Atraktori beskonačno dimenzionalnih disipativnih sustava

Abstract

Na početku rada, u poglavljima Uvod u dinamičke sustave i Orbite i invarijantni skupovi, obradili smo osnovne pojmove i u primjerima naveli neke dinamičke sustave. Definirali smo invarijantne i granične skupove, orbite, te važne vrste točaka sustava i njihova svojstva bitna za daljnje razumijevanje. U sljedećem poglavlju definirali smo pojam atraktora i potom tu definiciju proširili uz dodatne uvjete na nekoliko različitih tipova atraktora, od kojih su najpoznatiji Milnorov i Ilyashenkov. Zatim smo pokazali odnos među njima i pripremili temelje za sljedeća razmatranja. Nadalje, u četvrtom i petom poglavlju, proučavali smo apsorbirajuće skupove, objasnili pojam disipativnosti i kroz različite rezultate promatrali zahtjeve asimptotske kompaktnosti na atraktorima. Također, upoznali smo se s uvjetima za postojanje globalnog atraktora i njegovim svojstvima. U šestom smo se poglavlju, Struktura globalnog atraktora, bavili pojmom nestabilnog skupa i njegovom ulogom u atraktoru, objasnili termin Lyapunove funkcije i primjerom pokazali njeno korištenje pri dokazivanju egzistencije periodičke orbite u atraktoru. U posljednjem, sedmom poglavlju, obradili smo definicije stabilnosti, asimptotske stabilnosti te Poissonove stabilnosti i proučili uvjete potrebne da bi atraktor bio stabilan. Na samom kraju naveli smo i dokazali važan tehnički rezultat – Princip redukcije, koji nam omogućava smanjenje dimenzije faznog prostora, što je vrlo važna činjenica u proučavanju beskonačno dimenzionalnih sustava.At the beginning of this thesis the themes Introduction to dynamic systems and Trajectories and invariant sets are dealt with, including a few examples of dynamic systems. Defined are invariant and limit sets, trajectories, together with important points of the systems and their characteristics which are crucial for further understanding. The focus of the next chapter is on the concept of attractors, their definition being extended with additional conditions for a number of different types of attractors, Milnor’s and Ilyashenkov’s being the most famous. We then showed relations among them and introduced milestones for further considerations. In the fourth and fifth chapters absorbing sets are examined, the concept of dissipativity is explained and asymptotic compactness on attractors is contemplated by way of different results. Furthermore, conditions for the existence of a global attractor and its characteristics are introduced. The sixth chapter – The Structure of a Global Attractor – deals with the concept of unstable set and its role within an attractor, and the concept of Lyapun’s function is explained, followed by an example of its use at proving the existence of periodic trajectories in an attractor. In the final, seventh chapter, the definition of stability, asymptotic stability and Poisson’s stability is addressed and conditions necessary for an attractor to be stable are examined. At the very end an important technical result – Reduction Principle - is quoted and proven because it enables us to decrease the dimension of the phase space, this fact being very important for the study of infinitedimensional systems