Kapaciteter kan vara till stor nytta, bland annat då partiella differentialekvationer ska lösas. Kapaciteter är dock i många fall väldigt svåra att beräkna exakt, speciellt i viktade rum. Vad som istället kan göras är att försöka uppskatta kapaciteterna, vilket för ringar runt en fix punkt kan utföras med hjälp av fyra olika exponentmängder, \underline{Q}_0, \underline{S}_0, \overline{S}_0 och \overline{Q}_0, som beskriver hur vikten beter sig i närheten av denna punkt och i viss mån ger rummets lokala dimension. För att kunna dra nytta av exponentmängderna är det bra att veta vilka kombinationer av dessa som kan förekomma. För att få fram nya kombinationer använder vi olika sätt att mäta volym av klot med varierande radier. Dessa mått är definierade genom olika vikter. Det har tidigare funnits ett fåtal exempel på hur olika kombinationer av exponentmängderna kan se ut. Variationerna består av hur avstånden är i förhållande till varandra och om ändpunkterna tillhör mängderna eller inte. I denna rapport har vi tagit fram ytterligare fem nya kombinationer av mängderna, bland annat en där \underline{Q}_0 är öppen. / Capacities can be of great benefit, for instance when solving partial differential equations. In most cases, capacities can be difficult to calculate exactly, in particular on weighted spaces. In these cases, it can be sufficient with an estimation of the capacity instead. For annuli around a given point, the estimation can be done using four exponent sets \underline{Q}_0, \underline{S}_0, \overline{S}_0 and \overline{Q}_0, which describe how the weight behaves in a neighbourhood of that point and in some sense define the local dimension of the space. To be able to use the exponent sets, it is useful to know which combinations of them can exist. For this we use various measures, which are a way to measure volumes of balls with varying radii in Rn. These measures are defined by different weights. Earlier, there existed a few examples giving different combinations of exponent sets. The variations consist in their relationship to each other and if their endpoints belong to the set or not. In this thesis we present five new combinations of the exponent sets, amongst them one where \underline{Q}_0 is open.

Dado um sistema diferencial no plano, muito se questiona sobre o comportamento de suas soluções. Nas vizinhanças dos pontos singulares existem ferramentas que nos indicam o tipo e a estabilidade estrutural de cada um deles; são as chamadas formas normais. No entanto, o interesse vai mais além do conhecimento local das soluções em cada singularidade. Nesse trabalho apresentamos algumas ferramentas clássicas da teoria qualitativa das equações diferenciais ordinárias empregadas na investigação global dos campos de vetores polinomiais planares e as empregamos na investigação de duas famílias paramétricas de campos quadráticos encontradas no estudo dos campos com hipérboles invariantes. Dentre as ferramentas estudadas destacamos a classificação local das soluções em pontos singulares elementares e semi-elementares e a técnica de compactificação de Poincaré. / Given a planar differential system, many questions are raised about the behavior of their solutions. In the neighborhood of singular points there exist many tools which indicate their type and their structural stability; they are known as normal forms. However, the interest goes beyond the local behavior in the neighborhood of each singularity. In this dissertation we present some classical tools from the qualitative theory of ordinary differential equations which are usually applied to the global investigation of planar polinomial vector fields and we apply them to the investigation of two parametric families of quadratic fields from the study of the vector fields with invariant hyperbolas. Among the studied tools we highlight the local classification of the solutions around elementary and semi-elementary singular points and the technique known as Poincarés compactification.

Dado um sistema diferencial no plano, muito se questiona sobre o comportamento de suas soluções. Nas vizinhanças dos pontos singulares existem ferramentas que nos indicam o tipo e a estabilidade estrutural de cada um deles; são as chamadas formas normais. No entanto, o interesse vai mais além do conhecimento local das soluções em cada singularidade. Nesse trabalho apresentamos algumas ferramentas clássicas da teoria qualitativa das equações diferenciais ordinárias empregadas na investigação global dos campos de vetores polinomiais planares e as empregamos na investigação de duas famílias paramétricas de campos quadráticos encontradas no estudo dos campos com hipérboles invariantes. Dentre as ferramentas estudadas destacamos a classificação local das soluções em pontos singulares elementares e semi-elementares e a técnica de compactificação de Poincaré. / Given a planar differential system, many questions are raised about the behavior of their solutions. In the neighborhood of singular points there exist many tools which indicate their type and their structural stability; they are known as normal forms. However, the interest goes beyond the local behavior in the neighborhood of each singularity. In this dissertation we present some classical tools from the qualitative theory of ordinary differential equations which are usually applied to the global investigation of planar polinomial vector fields and we apply them to the investigation of two parametric families of quadratic fields from the study of the vector fields with invariant hyperbolas. Among the studied tools we highlight the local classification of the solutions around elementary and semi-elementary singular points and the technique known as Poincarés compactification.