Sistemas dinámicos estocásticos no autónomos y sistemas multivaluados

Uno de los conceptos más importantes para la descripción del comportamiento asintótico de ecuaciones en derivadas parciales de evolución disipativas es el de atractor global. Presentamos los trabajos realizados en esta línea para ecuaciones en derivadas parciales estocásticas (presencia de ruidos en alguno de los términos), para inclusiones diferenciales (es decir, en donde la variación de la variable incógnita no verifica una determinada expresión, sino un conjunto de expresiones), para ecuaciones con retardo (modelos que tienen en cuenta en cada instante parte de la dinámica pasada) y para ecuaciones en derivadas parciales no autónomas (es decir, donde los términos que afectan a la incógnita son dependientes del tiempo). Esta diversidad de situaciones nos ha obligado a adentrarnos en disciplinas a veces muy distintas del Análisis Funcional, las funciones multivaluadas o los procesos estocásticos, o teorías como la de la medida o la de la dimensión de conjuntos. En una cantidad importante de los trabajos que resumimos hay una idea motor de fondo que recorre todos ellos: el hecho de poder describir la dinámica infinito dimensional propia de estos modelos con sólo una cantidad finita de grados de libertad

Informational structures and informational fields as a prototype for the description of postulates of the integrated information theory

Informational Structures (IS) and Informational Fields (IF) have been recently introduced to deal with a continuous dynamical systems-based approach to Integrated Information Theory (IIT). IS and IF contain all the geometrical and topological constraints in the phase space. This allows one to characterize all the past and future dynamical scenarios for a system in any particular state. In this paper, we develop further steps in this direction, describing a proper continuous framework for an abstract formulation, which could serve as a prototype of the IIT postulates.National Science Center of PolandUMO-2016/22/A/ST1/00077Junta de AndalucíaMinisterio de Economia, Industria y Competitividad (MINECO). Españ

Semimartingale attractors for Allen-Cahn SPDEs driven by space-time white noise I: existence and finite dimensional asymptotic behavior

We delve deeper into the study of semimartingale attractors that we recently introduced in Allouba and Langa [4] H. Allouba and J.A. Langa, Semimartingale attractors for generalized Allen-Cahn SPDEs driven by space-time white noise, C. R. Acad. Sci. Paris, Ser. I 337 (2003), 201-206. In this article we focus on second order SPDEs of the Allen-Cahn type. After proving existence, uniqueness, and detailed regularity results for our SPDEs and a corresponding random PDE of Allen-Cahn type, we prove the existence of semimartingale global attractors for these equations. We also give some results on the finite dimensional asymptotic behavior of the solutions. In particular, we show the finite fractal dimension of this random attractor and give a result on determining modes, both in the forward and the pullback sense.National Security AgencyMinisterio de Educación y CienciaFondo Europeo de Desarrollo Regiona

Comparison of the long-time behaviour of linear Ito and Stratonovich partial differential equations

In this paper, we point out the different long-time behaviour of stochastic partial differential equations when one considers the stochastic term in the Ito or Stratonovich sense. In particular, we prove that the Stratonovich interpretation may not produce modification in the exponential stability of the deterministic model for a wide range of stochastic perturbations, while Ito’s one can give different results. In fact, some stabilization or destabilization effect can be obtained

Structure of the pullback attractor for a non-autonomous scalar differential inclusion

The structure of attractors for differential equations is one of the main topics in the qualitative theory of dynamical systems. However, the theory is still in its infancy in the case of multivalued dynamical systems. In this paper we study in detail the structure and internal dynamics of a scalar differential equation, both in the autonomous and nonautonomous cases. To this aim, we will also show a general result on the characterization of a pullback attractor for a multivalued process by the union of all the complete bounded trajectories of the system.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de AndalucíaBrazilian-European partnership in Dynamical SystemsFundación Séneca-Agencia de Ciencia y Tecnología de la Region de Murci

Determining asymptotic behavior from the dynamics on attracting sets

Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar results as in the theory of inertial manifolds. Furthermore, such sets are shown to be retracts of the phase space, which implies that they are simply connected.Ministerio de Educación y CienciaDepartamento de Ecuaciones Diferenciales y Análisis Numérico (Universidad de Sevilla

Flattening, squeezing and the existence of random attractors

The study of qualitative properties of random and stochastic differential equations is now one of the most active fields in the modern theory of dynamical systems. In the deterministic case, the properties of flattening and squeezing in infinite-dimensional autonomous dynamical systems require the existence of a bounded absorbing set and imply the existence of a global attractor. The flattening property involves the behaviour of individual trajectories while the squeezing property involves the difference of trajectories. It is shown here that the flattening property is implied by the squeezing property and is in fact weaker, since the attractor in a system with the flattening property can be infinite-dimensional, whereas it is always finite-dimensional in a system with the squeezing property. The flattening property is then generalized to random dynamical systems, for which it is called the pullback flattening property. It is shown to be weaker than the random squeezing property, but equivalent to pullback asymptotic compactness and pullback limit-set compactness, and thus implies the existence of a random attractor. The results are also valid for deterministic non-utonomous dynamical systems formulated as skew-product flows.Ministerio de Educación y Cienci

Tracking Properties of Trajectories On Random Attracting Sets

The theory of random attracting sets highlights interesting properties of the asymptotic behaviour of some stochastic differential equations. In this paper some results on the relation between the dynamics on random attractors and stochastic inertial manifolds, and the dynamics in the associated random dynamical system are studied. In particular, some tracking properties of trajectories on random attractors and a general result on the asymptotic completeness of stochastic inertial manifolds are shown

Existence and regularity of the pressure for the stochastic Navier-Stokes equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H−1(D))d, where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier-Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V0 of the divergence free subspace V of (H1 0(D))d, in general it is not possible to solve the stochastic Navier-Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier-Stokes equations could be meaningful for them.Ministerio de Ciencia y TecnologíaFondo Europeo de Desarrollo Regiona

Regularity and structure of pullback attractors for reaction-diffusion type systems without uniqueness

In this paper, we study the pullback attractor for a general reaction-diffusion system for which the uniqueness of solutions is not assumed. We first establish some general results for a multi-valued dynamical system to have a bi-spatial pullback attractor, and then we find that the attractor can be backwards compact and composed of all the backwards bounded complete trajectories. As an application, a general reaction-diffusion system is proved to have an invariant (H, V )-pullback attractor A = {A(τ)}τ∈R. This attractor is composed of all the backwards compact complete trajectories of the system, pullback attracts bounded subsets of H in the topology of V, and moreover ∪ s6τ A(s) is precompact in V, ∀τ ∈ R. A non-autonomous Fitz-Hugh-Nagumo equation is studied as a specific example of the reaction–diffusion system.State Scholarship Fund (China)Junta de AndalucíaBrazilian-European partnership in Dynamical SystemsEuropean UnionNational Natural Science Foundation of Chin