GLOBAL ATTRACTORS OF NON-AUTONOMOUS DISSIPATIVE DYNAMICAL SYSTEMS
CONTENTS 1 AUTONOMOUS DYNAMICAL SYSTEMS 1 1 SOME NOTIONS, NOTATIONS AND FACTS FROM THEORY OF DYNAMICAL SYSTEMS 1 2 LIMIT PROPERTIES OF DYNAMICAL SYSTEMS 1 3 CENTER OF LEVINSON 1 4 DISSIPATIVE SYSTEMS ON THE LOCAL COMPACT SPACES 1 5 CRITERIONS OF COMPACT DISSIPATIVITY 1 6 LOCAL DISSIPATIVE SYSTEMS 1 7 GLOBAL ATTRACTORS 1 8 ON A PROBLEM OF J HALE 1 9 CONNECTEDNESS OF THE LEVINSON’S CENTER 1 10 WEAK ATTRACTORS AND CENTER OF LEVINSON 1 11 ASYMPTOTIC STABILITY 2 NON-AUTONOMOUS DISSIPATIVE DYNAMICAL SYSTEMS 2 1 ON THE STABILITY OF LEVINSON’S CENTER 2 2 THE POSITIVELY STABLE SYSTEMS 2 3 BEHAVIOUR OF DISSIPATIVE DYNAMICAL SYSTEMS UNDER HOMOMORPHISMS 2 4 NON-AUTONOMOUS DYNAMICAL SYSTEMS WITH CONVERGENCE 2 5 TESTS FOR CONVERGENCE 2 6 GLOBAL ATTRACTORS OF NON-AUTONOMOUS DYNAMICAL SYSTEMS 2 7 GLOBAL ATTRACTOR OF COCYCLES 2 8 GLOBAL ATTRACTORS OF NON-AUTONOMOUS DYNAMICAL SYSTEM WITH MINIMAL BASE 2 9 HOMOGENEOUS DYNAMICAL SYSTEMS 2 10 POWER-LAW ASYMPTOTIC OF HOMOGENEOUS SYSTEMS 2 11 LINEAR SYSTEMS 3 ANALYTIC DISSIPATIVE SYSTEMS 3 1 SKEW-PRODUCT DYNAMICAL SYSTEMS AND COCYCLES 3 2 C-ANALYTIC SYSTEMS 3 3 CONVERSE OF LYAPUNOV’S THEOREM FOR C-ANALYTIC SYSTEMS 3 4 ON THE STRUCTURE OF COMPACT ATTRACTING SETS OF C-ANALYTIC SYSTEMS 3 5 DYNAMICAL SYSTEMS IN SPACES OF SECTIONS 3 6 QUASI-PERIODIC SOLUTIONS 3 7 THE ANALOGY OF CAMERON-JOHNSON’S THEOREM 3 8 ALMOST PERIODIC SOLUTIONS OF THE WEAK NONLINEAR DISSIPATIVE SYSTEMS 4 THE STRUCTURE OF THE LEVINSON CENTER OF SYSTEM WITH THE CONDITION OF THE HYPERBOLICITY 4 1 THE CHAIN RECURRENT MOTIONS 4 2 THE SPECTRAL DECOMPOSITION OF THE LEVINSON’S CENTER 4 3 ONE-DIMENSIONAL SYSTEMS WITH HYPERBOLIC CENTER 4 4 THE DISSIPATIVE CASCADES 4 5 THE PERIODIC DISSIPATIVE SYSTEMS 5 METHOD OF LYAPUNOV FUNCTIONS 5 1 CRITERIONS OF DISSIPATIVITY IN TERM OF LYAPUNOV FUNCTIONS 5 2 SOME CRITERIONS OF DISSIPATIVITY OF DI?ERENTIAL EQUATIONS 5 3 THEOREM OF BARBASHIN–KRASOVSKII FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS 5 4 EQUATIONS WITH CONVERGENCE 5 5 DISSIPATIVITY AND CONVERGENCE OF SOME EQUATIONS OF 2ND AND 3RD ORDER 5 6 CONSTRUCTION OF LYAPUNOV FUNCTION FOR HOMOGENEOUS SYSTEMS 5 7 DI?ERENTIABLE HOMOGENEOUS SYSTEMS 5 8 GLOBAL ATTRACTORS OF QUASI-HOMOGENEOUS SYSTEMS 6 DISSIPATIVITY OF SOME CLASSES OF EQUATIONS 6 1 DI?ERENCE EQUATIONS 6 2 EQUATIONS WITH IMPULSE 6 3 CONVERGENT PERIODIC EQUATIONS WITH IMPULSE 6 4 ASYMPTOTIC STABILITY OF LINEAR FUNCTIONAL DI?ERENTIAL EQUATIONS 6 5 CONVERGENCE OF MONOTONE EVOLUTIONARY EQUATIONS 6 6 GLOBAL ATTRACTORS OF NON-AUTONOMOUS LORENZ SYSTEMS NON-AUTONOMOUS LORENZ SYSTEMS NON-AUTONOMOUS DISSIPATIVE DYNAMICAL SYSTEMS AND THEIR ATTRACTORS ALMOST PERIODIC AND RECURRENT SOLUTIONS OF NON-AUTONOMOUS LORENZ SYSTEMS UNIFORM AVERAGING PRINCIPLE GLOBAL AVERAGING PRINCIPLE FOR THE NON-AUTONOMOUS LORENZ SYSTEMS 7 UPPER SEMI-CONTINUITY OF ATTRACTORS 7 1 INTRODUCTION 7 2 MAXIMAL COMPACT INVARIANT SETS 7 3 UPPER SEMI-CONTINUITY 7 4 CONNECTEDNESS 7 5 APPLICATIONS QUASI-HOMOGENEOUS SYSTEMS MONOTONE SYSTEMS QUASI-LINEAR SYSTEMS NON-AUTONOMOUSLY PERTURBED SYSTEMS NON-AUTONOMOUS 2D NAVIER-STOKES EQUATIONS QUASI-LINEAR FUNCTIONAL-DI?ERENTIAL EQUATIONS 8 THE RELATIONSHIP BETWEEN PULLBACK, FORWARD AND GLOBAL ATTRACTORS 8 1 PULLBACK, FORWARD AND GLOBAL ATTRACTORS 8 2 ASYMPTOTIC STABILITY IN ?-CONDENSING SEMI–DYNAMICAL SYSTEMS 8 3 UNIFORM PULLBACK ATTRACTORS AND GLOBAL ATTRACTORS 8 4 EXAMPLES OF UNIFORM PULLBACK ATTRACTORS PERIODIC DRIVING SYSTEMS PULLBACK ATTRACTORS WITH SINGLETON COMPONENT SETS DISTAL DYNAMICAL SYSTEMS 9 PULLBACK ATTRACTORS OF C-ANALYTIC SYSTEMS 9 1 C-ANALYTIC COCYCLES 9 2 SOME GENERAL FACTS ABOUT NON-AUTONOMOUS DYNAMICAL SYSTEMS 9 3 POSITIVELY UNIFORMLY STABLE COCYCLES 9 4 THE COMPACT GLOBAL PULLBACK ATTRACTORS OF C-ANALYTIC COCYCLES WITH COMPACT BASE 9 5 THE UNIFORM DISSIPATIVE COCYCLES WITH NONCOMPACT BASE 9 6 THE COMPACT AND LOCAL DISSIPATIVE COCYCLES WITH NONCOMPACT BASE 9 7 APPLICATIONS ODES CARATHEODORY DI?ERENTIAL EQUATIONS ODES WITH IMPULSES 10 PULLBACK ATTRACTORS UNDER DISCRETIZATION 10 1 NON-AUTONOMOUS DYNAMICAL SYSTEMS AND PULLBACK ATTRACTORS 10 2 NON-AUTONOMOUS QUASI-LINEAR DI?ERENTIAL EQUATION 10 3 COCYCLE PROPERTY 10 4 MAIN RESULT 10 4 1 EXISTENCE OF AN ABSORBING SET 10 4 2 UPPER SEMI–CONTINUITY OF THE PULLBACK ATTRACTOR COMPONENT SETS 10 4 3 UPPER SEMI–CONTINUOUS CONVERGENCE OF THE DISCRETIZED PULLBACK ATTRACTORS 10 4 4 UPPER SEMI–CONTINUOUS CONVERGENCE OF THE DISCRETIZED GLOBAL ATTRACTORS 10 5 SINGLETON SET-VALUED PULLBACK ATTRACTOR CASE 11 GLOBAL ATTRACTORS OF NON-AUTONOMOUS NAVIER-STOKES EQUATIONS 11 1 NON-AUTONOMOUS NAVIER-STOKES EQUATIONS 11 2 ATTRACTORS OF NON-AUTONOMOUS DYNAMICAL SYSTEMS 11 3 ALMOST PERIODIC AND RECURRENT SOLUTIONS OF NON-AUTONOMOUS NAVIER- STOKES EQUATIONS 11 4 UNIFORM AVERAGING FOR A ?NITE INTERVAL 11 5 THE GLOBAL AVERAGING PRINCIPLE FOR NAVIER-STOKES EQUATIONS 12 GLOBAL ATTRACTORS OF V -MONOTONE DYNAMICAL SYSTEMS 12 1 GLOBAL ATTRACTORS OF V -MONOTONE NDS 12 2 ON THE STRUCTURE OF LEVINSON CENTER OF V -MONOTONE NDS 12 3 ALMOST PERIODIC SOLUTIONS OF V -MONOTONE SYSTEMS 12 4 PULLBACK ATTRACTORS OF V -MONOTONE NDS 12 5 APPLICATIONS 12 5 1 FINITE-DIMENSIONAL SYSTEMS 12 5 2 CARATHEODORY’S DI?ERENTIAL EQUATIONS 12 5 3 ODES WITH IMPULSE 12 5 4 EVOLUTION EQUATIONS WITH MONOTONE OPERATORS 13 LINEAR ALMOST PERIODIC DYNAMICAL SYSTEMS 13 1 BOUNDED MOTIONS OF LINEAR SYSTEMS 13 2 BOUNDED SOLUTIONS OF LINEAR EQUATIONS 13 3 FINITE-DIMENSIONAL SYSTEMS 13 4 RELATIONSHIP BETWEEN DI?ERENT TYPES OF STABILITY 13 5 LINEAR ?-CONDENSING SYSTEMS 13 6 EXPONENTIAL STABLE SYSTEMS 13 7 LINEAR SYSTEM WITH A MINIMAL BASE 13 8 SOME CLASSES OF UNIFORMLY EXPONENTIALLY STABLE EQUATIONS 13 9 LINEAR PERIODIC SYSTEMS 13 9 1 EXPONENTIAL STABLE LINEAR PERIODIC DYNAMICAL SYSTEMS 13 9 2 SOME CLASSES OF LINEAR UNIFORMLY EXPONENTIALLY STABLE PERIODIC DI?ERENTIAL EQUATIONS 14 TRIANGULAR MAPS 14 1 TRIANGULAR MAPS AND NON-AUTONOMOUS DYNAMICAL SYSTEMS 14 2 LINEAR NON-AUTONOMOUS DYNAMICAL SYSTEMS 14 3 QUASI-LINEAR NON-AUTONOMOUS DYNAMICAL SYSTEMS 14 4 GLOBAL ATTRACTORS OF QUASI-LINEAR TRIANGULAR SYSTEMS 14 5 ALMOST PERIODIC AND RECURRENT SOLUTIONS 14 6 PSEUDO RECURRENT SOLUTIONS 14 7 CHAOS IN TRIANGULAR MAPS