Global phase diagrams of binary systems in pressure-temperature (PT) space are very useful. In this project the techniques to automatically develop global phase diagrams in PT space were created. The codes to compute different components of a global phase diagram in PT space were developed. These codes were then successfully incorporated into a single functional program. <p>To generate the binary PT phase diagram, the overall composition was varied from pure component 2, the least volatile component (LVC) to pure component 1, the most volatile component (MVC). The step size for changing mole fraction was varied in the calculation of different parts of a global phase diagram. When the points near the joining points between different parts were computed, the step size was set to a rather small value. The step size was then increased to twice of the last value for each subsequent point computed. When the MVC mole fraction was approaching one, the step size was set to a small value to obtain enough points needed to minimize the chances of missing important phenomena. <p>The techniques to set initial guesses for evaluation of different components of a global phase diagram were discussed. The code performance, including the number of iterations for different convergence criteria and the sensitivity of the algorithm were presented. Using the code developed, phase diagrams of type I, type II, type III and type V were generated using representative binary systems from the petroleum processing field. <p>The boundary states between different types of phase behaviour were also explored. It was observed that with the increase of the binary interaction parameters, the phase behaviour of the ethane + ethanol binary system changes from type I to type II to type III while the methane + n-hexane binary system changes from type V to type III. These conclusions matched the results of van Konynenburg and Scott (1980). It was also concluded that with the increase of the binary interaction parameter for a binary system, the system showed a trend to exhibit more liquid-liquid immiscibility.

Global phase diagrams of binary systems in pressure-temperature (PT) space are very useful. In this project the techniques to automatically develop global phase diagrams in PT space were created. The codes to compute different components of a global phase diagram in PT space were developed. These codes were then successfully incorporated into a single functional program. <p>To generate the binary PT phase diagram, the overall composition was varied from pure component 2, the least volatile component (LVC) to pure component 1, the most volatile component (MVC). The step size for changing mole fraction was varied in the calculation of different parts of a global phase diagram. When the points near the joining points between different parts were computed, the step size was set to a rather small value. The step size was then increased to twice of the last value for each subsequent point computed. When the MVC mole fraction was approaching one, the step size was set to a small value to obtain enough points needed to minimize the chances of missing important phenomena. <p>The techniques to set initial guesses for evaluation of different components of a global phase diagram were discussed. The code performance, including the number of iterations for different convergence criteria and the sensitivity of the algorithm were presented. Using the code developed, phase diagrams of type I, type II, type III and type V were generated using representative binary systems from the petroleum processing field. <p>The boundary states between different types of phase behaviour were also explored. It was observed that with the increase of the binary interaction parameters, the phase behaviour of the ethane + ethanol binary system changes from type I to type II to type III while the methane + n-hexane binary system changes from type V to type III. These conclusions matched the results of van Konynenburg and Scott (1980). It was also concluded that with the increase of the binary interaction parameter for a binary system, the system showed a trend to exhibit more liquid-liquid immiscibility.

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.

No description available.

No description available.

Sistemas diferenciais quadráticos planares estão presentes em muitas áreas da matemática aplicada. Embora mais de mil artigos tenham sido publicados sobre os sistemas quadráticos ainda resta muito a se conhecer sobre esses sistemas. Problemas clássicos, e em particular o XVI problema de Hilbert, estão ainda em aberto para essa família. Um dos objetivos dos pesquisadores contemporâneos é obter a classificação topológica completa dos sistemas quadráticos. Devido ao grande número de parâmetros (essa família possui doze parâmetros e, aplicando transformações afins e reescala do tempo, reduzimos esse número a cinco, sendo ainda um número grande para se trabalhar) usualmente subclasses são consideradas nas investigações realizadas. Quando características específicas são levadas em consideração, o número de parâmetros é reduzido e o estudo se torna possível. Nesta tese estudamos principalmente duas subfamílias de sistemas quadráticos: a primeira possuindo um nó triplo semielemental e a segunda possuindo uma selanó semi elemental finita e uma selanó semielemental infinita formada pela colisão de uma sela infinita com um nó infinito. Os diagramas de bifurcação para ambas as famílias são tridimensionais. A família tendo um nó triplo gera 28 retratos de fase topologicamente distintos, enquanto o fecho da família tendo as selasnós dentro do espaço de bifurcação de sua forma normal gera 417. Polinômios invariantes são usados para construir os conjuntos de bifurcação e os retratos de fase topologicamente distintos são representados no disco de Poincaré. Os conjuntos de bifurcação são a união de superfícies algébricas e superfícies cuja presença foi detectada numericamente. Ainda nesta tese, apresentamos todos os retratos de fase de um sistema diferencial conhecido como modelo do tipo SIS (sistema suscetívelinfectadosuscetível, muito comum na matemática aplicada) e a classificação dos sistemas quadráticos possuindo hipérboles invariantes. Ambos sistemas foram investigados usando de polinômios invariantes afins. / Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilberts 16th problem, are still open for this family. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a large number), many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study. In this thesis we mainly study two subfamilies of quadratic systems: the first one possessing a finite semielemental triple node and the second one possessing a finite semielemental saddlenode and an infinite semielemental saddlenode formed by the collision of an infinite saddle with an infinite node. The bifurcation diagram for both families are tridimensional. The family having the triple node yields 28 topologically distinct phase portraits, whereas the closure of the family having the saddlenodes within the bifurcation space of its normal form yields 417. Invariant polynomials are used to construct the bifurcation sets and the phase portraits are represented on the Poincaré disk. The bifurcation sets are the union of algebraic surfaces and surfaces whose presence was detected numerically. Moreover, we also present the analysis of a differential system known as SIS model (this kind of systems are easily found in applied mathematics) and the complete classification of quadratic systems possessing invariant hyperbolas.

Sistemas diferenciais quadráticos planares estão presentes em muitas áreas da matemática aplicada. Embora mais de mil artigos tenham sido publicados sobre os sistemas quadráticos ainda resta muito a se conhecer sobre esses sistemas. Problemas clássicos, e em particular o XVI problema de Hilbert, estão ainda em aberto para essa família. Um dos objetivos dos pesquisadores contemporâneos é obter a classificação topológica completa dos sistemas quadráticos. Devido ao grande número de parâmetros (essa família possui doze parâmetros e, aplicando transformações afins e reescala do tempo, reduzimos esse número a cinco, sendo ainda um número grande para se trabalhar) usualmente subclasses são consideradas nas investigações realizadas. Quando características específicas são levadas em consideração, o número de parâmetros é reduzido e o estudo se torna possível. Nesta tese estudamos principalmente duas subfamílias de sistemas quadráticos: a primeira possuindo um nó triplo semielemental e a segunda possuindo uma selanó semi elemental finita e uma selanó semielemental infinita formada pela colisão de uma sela infinita com um nó infinito. Os diagramas de bifurcação para ambas as famílias são tridimensionais. A família tendo um nó triplo gera 28 retratos de fase topologicamente distintos, enquanto o fecho da família tendo as selasnós dentro do espaço de bifurcação de sua forma normal gera 417. Polinômios invariantes são usados para construir os conjuntos de bifurcação e os retratos de fase topologicamente distintos são representados no disco de Poincaré. Os conjuntos de bifurcação são a união de superfícies algébricas e superfícies cuja presença foi detectada numericamente. Ainda nesta tese, apresentamos todos os retratos de fase de um sistema diferencial conhecido como modelo do tipo SIS (sistema suscetívelinfectadosuscetível, muito comum na matemática aplicada) e a classificação dos sistemas quadráticos possuindo hipérboles invariantes. Ambos sistemas foram investigados usando de polinômios invariantes afins. / Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular Hilberts 16th problem, are still open for this family. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible in the whole class due to the large number of parameters (twelve, but, after affine transformations and time rescaling, we arrive at families with five parameters, which is still a large number), many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study. In this thesis we mainly study two subfamilies of quadratic systems: the first one possessing a finite semielemental triple node and the second one possessing a finite semielemental saddlenode and an infinite semielemental saddlenode formed by the collision of an infinite saddle with an infinite node. The bifurcation diagram for both families are tridimensional. The family having the triple node yields 28 topologically distinct phase portraits, whereas the closure of the family having the saddlenodes within the bifurcation space of its normal form yields 417. Invariant polynomials are used to construct the bifurcation sets and the phase portraits are represented on the Poincaré disk. The bifurcation sets are the union of algebraic surfaces and surfaces whose presence was detected numerically. Moreover, we also present the analysis of a differential system known as SIS model (this kind of systems are easily found in applied mathematics) and the complete classification of quadratic systems possessing invariant hyperbolas.