El objetivo de este trabajo es estudiar sistemas dinámicos multivaluados. En particular, pretendemos obtener resultados relacionados con la estructura de los atractores para describir el comportamiento de las soluciones de diferentes ecuaciones. Por tanto, nuestra investigación puede situarse en el área de Matemática Aplicada. Más concretamente, el Capítulo 1 versa sobre la robustez de los semiflujos multivaluados dinámicamente gradientes. Para aplicar este resultado describimos las propiedades dinámicas de una familia de problemas Chafee-Infante aproximando una inclusión diferencial, demostrando que las soluciones débiles de estos problemas generan un semiflujo multivaluado dinámicamente gradiente con respecto a unos conjuntos de Morse. El Capítulo 2 se centra en una ecuación más general llamada ecuación de reacción-difusión no local, donde el término de difusión depende del gradiente de la solución. En primer lugar, demostramos la existencia y unicidad de soluciones regulares y fuertes. En segundo lugar, obtenemos la existencia de atractores globales en ambas situaciones bajo supuestos bastante débiles al definir un semiflujo multivaluado. En el último capítulo estudiamos la estructura del atractor global para el semiflujo multivaluado generado por una ecuación de reacción-difusión no local donde no podemos garantizar la unicidad del problema de Cauchy. Comenzamos analizando la existencia y propiedades de los puntos estacionarios, mostrando que el problema sufre la misma cascada de bifurcaciones que en la ecuación de Chafee-Infante. Para concluir, estudiamos la estabilidad de los puntos fijos y establecemos que el semiflujo es dinámicamente gradiente. Además, probamos que el atractor está formado por los puntos estacionarios y sus conexiones heteroclínicas y analizamos algunas de las posibles conexiones.

The book treats the theory of attractors for non-autonomous dynamical systems. The aim of the book is to give a coherent account of the current state of the theory, using the framework of processes to impose the minimum of restrictions on the nature of the non-autonomous dependence. The book is intended as an up-to-date summary of the field, but much of it will be accessible to beginning graduate students. Clear indications will be given as to which material is fundamental and which is more advanced, so that those new to the area can quickly obtain an overview, while those already involved can pursue the topics we cover more deeply.

Filled with a wealth of examples to illustrate concepts, this title presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms - algorithms that feature logic, timers, or combinations of digital and analog components.

The study of attractors of dynamical systems occupies an important position in the modern qualitative theory of differential equations. This engaging volume presents an authoritative overview of both autonomous and non-autonomous dynamical systems, including the global compact attractor. From an in-depth introduction to the different types of dissipativity and attraction, the book takes a comprehensive look at the connections between them, and critically discusses applications of general results to different classes of differential equations.The new Chapters 15-17 added to this edition include some results concerning Control Dynamical Systems — the global attractors, asymptotic stability of switched systems, absolute asymptotic stability of differential/difference equations and inclusions — published in the works of author in recent years.

This two-volume work presents state-of-the-art mathematical theories and results on infinite-dimensional dynamical systems. Inertial manifolds, approximate inertial manifolds, discrete attractors and the dynamics of small dissipation are discussed in detail. The unique combination of mathematical rigor and physical background makes this work an essential reference for researchers and graduate students in applied mathematics and physics. The main emphasis in the fi rst volume is on the existence and properties for attractors and inertial manifolds. This volume highlights the use of modern analytical tools and methods such as the geometric measure method, center manifold theory in infinite dimensions, the Melnihov method, spectral analysis and so on for infinite-dimensional dynamical systems. The second volume includes the properties of global attractors, the calculation of discrete attractors, structures of small dissipative dynamical systems, and the existence and stability of solitary waves. Contents Discrete attractor and approximate calculation Some properties of global attractor Structures of small dissipative dynamical systems Existence and stability of solitary waves

This book provides a comprehensive study of how attractors behave under perturbations for both autonomous and non-autonomous problems. Furthermore, the forward asymptotics of non-autonomous dynamical systems is presented here for the first time in a unified manner. When modelling real world phenomena imprecisions are unavoidable. On the other hand, it is paramount that mathematical models reflect the modelled phenomenon, in spite of unimportant neglectable influences discounted by simplifications, small errors introduced by empirical laws or measurements, among others. The authors deal with this issue by investigating the permanence of dynamical structures and continuity properties of the attractor. This is done in both the autonomous (time independent) and non-autonomous (time dependent) framework in four distinct levels of approximation: the upper semicontinuity, lower semicontinuity, topological structural stability and geometrical structural stability. This book is aimed at graduate students and researchers interested in dissipative dynamical systems and stability theory, and requires only a basic background in metric spaces, functional analysis and, for the applications, techniques of ordinary and partial differential equations.

This book provides analytical and numerical methods for the estimation of dimension characteristics (Hausdorff, Fractal, Carathéodory dimensions) for attractors and invariant sets of dynamical systems and cocycles generated by smooth differential equations or maps in finite-dimensional Euclidean spaces or on manifolds. It also discusses stability investigations using estimates based on Lyapunov functions and adapted metrics. Moreover, it introduces various types of Lyapunov dimensions of dynamical systems with respect to an invariant set, based on local, global and uniform Lyapunov exponents, and derives analytical formulas for the Lyapunov dimension of the attractors of the Hénon and Lorenz systems. Lastly, the book presents estimates of the topological entropy for general dynamical systems in metric spaces and estimates of the topological dimension for orbit closures of almost periodic solutions to differential equations.