Cheban D.N. Global Attractors Of Non-autonomous Dissipative Dynamical Systems

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Index

− cocycle, 451

(M), 18

−

, 311

, 310

H-class, 7, 135

(M), 18

V - monotone, 386

(x), 174

V −monotone cocycle, 394

α-condensing, 295, 435

α-condensing cocycle, 299

α-contraction, 436

α-limit, 2

λ-condensing, 35

λ-contraction, 33

C-analytic dissipative dynamical system,

124

C-analytic system, 123

K, 390

, 146

, 310

−

, 310

M-dissipative, 10

, 475

ω-limit, 2

ε > 0 shift (almost period) of point

x ∈ X, 2

r-cycle, 156

A, 173

1-cycle, 156

absorbing (uniformly absorbing), 270

almost recurrent (almost periodic), 2

asymptotic stable, 3

asymptotically τ -periodic, 74

asymptotically almost periodic, 74

asymptotically compact, 31

asymptotically recurrent, 74

asymptotically stationary, 74

attracting, 3

autonomous Lorenz systems, 245

base of extension, 6

Bebutov’s dynamical system, 5

Bohr’s almost periodic function, 327

bounded, 36

bounded k (b)-dissipativity, 11

bounded dissipative, 10

cascade (discrete ﬂow), 1

Cauchy’s matrix, 226

center of Levinson, 11, 12

chain recurrent, 155

chaotic, 478

cocycle, 7

collectively ( uniformly collectively)

asymptotic compact, 270

collectively compact dissipative, 268

compact k (b)-dissipative, 11

compact k (b)-dissipativity, 11

compact bounded, 36

compact dissipative, 10

compact dissipative cocycle, 249

comparable in limit, 74

completely continuous, 30

condensing, 34

condition (C), 53

condition (A), 70

condition of Ladyzhenskaya, 30

conditional α-contraction, 436

conditionally α-condensing, 435

501

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502 Global Attractors of Non-autonomous Dissipative Dynamical Systems

conditionally relatively compact, 310

connectedness, 38

continuous section, 67

convergent, 70

direct product of the dynamical systems,

407

dissipativity of the equation, 120

distal, 304

dynamical system, 1

dynamical system of translations, 5

exponential dichotomy, 463

extension, 6

factor of dynamical system, 6

ﬂow, 1

forward attractor, 290

generalized homoclinic contour, 156

global asymptotic stable, 3

global attractor, 29

global attractor of the cocycle, 93

globally asymptotically stable in the sense

of Lyapunov-Barbashin, 54

group (semi-group) system, 1

homogeneous, 101

homomorphism, 6

hyperbolic, 157

hyperbolic structure, 157

indecomposable, 2, 38

invariant section, 67

invariant with respect to a cocycle, 290

isomorphism, 6

jointly recurrent, 408

linear non-autonomous dynamical system,

113

linear non-homogeneous system, 115

local k (b)-dissipativity, 11

local attractor, 294, 379

local condition of Lipschitz, 179

local forward attractor, 379

locally bounded, 35

locally compact, 73

locally completely continuous, 16

locally dissipative, 10

locally maximal, 295

maximal compact invariant set of cocycle,

264

maximal monotone operator, 239

measure of non-compactness of

Kuratowsky, 9

metric space with segments, 389

minimal set, 2

monotone, 239

monotone operator, 239

motion, 1

multiplicator, 454

non-autonomous dynamical system, 6

non-autonomous dynamical system,

generated by cocycle, 7

non-autonomous Lorenz, 245

non-autonomous Navier-Stokes equation,

361

nonlinear elliptic operator, 403

operator of monodromy, 454

orbital stable, 3

orbitally stable with respect to the

non-autonomous system, 54

Poincare transformation, 169

point k (b)-dissipative, 11

point k(b)-dissipativity, 11

point dissipative, 10

Poisson stable, 2

positively invariant (negatively invariant,

invariant), 1

power-law asymptotic, 107

Problem of J. Hale, 33

projection, 463

property (S), 43, 272

pseudo recurrent, 369, 477

pullback attractor, 93

pullback dissipative, 327

quasi-minimal, 165

recurrent, 2

relation of partial order, 156

sectorial operator, 358

semi-continuous, 239

shifts dynamical system, 5

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Index 503

skew-product dynamical system, 7

stable (unstable) manifold, 3

stable in the sense of Poisson in the

negative direction, 2

stable in the sense of Poisson in the

positive direction, 2

stationary (τ -periodic, τ > 0, τ ∈ T)

point, 2

Stepanov almost periodic function, 330

strictly metric-convex space, 389

systems of hydrodynamic type, 245

trajectory, 1

transitive, 369, 478

triangular map, 462

tubular neighborhood, 189

uniform attracting, 3

uniform forward attractor, 290

uniform pullback attractor, 93

uniform stable in the positive direction, 61

uniformly collectively compact dissipative,

268

uniformly compatible, 146

uniformly dissipative, 121, 320

uniformly monotone, 239

uniformly stable in the positive direction,

uniformly stable in the sense of Lagrange,

upper semi–continuous, 292

weak attractor, 16

weakly b-dissipative, 30

weakly dissipative, 16

weakly regular, 146

whole trajectory, 3