DCDS-B
Almost periodic and asymptotically almost periodic solutions of Liénard equations
Tomás Caraballo David Cheban
Discrete & Continuous Dynamical Systems - B 2011, 16(3): 703-717 doi: 10.3934/dcdsb.2011.16.703
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
Attractivity for neutral functional differential equations
Tomás Caraballo Gábor Kiss
Discrete & Continuous Dynamical Systems - B 2013, 18(7): 1793-1804 doi: 10.3934/dcdsb.2013.18.1793
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.
keywords: pullback attractor Non-autonomous dynamics neutral equation.
DCDS-S
Non-autonomous attractors for integro-differential evolution equations
Tomás Caraballo P.E. Kloeden
Discrete & Continuous Dynamical Systems - S 2009, 2(1): 17-36 doi: 10.3934/dcdss.2009.2.17
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.
keywords: Integro-differential equation set-valued process differential equation with infinite delay pullback attractor. set-valued non-autonomous dynamical system
DCDS-S
New trends on nonlinear dynamics and its applications
Tomás Caraballo Juan L. G. Guirao
Discrete & Continuous Dynamical Systems - S 2015, 8(6): i-ii doi: 10.3934/dcdss.2015.8.6i
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
Stability of delay evolution equations with stochastic perturbations
Tomás Caraballo Leonid Shaikhet
Communications on Pure & Applied Analysis 2014, 13(5): 2095-2113 doi: 10.3934/cpaa.2014.13.2095
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
CPAA
On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence
Tomás Caraballo David Cheban
Communications on Pure & Applied Analysis 2013, 12(1): 281-302 doi: 10.3934/cpaa.2013.12.281
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
Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay
Rodrigo Samprogna Tomás Caraballo
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 509-523 doi: 10.3934/dcdsb.2017195

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.

keywords: Pullback attractors unbounded delays multivalued process dynamical boundary p-Laplacian asymptotic behavior of solutions
DCDS
Morse decomposition of global attractors with infinite components
Tomás Caraballo Juan C. Jara José A. Langa José Valero
Discrete & Continuous Dynamical Systems - A 2015, 35(7): 2845-2861 doi: 10.3934/dcds.2015.35.2845
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
DCDS
Non--autonomous and random attractors for delay random semilinear equations without uniqueness
Tomás Caraballo M. J. Garrido-Atienza B. Schmalfuss José Valero
Discrete & Continuous Dynamical Systems - A 2008, 21(2): 415-443 doi: 10.3934/dcds.2008.21.415
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
DCDS
Random pullback exponential attractors: General existence results for random dynamical systems in Banach spaces
Tomás Caraballo Stefanie Sonner
Discrete & Continuous Dynamical Systems - A 2017, 37(12): 6383-6403 doi: 10.3934/dcds.2017277

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.

keywords: Random dynamical system exponential attractor random pullback attractor fractal dimension ε-entropy

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