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-B

We study the long term dynamics of non-autonomous functional differential equations. Namely, we establish existence results on pullback attractors for non-linear neutral functional differential equations with time varying delays. The two main results differ in smoothness properties of delay functions.

DCDS-S

We show that infinite-dimensional integro-differential equations
which involve an integral of the solution over the time interval
since starting can be formulated as non-autonomous delay
differential equations with an infinite delay. Moreover, when conditions
guaranteeing uniqueness of solutions do not hold, they
generate a non-autonomous (possibly) multi-valued dynamical system
(MNDS). The pullback attractors here are defined with respect to
a universe of subsets of the state space with sub-exponetial
growth, rather than restricted to bounded sets. The theory of
non-autonomous pullback attractors is extended to such MNDS in a
general setting and then applied to the original
integro-differential equations. Examples based on the logistic
equations with and without a diffusion term are considered.

DCDS-S

Nonlinear Dynamics and Complexity is a wide area of research which is experiencing a big development nowadays. For this reason, we thought that it could be very helpful to publish a theme issue which could provide an overview of the recent developments, discoveries and progresses on this fascinating field of Nonlinear Dynamics and Complexity. Therefore, the main aims of this issue are to present the fundamental and frontier theories and techniques for modern science and technology which can also stimulate more research interest for exploration of nonlinear science and complexity and, to directly pass the new knowledge to the young generation, engineers and technologists in the corresponding fields. Consequently, the contributions which have been accepted for this issue focus on recent developments, findings and progresses on fundamental theories and principles, analytical and symbolic approaches, computational techniques in nonlinear physical science and nonlinear mathemati
cs. Amongst others, the main topics of interest in Nonlinear Dynamics and Complexity treated in these papers include, but are not limited to:

For more information please click the “Full Text” above.

keywords:

CPAA

The investigation of stability for hereditary
systems is often related to the construction of Lyapunov
functionals. The general method of Lyapunov functionals
construction, which was proposed by V.Kolmanovskii and L.Shaikhet, is used here to
investigate the stability of stochastic delay evolution equations,
in particular, for stochastic partial differential equations. This method had already
been successfully used for functional-differential
equations, for difference equations with discrete time, and for
difference equations with continuous time.
It is shown that the stability conditions obtained for stochastic 2D Navier-Stokes model with delays are essentially better than the known ones.

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

In this work we prove the existence of solution for a p-Laplacian non-autonomous problem with dynamic boundary and infinite delay. We ensure the existence of pullback attractor for the multivalued process associated to the non-autonomous problem we are concerned.

DCDS

In this paper we describe some dynamical properties of a Morse decomposition
with a countable number of sets. In particular, we are able to prove that the gradient dynamics on
Morse sets together with a separation assumption is equivalent to the
existence of an ordered Lyapunov function associated to the Morse sets and
also to the existence of a Morse decomposition -that is, the global attractor
can be described as an increasing family of local attractors and their
associated repellers.

DCDS

We first prove the existence and uniqueness of pullback and random
attractors for abstract multi-valued non-autonomous and random
dynamical systems. The standard assumption of compactness of these
systems can be replaced by the assumption of asymptotic
compactness. Then, we apply the abstract theory to handle a random
reaction-diffusion equation with memory or delay terms which can
be considered on the complete past defined by $\mathbb{R}^{-}$. In
particular, we do not assume the uniqueness of solutions of these
equations.

DCDS

We derive general existence theorems for random pullback exponential attractors and deduce explicit bounds for their fractal dimension. The results are formulated for asymptotically compact random dynamical systems in Banach spaces.