Minimality properties of set-valued processes and their pullback attractors

We discuss the existence of pullback attractors for multivalued dynamical systems on metric spaces. Such attractors are shown to exist without any assumptions in terms of continuity of the solution maps, based only on minimality properties with respect to the notion of pullback attraction. When invariance is required, a very weak closed graph condition on the solving operators is assumed. The presentation is complemented with examples and counterexamples to test the sharpness of the hypotheses involved, including a reaction-diffusion equation, a discontinuous ordinary differential equation and an irregular form of the heat equation.Comment: 33 pages. A few typos correcte

Asymptotic behaviour of the non-autonomous 3D Navier-Stokes problem with coercive force

We construct pullback attractors to the weak solutions of the three-dimensional Dirichlet problem for the incompressible Navier-Stokes equations in the case when the external force may become unbounded as time goes to plus or minus infinity.Comment: 22 page

Trajectory and global attractors for generalized processes

In this work the theory of generalized processes is used to describe the dynamics of a nonautonomous multivalued problem and, through this approach, some conditions for the existence of trajectory attractors are proved. By projecting the trajectory attractor on the phase space, the uniform attractor for the multivalued process associated to the problem is obtained and some conditions to guarantee the invariance of the uniform attractor are given. Furthermore, the existence of the uniform attractor for a class of p-Laplacian nonautonomous problems with dynamical boundary conditions is established.Conselho Nacional de Desenvolvimento Científico e Tecnológico. BrasilEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaJunta de Andalucí

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

Rate-induced transitions for parameter shift systems

Rate-induced transitions have recently emerged as an identifiable type of instability of attractors in nonautonomous dynamical systems. In most studies so far, these attractors can be associated with equilibria of an autonomous limiting system, but this is not necessarily the case. For a specific class of systems with a parameter shift between two autonomous systems, we consider how the breakdown of the quasistatic approximation for attractors can lead to rate-induced transitions, where nonautonomous instability can be characterised in terms of a critical rate of the parameter shift. We find a number of new phenomena for non-equilibrium attractors: weak tracking where the pullback attractor of the system limits to a proper subset of the attractor of the future limit system, partial tipping where certain phases of the pullback attractor tip and others track the quasistatic attractor, em invisible tipping where the critical rate of partial tipping is isolated and separates two parameter regions where the system exhibits end-point tracking. For a model parameter shift system with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system. Considering weak tracking for a nonautonomous Rossler system, we show that there are infinitely many critical rates at which a pullback attracting solution of the system tracks an embedded unstable periodic orbit of the future chaotic attractor

On cocycle and uniform attractors for multi-valued and random non-autonomous dynamical systems

En esta tesis estudiamos el comportamiento a largo plazo de sistemas dinámicos multivaluados y aleatorios en términos de sus atractores globales. Comenzamos con el estudio de los atractores cociclo, pullback y uniforme para sistemas dinámicos no autónomos multivaluados. En primer lugar consideramos la relación entre estos tres tipos de atractores para encontrar que, bajo condiciones adecuadas, se implican entre sí. Encontramos además que estos atractores pueden caracterizarse por trayectorias (soluciones globales), lo que implica que el atractor uniforme tiene una propiedad de invarianza (lifted invariance), aunque, por definición, no posee la invarianza estándar. Finalmente, estudiamos tanto la semicontinuidad superior como inferior de estos atractores. Se introduce un equi-atracción débil para estudiar la semicontinuidad inferior, y se muestra con un ejemplo las ventajas de este método. Un sistema de reaccióndifusión y una inclusión diferencial ordinaria escalar se estudian como aplicaciones. A continuación estudiamos el caso aleatorio (pero univaluado), en el marco de los sistemas dinámicos aleatorios (RDS, por sus siglas en inglés). En primer lugar, se estudian los atractores cociclo para RDS y sistemas dinámicos aleatorios no autónomos (NRDS) con sólo una continuidad llamada cuasi fuerte a débil (abreviadamente cuasi-S2W). Esta continuidad se muestra heredable: si una aplicación es cuasi-S2W continua en algún espacio, entonces lo es automáticamente en espacios más regulares. Además, al establecer algunos criterios de existencia para los atractores cociclo, vemos que la continuidad cuasi- S2W es suficiente para derivar la medibilidad del atractor cociclo. Estas observaciones generalizan los teoremas de existencia conocidos para los atractores cociclo, por un lado, y, por otro, nos permiten estudiar estos atractores en espacios regulares sin demostrar la continuidad del sistema. Aplicando estos resultados a la teoría bi-espacial de atractores cociclos, establecemos un teorema de existencia que indica que la medibilidad de los atractores bi-espaciales es válida en espacio más regulares, no sólo en el espacio de fases básico como previamente en la literatura. En segundo lugar, para NRDS se comparan los atractores cociclos con universos de atracción autónomos y no autónomos, y luego para universos autónomos se establecen algunos criterios de existencia y caracterización. También estudiamos la semicontinuidad superior de estos atractores con respecto a los símbolos no autónomos, para hallar que un atractor cociclo es semicontinuo superiormente respecto a los símbolos si y sólo si es uniformemente compacto. En tercer lugar, establecemos una teoría de atractores uniformes (aleatorios) para NRDS. Definimos un atractor uniforme como el menor conjunto aleatorio compacto uniformemente atrayente. En cuanto a la definición, observamos que la propiedad de atracción uniforme de un atractor uniforme, de hecho, implica una atracción uniforme hacia adelante en probabilidad, e implica también una atracción pullback casi uniforme para sucesiones de tiempo discretas. Aunque no se requiere invarianza por definición, el atractor uniforme posee una semi-invarianza negativa. Estudiamos la existencia de atractores uniformes, y la relación entre los atractores uniformes y los atractores para los productos cruzados aleatorios (random skew-products). Para superar la dificultad de la medibilidad de los conjuntos aleatorios, se requiere que el espacio de símbolos sea Polish, que se tiene para funciones localmente integrables cuando el espacio de símbolos se define como la clausura de las mismas. Para la relación entre los atractores uniformes y cociclos encontramos, por un lado, que el atractor uniforme de un NRDS continuo se compone de estados involucrados en el atractor cociclo, y que, por el otro, puede ser descrito como el atractor cociclo de un RDS multivaluado (pero autónomo). Además, los atractores uniformes para NRDS continuos aparecen determinados (como en el caso de RDS aut\'onomos) por la atracción uniforme de conjuntos compactos no aleatorios. Como aplicaciones se estudian la existencia y caracterización de atractores cociclo y uniformes para la ecuación de reacción-difusión, la ecuación de Ginzburg- Landau y la ecuación bidimensional de Navier-Stokes con ruido blanco escalar

Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is stud ied, for which the finite-dimensionality in L2 is established by the squeezing approach and that in H1 0 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed

On non-autonomously forced Burgers equation with periodic and Dirichlet boundary conditions

We study the non-autonomously forced Burgers equation $$u_t(x,t) + u(x,t)u_x(x,t) - u_{xx}(x,t) = f(x,t)$$ on the space interval $(0,1)$ with two sets of the boundary conditions: the Dirichlet and periodic ones. For both situations we prove that there exists the unique $H^1$ bounded trajectory of this equation defined for all $t\in \mathbb{R}$. Moreover we demonstrate that this trajectory attracts all trajectories both in pullback and forward sense. We also prove that for the Dirichlet case this attraction is exponential

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