New trends on nonlinear dynamics and its applications
Tomás Caraballo Juan L. G. Guirao
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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Stability of delay evolution equations with stochastic perturbations
Tomás Caraballo Leonid Shaikhet
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.
keywords: stochastic evolution equations stochastic 2D Navier-Stokes model with delays. stochastic partial differential equations exponential stability Method of Lyapunov functionals construction
On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence
Tomás Caraballo David Cheban
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
Morse decomposition of global attractors with infinite components
Tomás Caraballo Juan C. Jara José A. Langa José Valero
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.
keywords: Morse decomposition infinite components gradient dynamics gradient-like semigroup. Lyapunov function
Non--autonomous and random attractors for delay random semilinear equations without uniqueness
Tomás Caraballo M. J. Garrido-Atienza B. Schmalfuss José Valero
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.
keywords: multi-valued non-autonomous and random dynamical systems pullback and random attractors delay differential equations. Cocycles
Equi-attraction and continuity of attractors for skew-product semiflows
Tomás Caraballo Alexandre N. Carvalho Henrique B. da Costa José A. Langa
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.
keywords: skew-product semiflows. Equi-attraction continuity of attractors
Asymptotic behaviour of a non-autonomous Lorenz-84 system
María Anguiano Tomás Caraballo
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.
keywords: pullback attractor Lorenz system non-autonomous equation Hausdorff dimensionn. uniform attractor
Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions
Tomás Caraballo María J. Garrido–Atienza Björn Schmalfuss José Valero
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.
keywords: Multivalued non-autonomous and random dynamical systems functional stochastic equations conjugacy method. pullback and random attractors
A survey on Navier-Stokes models with delays: Existence, uniqueness and asymptotic behavior of solutions
Tomás Caraballo Xiaoying Han
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
Attractors for a random evolution equation with infinite memory: Theoretical results
Tomás Caraballo María J. Garrido-Atienza Björn Schmalfuss José Valero

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.

keywords: Pullback and random attractor random dynamical system random delay equation infinite delay

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