes equations......Page 403
12.1 Global attractors of V -monotone NDS......Page 409
12.2 On the structure of Levinson center of V -monotone NDS......Page 413
12.3 Almost periodic solutions of V -monotone systems......Page 415
12.4 Pullback attractors of V -monotone NDS......Page 418
12.5.1 Finite-dimensional systems......Page 419
12.5.2 Caratheodory's differential equations......Page 423
12.5.3 ODEs with impulse......Page 424
12.5.4 Evolution equations with monotone operators......Page 426
13.1 Bounded motions of linear systems......Page 431
13.2 Bounded solutions of linear equations......Page 440
13.3 Finite-dimensional systems......Page 446
13.4 Relationship between different types of stability......Page 450
13.5 Linear  alpha-condensing systems......Page 458
13.6 Exponential stable systems......Page 461
13.7 Linear system with a minimal base......Page 463
13.8 Some classes of uniformly exponentially stable equations......Page 465
13.9 Linear periodic systems......Page 474
13.9.1 Exponential stable linear periodic dynamical systems......Page 475
13.9.2 Some classes of linear uniformly exponentially stable periodic di......Page 480
Triangular maps......Page 485
14.1 Triangular maps and non-autonomous dynamical systems......Page 486
14.2 Linear non-autonomous dynamical systems......Page 487
14.3 Quasi-linear non-autonomous dynamical systems......Page 491
14.4 Global attractors of quasi-linear triangular systems......Page 493
14.5 Almost periodic and recurrent solutions......Page 499
14.6 Pseudo recurrent solutions......Page 501
14.7 Chaos in triangular maps......Page 502
Bibliography......Page 505
Index......Page 525