DCDS-B

The aim of this paper is to study the almost periodic and
asymptotically almost periodic solutions on $(0,+\infty)$ of the
Liénard equation
$ x''+f(x)x'+g(x)=F(t), $

where $F: T\to R$ ($ T= R_+$ or
$R$) is an almost periodic or asymptotically almost
periodic function and $g:(a,b)\to R$ is a strictly
decreasing function. We study also this problem for the vectorial
Liénard equation.

We analyze this problem in the framework of general non-autonomous
dynamical systems (cocycles). We apply the general results
obtained in our early papers [3, 7] to prove the existence of almost periodic
(almost automorphic, recurrent, pseudo recurrent) and
asymptotically almost periodic (asymptotically almost automorphic,
asymptotically recurrent, asymptotically pseudo recurrent)
solutions of Liénard equations (both scalar and vectorial).

keywords:
recurrent solutions
,
Non-autonomous dynamical systems
,
asymptotically almost periodic solutions
,
Lienard equation.
,
convergent
systems
,
almost automorphic
,
global attractor
,
skew-product systems
,
quasi-periodic
,
almost periodic
,
cocycles
DCDS

The paper is dedicated to the study of the problem of continuous
dependence of compact global attractors on parameters of
non-autonomous dynamical systems and infinite iterated function
systems (IIFS). We prove that if a family of non-autonomous
dynamical systems ‹$(X,\mathbb
T_1,\pi_{\lambda}),(Y,\mathbb T_{2},\sigma),h $›depending
on parameter $\lambda\in\Lambda$ is uniformly contracting (in the
generalized sense), then each system of this family admits a
compact global attractor $J^{\lambda}$ and the mapping $\lambda
\to J^{\lambda}$ is continuous with respect to the Hausdorff
metric. As an application we give a generalization of well known
Theorem of Bransley concerning the continuous dependence of
fractals on parameters.

PROC

Please refer to Full Text.

CPAA

The aim of this paper is to describe the structure of global
attractors for infinite-dimensional non-autonomous dynamical
systems with recurrent coefficients. We consider a special class
of this type of systems (the so--called weak convergent systems).
We study this problem in the framework of general non-autonomous
dynamical systems (cocycles). In particular, we apply the general results
obtained in our previous paper [6] to study the almost periodic (almost
automorphic, recurrent, pseudo recurrent) and asymptotically
almost periodic (asymptotically almost automorphic, asymptotically
recurrent, asymptotically pseudo recurrent) solutions of different
classes of differential equations (functional-differential
equations, evolution equation with monotone operator, semi-linear
parabolic equations).

keywords:
Non-autonomous dynamical systems
,
almost periodic
,
quasi-periodic
,
asymptotically almost periodic solutions
,
convergent systems
,
almost automorphic
,
functional differential equations; evolution equations with monotone operators
,
cocycles
,
global attractor
,
dissipative
systems
,
recurrent solutions
,
skew-product systems
DCDS-B

The aim of this paper is study the problem of global asymptotic
stability of solutions for $\mathbb C$-analytical dynamical
systems (both with continuous and discrete time). In particular
we present some new results for the $C$-analytical version of
Belitskii--Lyubich conjecture. Some applications of these results for
periodic $\mathbb C$-analytical differential/difference equations
and the equations with impulse are given.

DCDS

We analyze the existence of almost periodic (respectively,
almost automorphic, recurrent) solutions of a linear
non-homogeneous differential (or difference) equation in a Banach space,
with almost periodic (respectively, almost automorphic, recurrent)
coefficients. Under some conditions we prove that one of the
following alternatives is fulfilled:

(i) There exists a complete trajectory of the corresponding homogeneous equation with
constant positive norm;

(ii) The trivial solution of the homogeneous
equation is uniformly asymptotically stable.

If the second alternative holds, then the
non-homogeneous equation with almost periodic (respectively, almost
automorphic, recurrent) coefficients possesses a unique almost
periodic (respectively, almost automorphic, recurrent) solution.
We investigate this problem within the framework of
general linear nonautonomous dynamical systems. We apply our
general results also to the cases of functional-differential
equations and difference equations.

CPAA

The aim of this paper is to describe the structure of global
attractors for non-autonomous dynamical systems with recurrent
coefficients (with both continuous and discrete time). We consider
a special class of this type of systems (the so--called weak
convergent systems). It is shown that, for weak convergent
systems, the answer to Seifert's question (Does an almost periodic
dissipative equation possess an almost periodic solution?) is
affirmative, although, in general, even for scalar equations, the
response is negative. We study this problem in the framework of
general non-autonomous dynamical systems (cocycles). We apply the
general results obtained in our paper to the study of almost
periodic (almost automorphic, recurrent, pseudo recurrent) and
asymptotically almost periodic (asymptotically almost automorphic,
asymptotically recurrent, asymptotically pseudo recurrent)
solutions of different classes of differential equations.

keywords:
dissipative systems
,
convergent
systems
,
skew-product systems
,
almost periodic
,
global attractor
,
Non-autonomous dynamical systems
,
cocycles
,
almost automorphic
,
quasi-periodic
,
asymptotically almost periodic solutions.
,
recurrent solutions
DCDS-B

We prove the existence of recurrent or Poisson stable
motions in the Navier-Stokes
fluid system under recurrent or Poisson stable forcing, respectively.
We use an approach
based on nonautonomous dynamical systems ideas.