Nonautonomous Dynamics - Nonlinear Oscillations and Global Attractors

Preface......Page 7
Contents......Page 12
Notation......Page 17
1 Almost Periodic Motions of Dynamical Systems......Page 21
 1.1 Some Notions, Notation, and Facts from the Theory of Dynamical Systems......Page 22
  1.2.1 Minimal Sets......Page 31
  1.2.2 Almost Recurrent Motions......Page 33
  1.2.3 Recurrent Motions......Page 35
 1.3 Lyapunov Stable Sets......Page 38
 1.4 Almost Periodic Motions......Page 41
  1.5.1 Adjoint Dynamical System......Page 46
  1.5.2 Measure on the Compact Space......Page 48
  1.5.3 Ergodicity of Almost Periodic Systems......Page 51
  1.5.4 Existence a Unique Invariant Measure for Almost Periodic Minimal Dynamical Systems......Page 52
 1.6 Two Definitions of Almost Periodic Motions in Semigroup …......Page 56
  1.7.1 B. A. Shcherbakov's Principle of Comparability of Motions by Their Character of Recurrence......Page 64
  1.7.2 Comparability by Character of Recurrence of Almost Periodic Motions......Page 70
  1.7.3 Some Generalization of B. A. Shcherbakov's Results......Page 73
  1.7.4 Strongly Comparability of Motions by Their Character of Recurrence......Page 76
  1.8.1 Poisson Asymptotically Stable Motions......Page 77
  1.8.2 Criterion of Asymptotical Almost Periodicity......Page 79
  1.8.3 Asymptotically Periodic Motions......Page 83
  1.8.4 Comparability by the Character of Recurrence in the Limit of Asymptotically Poisson Stable Motions......Page 87
  1.9.1 Bohr Almost Periodic Functions......Page 89
  1.9.2 Sp-Almost Periodic Functions......Page 92
  1.9.3 Fréchet Asymptotically Almost Periodic Functions......Page 96
  1.9.4 Asymptotically Sp Almost Periodic Functions......Page 98
2 Compact Global Attractors......Page 100
 2.1 Limit Properties of Dynamical Systems......Page 101
 2.2 Levinson Center......Page 103
 2.3 Dissipative Systems on Local Compact Spaces......Page 109
 2.4 Criteria for Compact Dissipativity......Page 111
 2.5 Local Dissipative Dynamical Systems......Page 118
 2.6 Global Attractors......Page 122
 2.7 Global Attractors of Nonautonomous Dynamical Systems......Page 126
 2.8 Global Attractors of Cocycles......Page 130
 2.9 Global Attractors of Nonautonomous Dynamical Systems with Minimal Base......Page 135
 2.10 The Relationship Between Pullback, Forward, and Global Attractors......Page 138
  2.10.1 Pullback, Forward, and Global Attractors......Page 139
  2.10.2 Asymptotic Stability in α-Condensing Semidynamical Systems......Page 146
  2.10.3 Uniform Pullback and Global Attractors......Page 150
  2.10.4 Examples of Uniform Pullback Attractors......Page 152
3 Analytical Dissipative Systems......Page 157
 3.1 Skew-Product Dynamical Systems and Cocycles......Page 158
 3.2 Positively Stable Systems......Page 162
 3.3 mathbbC-Analytic Systems......Page 166
  3.4.1 mathbbC-Analytic Cocycles......Page 171
  3.4.2 Some General Facts About Nonautonomous Dynamical Systems......Page 172
  3.4.3 Positively Uniformly Stable Cocycles......Page 179
  3.4.4 The Compact Global Pullback Attractors of mathbbC-Analytic Cocycles with Compact Base......Page 182
  3.4.5 Uniform Dissipative Cocycles with Noncompact Base......Page 185
  3.4.6 Compact and Local Dissipative Cocycles with Noncompact Base......Page 189
  3.4.7 Applications......Page 192
  3.5.1 Introduction......Page 196
  3.5.2 The Belitskii–Lyubich Conjecture......Page 197
  3.5.3 Belitskii–Lyubich Conjecture for Holomorphic Flows......Page 203
  3.5.4 Holomorphic Dissipative Dynamical Systems......Page 206
  3.5.5 Some Applications......Page 208
4 Almost Periodic Solutions of Linear Differential Equations......Page 214
  4.1.1 Nonautonomous Version of Birkhoff's Theorem......Page 217
  4.1.2 Strongly Poisson Stable Motions......Page 219
 4.2 Semigroup Dynamical Systems......Page 222
 4.3 Invariant and Minimal Sets for Some Abstract Semigroups …......Page 226
 4.4 Some Tests of Comparability and Uniform Comparability of Motions......Page 230
  4.5.1 Linear Differential Equations......Page 239
  4.5.2 Linear Difference Equations......Page 244
  4.5.3 Linear Functional Differential/Difference Equations......Page 247
 4.6 Almost Periodic Solutions of Linear Almost Periodic Systems......Page 259
  4.6.1 Bounded Motions of Linear Systems......Page 262
  4.6.2 Linear Contractive Systems......Page 265
  4.6.3 Applications......Page 272
 4.7 Almost Periodic Solutions of Semilinear Dissipative Systems......Page 283
5 Almost Periodic Solutions of Monotone Differential Equations......Page 292
 5.1 Bohr/Levitan Almost Periodic Solutions  of Scalar Differential Equations......Page 294
  5.1.1 Cocycles, Skew-Product Dynamical Systems and Nonautonomous Dynamical Systems Generated by Differential Equations......Page 296
  5.1.2 Some Criteria for the Existence of Fixed Points for a Semigroup of Transformations......Page 298
  5.1.3 One-Dimensional Nonautonomous Dynamical Systems......Page 299
  5.1.4 Scalar Differential Equations......Page 304
 5.2 Levitan/Bohr Almost Periodic Solutions  of Second-Order Differential Equations......Page 308
  5.2.1 Some Nonautonomous Dynamical Systems Generated by Second-Order Differential Equations......Page 310
  5.2.2 Levitan Almost Periodic and Almost Automorphic Solutions of Second-Order Differential Equations......Page 311
  5.2.3 Quasiperiodic, Bohr Almost Periodic, Almost Automorphic, and Recurrent Solutions of Second-Order Differential Equations......Page 314
  5.2.4 Some Generalizations......Page 316
 5.3 Monotone Second-Order Differential Equations in Hilbert Spaces......Page 318
  5.3.1 Some Nonautonomous Dynamical Systems Generated by Differential Equations in Banach Spaces......Page 319
  5.3.2 Invariant Sections (Manifolds) of Second-Order Differential Equations......Page 321
  5.3.3 Almost Automorphic Solutions of Monotone Second-Order Differential Equations......Page 327
  5.4.1 Structure of the ω-Limit Set......Page 331
  5.4.2 Comparable Motions by the Nature of Their Recurrence for Monotone Nonautonomous Dynamical Systems......Page 338
  5.4.3 Applications......Page 344
 5.5 I. U. Bronshtein's Conjecture for Monotone Nonautonomous Dynamical Systems......Page 354
  5.5.1 Global Attractors of Cocycles......Page 355
  5.5.2 I. U. Bronshtein's Conjecture for Cocycles......Page 357
  5.5.3 Applications......Page 359
  5.5.4 Ordinary Differential Equations......Page 360
  5.5.5 Difference Equations......Page 364
6 Gradientlike Dynamical Systems......Page 367
 6.1 Compact Global Attractors of Nonautonomous Gradientlike Dynamical Systems......Page 368
 6.2 Gradient Systems with Bounded  Set of Fixed Points......Page 370
  6.2.1 Gradient Systems with Finite Number of Fixed Points......Page 372
  6.2.2 Chain Recurrent Motions of Gradient Systems......Page 375
  6.2.3 Nonautonomous Gradientlike Dynamical Systems......Page 376
 6.3 Relation Between Levinson Center, Chain Recurrent Set, and Birkhoff Center......Page 381
 6.4 Chain Recurrent Motions of Compact Dissipative Dynamical Systems......Page 391
  6.5.1 Linear Nonautonomous Dynamical Systems with Exponential Dichotomy......Page 397
  6.5.2 Linear Inhomogeneous (Affine) Dynamical Systems......Page 401
  6.5.3 Invariant Sections (Manifolds)......Page 403
  6.5.4 Dependence on Parameter of Green's Function......Page 404
  6.5.5 Relationship Between Different Definitions of Hyperbolicity......Page 406
  6.5.6 Uniformly Compatible Solutions......Page 411
  6.6.1 Local Existence and Uniqueness......Page 413
  6.6.2 Global Existence and Uniqueness......Page 416
  6.6.3 Invariant Sections (Manifolds) of Semilinear Equations......Page 418
 6.7 Perturbation of Gradient Systems......Page 422
 BookmarkTitle:......Page 430
Index......Page 444