ORDINARY DIFFERENTIAL EQUATIONS

Author: BRAVO TRINIDAD JOSÉ LUIS.
Year: 2003.
University: EXTREMADURA [www.unex.es].
Place of defense: FACULTAD DE CIENCIAS.
Place of preparation: FACULTAD DE CIENCIAS.

Author: CID ARAÚJO JOSÉ ÁNGEL.
Year: 2004.
University: SANTIAGO DE COMPOSTELA [www.usc.es].
Place of defense: FACULTAD DE MATEMÁTICAS.
Place of preparation: FACULTAD DE MATEMÁTICAS.

Summary: This report consists of three chapters and an apéndice.En the first chapter results are obtained for the existence of discrete solution to problems of the first order and results of uniqueness for the case continuo.En the second chapter test the existence of a solution to a problem the initial value and multiple solutions to a problem of border newspaper, for both equations of second order with discontinuidades.En the third chapter includes several original contributions to the theory of fixed points and provides multiple applications of these results the princiupales properties the functions absolutely continual and some fixed-point theorems for discontinuous growing operators that are fundamental to the results presented in this report.

Author: SOROLLA BARDAJÍ JORDI.
Year: 2004.
University: AUTÓNOMA DE BARCELONA [www.uab.es].
Place of defense: FACULTAT DE CIÈNCIES.
Place of preparation: ESCUELA DE POSTGRADO.

Summary: We found all systems quadratic (grade 2) we can get a curve invariant grade less than or equal to 4 containing an oval that is, in turn, cilclo limit of the system. First we study the potential curves in terms of their points autoinsercción (multiple items). Then, we came to approximating the shape of the curve from certain properties of the indexes at the points of intersection and its unique location. Finally, we see if it can be invariant under a system square and just adjust the parameters that are free. They also studied the cycles limit from the point of view of systems rather than from curves invariants. So, we quadratic systems that can limit cycles, specifically the classification of China (families I, II and III). To the family I seek invariant curves and inverse integral factor: for families looking II and III of inverse integral. We just showing that I have no family cycles limits algebracios. Finally, we study the coexistence of two algebraic cycles limits, which belong to different irreducible invariant curves, in a square. It shows that these cycles a limit should be contained inside the other. The demonstration passes to see whether these cycles defnieran regions not intersecasen, then studying the values of cofactor in the singular points we see that could build a reverse factor integral polynomial to be the product of the two curves, and that would lead to an integral Darboux first type, which leads to a contradiction.

Author: PASCUAL LERÍA ANA ISABEL.
Year: 2004.
University: LA RIOJA [www.unirioja.es].
Place of defense: UNIVERSIDAD DE LA RIOJA.
Place of preparation: UNIVERSIDAD DE LA RIOJA.

Summary: The question of the stability of systems hamiltonianos is a key element in the study of some problems in various scientific fields, such as Classical Mechanics, Celestial Mechanics, atomic physics, etc.. Moreover, it is a topic of great interest mathematician. However, the problem is difficult to tackle even for systems of two degrees of freedom where, despite being the simplest case where there are more studies done, there are still special situations unresolved. Despite the existence of many problems of implementation and results for some individual cases, until 1999 did not set forth any theorem. On this date, Cabral and Meyer establish an approach to solving the stability of a system hamiltoniano two degrees of freedom in the presence of resonances which encompasses most of the classical results. The main contribution of this thesis is to obtain a theorem that considers assumptions weaker than the theorem Cabral and Meyer, so that it generalizes and to solve the question of stability in more general terms. In addition, we geometrical interpretation of this result, thus establishing a geometric approach. From this approach it is possible to obtain new results of stability for some cases that the theorem does not solve Cabral and Meyer called degenerate cases. The process for drawing these conclusions is complicated and requires the use of a simpler model, the normal way of hamiltoniano. This involves techniques standardization we have done with a given set of variables, variables Lissajous. In this regard, another contribution is a compact characterization of the normal way in terms of some variables, called invariants, linked to the variables of Lissajous. In addition to the study of the stability is important characterization flow phasic as the strong relative tradeoffs is associated with the presence of families of periodic orbits. These families are of great interest, especially in certain problems of Celestial Mechanics. This issue is also being addressed in this thesis and, in particular, looks at what happens in the case of a resonance of order 4, characterizing the different types of flow phasic. These balances are determined by the relative branching including parametric, whose calculation is related to the number of real roots of a polynomial in a closed interval.

Author: GARCÍA QUESADA JESÚS.
Year: 2005.
University: LAS PALMAS DE GRAN CANARIA [www.ulpgc.es].
Place of defense: FACULTAD DE INFORMÁTICA.
Place of preparation: FACULTAD DE INFORMÁTICA.

Summary: This thesis contains findings on the dynamic behavior of five increasingly complex mathematical models on environmental quality of urban settlements. The models were proposed in [1] and in this work are completed studies on branching that began in that publication. In Chapter 1 relates some key results of the theory of dynamic systems and are some methods for simplifying the study of such systems. In the following chapter on the models presented different models and examines the dynamics that are presented by Planar four models introduced in [1], giving the bifurcation diagrams where they are needed. In most cases is sufficient analytical study, which allows specifying the behavior they present. In chapter 3, is studied in detail the fifth model. In this model the three-dimensional state variables representing quality, population and poor respectively. It consists of three coupled differential equations. For each of the model parameters, provides a range of numerical values "reasonable". These ranges form the basis of the numerical studies. Get singular points and made a study of their stability. Then we study the branching uniparamétricas and cycles resulting from the limits of Hopf bifurcation, and then obtain the behavior of the rest of parameters not seen so far. After studying the branching biparamétricas get the full branching diagram respect to the plane of parameters chosen.

Author: Alvarez Torres María Jesús.
Year: 2005.
University: AUTÓNOMA DE BARCELONA [www.uab.es].
Place of defense: Dpt.de Matemàtiques.
Place of preparation: Departament de Matemàtiques (Universitat Autònoma de Barcelona).

Summary: In this thesis we study systems of differential equations in the plane. We worked two of the unique solutions that have these dynamic systems: critical points and periodic orbits. Regarding apuntos critics, we focus mainly on the nilpotentes and studying the problem centro-foco. Regarding periodic orbits, we work with equations Abel and other systems in the cylinder. To give them different criteria to limit the number of periodic orbits that can have this kind of sisitemas and also obtain lower bounds for the number of periodic orbits.

Author: FERRAGUT I AMENGUAL ANTONI MANEL.
Year: 2005.
University: AUTÓNOMA DE BARCELONA [www.uab.es].
Place of defense: FACULTAT DE CIÈNCIES.
Place of preparation: DEPARTAMENT DE MATEMÁTIQUES, UNIVERSITAT AUTÓNOMA DE BARCELONA.