Nonunique Equilibria Of Projected Dynamical Systems And Their Applications

Projected dynamical systems were formulated in the 1990s by Dupuis and Nagurney. In contrast to classical dynamical systems, projected dynamical systems have discontinuous right hand sides that are associated with a projection operator. By the projection mecha- nism, the whole Hilbert space H is projected onto a non-empty, closed convex set K ⊂ H. Critical points of a projected dynamical system coincide with the solutions of a corresponding variational inequality problem; therefore, the applications of projected dynamical systems have been found in various fields such as economics, operations research and engineering. In this thesis, we study applications of equilibria of projected dynamical systems for market equilibrium problems, games and compartmental population models. First, the well-known market disequilibrium model with excess supply and demand is investigated to determine if it exhibits changes in the structure and the number of equilibrium states for specific choices of parameter values. We study the bifurcation problem (i.e., a qualitative change in equilib- rium states) as a parameterized variational inequality problem. We conduct our analysis by modeling the markets via a projected dynamical system. Second, we present a combination of theoretical and computational results meant to give insights into the question of the existence of nonunique Nash equilibria for N-player nonlinear games. Our inquiries make use of the theory of variational inequalities and projected systems to classify cases where multiplayer Nash games with parameterized payoffs exhibit changes in the number of Nash equilibria, depending on given parameter values. Finally, we use the compartmental popu- lation model, namely, the deterministic Susceptible-Exposed-Infectious-Recovered model, to analyze the dynamics of influenza infection of a farrow-to-finish swine farm, and we explore the reinfection at the farm level. We further examine the effectiveness of two control strate- gies: vaccination and reduction of indirect contact. In this case, we show that the model is a projected dynamical system but the projection does not add any relevant applied meaning to the results. Therefore, we show that the problem can be studied using classical dynamical systems.

Equilibrium is a concept used in operations research and economics to understand the interplay of factors and problems arising from competitive systems in the economic world. The problems in this area are large and complex and have involved a variety of mathematical methodologies. In this monograph, the authors have widened the scope of theoretical work with a new approach, `projected dynamical systems theory', to previous work in variational inequality theory. While most classical work in this area is static, the introduction to the theory of projected dynamical systems will allow many real-life dynamic situations and problems to be handled and modeled. This monograph includes: a new theoretical approach, `projected dynamical system', which allows the researcher to model real-life situations more accurately; new mathematical methods allowing researchers to combine other theoretical approaches with the projected dynamical systems approach; a framework in which research can adequately model natural, financial and human (real life) situations in competitive equilibrium problems; the computational and numerical methods for the implementation of the methods and theory discussed in the book; stability analysis, algorithms and computational procedures are offered for each set of applications.

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.

The communication of knowledge on nonlinear dynamical systems, between the mathematicians working on the analytic approach and the scientists working mostly on the applications and numerical simulations has been less than ideal. This volume hopes to bridge the gap between books written on the subject by mathematicians and those written by scientists. The second objective of this volume is to draw attention to the need for cross-fertilization of knowledge between the physical and biological scientists. The third aim is to provide the reader with a personal guide on the study of global nonlinear dynamical systems.

An introduction to aspects of the theory of dynamical systems based on extensions of Liapunov's direct method. The main ideas and structure for the theory are presented for difference equations and for the analogous theory for ordinary differential equations and retarded functional differential equations.

An authoritative treatment by leading researchers covering theory and optimal estimation, along with practical applications.

The 1995 observation of Bose-Einstein condensation in dilute atomic vapours spawned the field of ultracold, degenerate quantum gases. Unprecedented developments in experimental design and precision control have led to quantum gases becoming the preferred playground for designer quantum many-body systems.This self-contained volume provides a broad overview of the principal theoretical techniques applied to non-equilibrium and finite temperature quantum gases. Covering Bose-Einstein condensates, degenerate Fermi gases, and the more recently realised exciton-polariton condensates, it fills a gap by linking between different methods with origins in condensed matter physics, quantum field theory, quantum optics, atomic physics, and statistical mechanics. Thematically organised chapters on different methodologies, contributed by key researchers using a unified notation, provide the first integrated view of the relative merits of individual approaches, aided by pertinent introductory chapters and the guidance of editorial notes.Both graduate students and established researchers wishing to understand the state of the art will greatly benefit from this comprehensive and up-to-date review of non-equilibrium and finite temperature techniques in the exciting and expanding field of quantum gases and liquids./a