magistrsko delo
Metka Majcen (Author), Brigita Ferčec (Mentor)

Abstract

V magistrskem delu so obravnavani avtonomni ravninski sistemi navadnih diferencialnih enačb. Na začetku so opisani osnovni pojmi, kot so obstoj in enoličnost rešitev ter geometrijska predstavitev krivuljnih rešitev sistema. V nadaljevanju so obravnavani avtonomni sistemi v ravnini in konstrukcija njihovih faznih portretov. Drugo poglavje je namenjeno linearnim sistemom diferencialnih enačb, kjer so na začetku opisani t. i. nepovezani linearni sistemi, diagonalizacija in Jordanova forma matrike. Predvsem je poudarek na dvorazsežnih matrikah, kajti v nadaljevanju so navedene lastnosti enostavnih in ne enostavnih ravninskih linearnih sistemov, njihovi fazni portreti ter tipi in stabilnost singularne točke v izhodišču. Tretje poglavje je namenjeno avtonomnim nelinearnim sistemom v ravnini in njihovim faznim portretom. Na začetku je opisana linearizacija nelinearnega sistema v okolici singularne točke in z njo povezan linearizacijski izrek. Potem je podrobneje obravnavana stabilnost singularnih točk (tudi glede Liapunove funkcije). Nato so navedeni in na kratko opisani še nekateri drugi objekti, ki lahko ob singularnih točkah nastopijo v faznih portretih nelinearnih sistemov: navadne točke, limitni cikli (obravnavani sta tudi Hopfova in sedlo-vozel bifurkacija), homoklinične in heteroklinične orbite. Zadnji razdelek tega poglavja je namenjen enemu izmed osrednjih problemov v kvalitativni teoriji sistemov navadnih diferencialnih enačb, t.j. problemu centra in fokusa. Ker je le-ta povezan z obstojem prvega integrala določene oblike, je navedena tudi definicija prvega integrala sistema diferencialnih enačb.

Keywords

magistrska dela;navadne diferencialne enačbe;singularne točke;fazni portreti;trajektorije;limitni cikli;stabilnost;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UM FNM - Faculty of Natural Sciences and Mathematics
Publisher: [M. Majcen]
UDC: 517.91(043.2)
COBISS: 24181000 Link will open in a new window
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Downloads: 154
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Other data

Secondary language: English
Secondary title: Singular points and phase portraits in polynominal planar systems of ordinary differential equations
Secondary abstract: In this master thesis the autonomous planar systems of ordinary differential equations are studied. At the beginning, the basic notions such as the existence and uniqueness of the solution and the geometrical interpretation of curve solutions are described. Then, the autonomous systems in the plane and the construction of their phase portraits is considered. The second chapter is devoted to linear systems of differential equations, where at the beginning the so-called uncoupled linear systems, diagonalization and Jordan form of the matrix are studied. Especially, the focus is on the simple and non-simple planar linear systems, their phase portraits and the types and the stability of the singular point at the origin are mentioned. The third chapter is dedicated to autonomous nonlinear systems in the plane and their phase portraits. At the beginning, the linearization of nonlinear system in the neighbourhood of singular point and the Linearization theorem connected with it are stated. Then the stability of singular points (also in terms of Liapunov function) is discussed in more detail. Furthermore, some other elements are mentioned and briefly described, which can occur in the phase portraits of nonlinear systems beside singular points: ordinary points, limit cycles (Hopf bifurcation and saddle-node bifurcation are considered), homoclinic and heteroclinic orbits. The last section of this chapter is devoted to one of the main problems in qualitative theory of systems of ordinary differential equations, i.e. center-focus problem. Since this problem is connected with the existence of first integral of the certain form the definition of first integral of system of differential equations is stated, too.
Secondary keywords: master theses;ordinary differential equations;singular points;phase portraits;trajectories;limit cycles;stability;
URN: URN:SI:UM:
Type (COBISS): Master's thesis/paper
Thesis comment: Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages: XIV, 74 f.
ID: 10959029