## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
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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:

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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

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-B

In this paper we prove the equivalence between equi-attraction and continuity of attractors for skew-product semi-flows, and equi-attraction and continuity of uniform and cocycle attractors associated to non-autonomous dynamical systems. To this aim proper notions of equi-attraction have to be introduced in phase spaces where the driving systems depend on a parameter. Results on the upper and lower-semicontinuity of uniform and cocycle attractors are relatively new in the literature, as a deep understanding of the internal structure of these sets is needed, which is generically difficult to obtain. The notion of lifted invariance for uniform attractors allows us to compare the three types of attractors and introduce a common framework in which to study equi-attraction and continuity of attractors. We also include some results on the rate of attraction to the associated attractors.

DCDS

The so called Lorenz-84 model has been used in climatological studies, for example by coupling it with a low-dimensional model for ocean dynamics. The behaviour of this model has been studied extensively since its introduction by Lorenz in 1984. In this paper we study the asymptotic behaviour of a non-autonomous Lorenz-84 version with several types of non-autonomous features. We prove the existence of pullback and uniform attractors for the process associated to this model. In particular we consider that the non-autonomous forcing terms are more general than almost periodic. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate some examples of pullback attractors by numerical simulations.

DCDS-B

In this work we present the existence and uniqueness of pullback
and random attractors for stochastic evolution equations with
infinite delays when the uniqueness of solutions for these
equations is not required. Our results are obtained by means of
the theory of set-valued random dynamical systems and their
conjugation properties.

DCDS-S

In this survey paper we review several aspects related to Navier-Stokes models when some hereditary characteristics (constant, distributed or variable delay, memory, etc) appear in the formulation. First some results concerning existence and/or uniqueness of solutions are established. Next the local stability analysis of steady-state solutions is studied by using the theory of Lyapunov functions, the Razumikhin-Lyapunov technique and also by constructing appropriate Lyapunov functionals. A Gronwall-like lemma for delay equations is also exploited to provide some stability results. In the end we also include some comments concerning the global asymptotic analysis of the model, as well as some open questions and future lines for research.

keywords:
Galerkin
,
variable delay
,
stability
,
measurable delay.
,
Navier-Stokes equations
,
distributed delay

DCDS-B

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

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