As equações de diferenças (ou equações discretas) desempenham papel fundamental na modelagem de problemas em que o tempo é medido em intervalos discretos, por exemplo, dia, mês, ano. Elas estão presentes em sistemas físicos, químicos, biológicos, sociais e econômicos. O objetivo desse trabalho é estudar as equações de diferenças de primeira ordem, focando aspectos teóricos, análise do comportamento assintótico das soluções através de técnicas analíticas (teoremas de estabilidade) e técnicas gráficas (diagramas de Cobweb). Também são desenvolvidas algumas aplicações. Além disso, são apresentadas três propostas didáticas relacionadas ao tema para serem trabalhadas no Ensino Médio. / The difference equations (or discrete equations) play a key role in shaping problems in which time is measured at discrete intervals, e.g., day, month, year. They may be applied to physical, chemical, biological, social and economic systems. The aim of this work is to study the first-order difference equations, focusing on theoretical aspects, asymptotic behavior of solutions throughout analytical techniques (stability theorems) and graphical techniques (Cobweb diagrams). Some applications are also shown. Three teaching proposals related to the theme are presented in order to be developed in High School.

[pt] Este trabalho tem por objetivo apresentar novas técnicas de análise e julgamento para a tomada de decisão no que concerne ao planejamento estratégico das empresas de energia elétrica. Procura abordar a tomada de decisão e a comercialização da empresa, de uma maneira totalmente inovadora, onde os agentes baseiam suas decisões em seu conhecimento e sua criatividade. Para isso faz-se uso da Teoria de FUZZY SETS, de forma simples e consistente, fornecendo resultados mais estruturados para a análise de investimento. É proposta a utilização da Teoria dos Jogos Não- cooperativos, onde as incertezas são modeladas através de conjuntos fuzzy, sua posterior desfuzificação e finalmente a determinação do ponto de equilíbrio. Alguns exemplos ilustram o modelo proposto ao cálculo do preço da energia elétrica; também é abordado o estudo de um sistema real da América Latina. / [en] The objective of this work is to present new analysis and judgment techniques to help decisions actions in the field of strategic planning of the electric utility industries. It intends to help the industry with business trade, in an innovative way, where the decisions are based on the agents previous knowledge and creativity. In order to do so the FUZZT SETS THEORY is used, in a simple and consistent manner, to generated more structure results for investment analysis. The theory of non-Cooperative Games is proposed, where the uncertain are modeled through fuzzy sets, their defuzzification and finally the determination of the equilibrium point. Some examples show the usage of the proposed model to the calculation of electric energy price; also it is discussed a study of a real latin America System.

abstract: For a (N+1)-bus power system, possibly 2N solutions exists. One of these solutions is known as the high-voltage (HV) solution or operable solution. The rest of the solutions are the low-voltage (LV), or large-angle, solutions. In this report, a recently developed non-iterative algorithm for solving the power- flow (PF) problem using the holomorphic embedding (HE) method is shown as being capable of finding the HV solution, while avoiding converging to LV solutions nearby which is a drawback to all other iterative solutions. The HE method provides a novel non-iterative procedure to solve the PF problems by eliminating the non-convergence and initial-estimate dependency issues appeared in the traditional iterative methods. The detailed implementation of the HE method is discussed in the report. While published work focuses mainly on finding the HV PF solution, modified holomorphically embedded formulations are proposed in this report to find the LV/large-angle solutions of the PF problem. It is theoretically proven that the proposed method is guaranteed to find a total number of 2N solutions to the PF problem and if no solution exists, the algorithm is guaranteed to indicate such by the oscillations in the maximal analytic continuation of the coefficients of the voltage power series obtained. After presenting the derivation of the LV/large-angle formulations for both PQ and PV buses, numerical tests on the five-, seven- and 14-bus systems are conducted to find all the solutions of the system of nonlinear PF equations for those systems using the proposed HE method. After completing the derivation to find all the PF solutions using the HE method, it is shown that the proposed HE method can be used to find only the of interest PF solutions (i.e. type-1 PF solutions with one positive real-part eigenvalue in the Jacobian matrix), with a proper algorithm developed. The closet unstable equilibrium point (UEP), one of the type-1 UEP’s, can be obtained by the proposed HE method with limited dynamic models included. The numerical performance as well as the robustness of the proposed HE method is investigated and presented by implementing the algorithm on the problematic cases and large-scale power system. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2015

As equações de diferenças (ou equações discretas) desempenham papel fundamental na modelagem de problemas em que o tempo é medido em intervalos discretos, por exemplo, dia, mês, ano. Elas estão presentes em sistemas físicos, químicos, biológicos, sociais e econômicos. O objetivo desse trabalho é estudar as equações de diferenças de primeira ordem, focando aspectos teóricos, análise do comportamento assintótico das soluções através de técnicas analíticas (teoremas de estabilidade) e técnicas gráficas (diagramas de Cobweb). Também são desenvolvidas algumas aplicações. Além disso, são apresentadas três propostas didáticas relacionadas ao tema para serem trabalhadas no Ensino Médio. / The difference equations (or discrete equations) play a key role in shaping problems in which time is measured at discrete intervals, e.g., day, month, year. They may be applied to physical, chemical, biological, social and economic systems. The aim of this work is to study the first-order difference equations, focusing on theoretical aspects, asymptotic behavior of solutions throughout analytical techniques (stability theorems) and graphical techniques (Cobweb diagrams). Some applications are also shown. Three teaching proposals related to the theme are presented in order to be developed in High School.

[pt] É considerado um protocolo de múltiplo acesso de alocação híbrida e adaptativa, que pode ser utilizado em redes VSAt. Neste protocolo as transmissão são feitas em um canal S-Aloha ou em um canal TDMA de alocação por reserva. As estações remotas dispõem de buffer para armazenamento de mensagens. O desempenho do protocolo é avaliado em termos da vazão e do retardo médio, através da técnica de Análise do Ponto de Equilíbrio (EPA). A estabilidade do protocolo é também examinada. É feita uma comparação deste protocolo com um protocolo S-Aloha, ambos ocupando igual largura de faixa de freqüências. Variações do protocolo básico, considerando o canal TDMA de alocação por reserva com largura de faixa menor que a alocada para o canal S- Aloha são também analisadas. Outra variação importante abordada é aquela na qual W canais S-Aloha com grupos de equações remotas operando separando em cada um deles se servem de um único canal TDMA de alocação por reserva. Para esta variação são indicadas as modificações nas equações obtidas para o protocolo básico que permitem escrever as expressões da vazão e do retardo médio. Resultados numéricos da vazão e do retardo médio para os diversos protocolos são apresentados. / [en] A multiple acess protocol, which can be used in a VSAT network, with an adaptive and hybrid assignment is considered. In this protocol the transmission can be done in either an S-Aloha channel or in a demand assignment TDMA channel. The remote stations have finite buffer capacity to store arriving messages. The performance is evaluated in terms of throughput and expected delay, by means of the so called Equilibrium Point Analysis (EPA). Stability of the protocol is also examined. A comparison between this protocol and an S-Aloha one, both with the same bandwidth, is also carried out. Variations of the basic protocol considering the demand assignment TDMA channel with a smaller bandwidth than that one of the S- Aloha channel are also considered. Another important variation examined is the one in which W S-Aloha channels serve W separate groups of remote stations and all share a demand assignment TDMA channel. For this variation, modification in the equilibrium equations obtained for the basic protocol as well as for expression of throughput and expected delay are indicated. Numerical results of throughput and expect delay for the several protocols are given.

In this thesis an age structured model of fish and harvesting of fish are studied. This includes the basic properties as well as the analysis of equilibria and their stability for various recruitment functions. The only nonlinearity of this model lies in the juvenile stage in form of the recruitment function. Recruitment is considered as recruitment into the first year fertile class. The Ricker recruitment function will mostly be used . But we also discuss the Beverton-Holt recruitment and compare its impact. The main result is that monotone recruitment functions lead to a globally stable equilibrium. For Ricker recruitment functions periodic solutions are possible, but this model for cod and herring always leads to stable equilibria. Harvesting occurs in all classes separately with no harvesting in the first class. We describe harvesting in form of harvesting effort, resulting in an increase of mortality. This approach allows to model the interference of natural mortality and fishing mortality. Harvesting cost will be considered as a linear function of effort and the corresponding maxima are determined. This has the advantage to keep the number of parameters low. This makes the model more tractable and easier to apply to other stocks or species of fish. Optimization of the harvest is determined and analyzed. It is based on harvesting intensity and mesh size. We also show that harvesting increases stability of the static model. This model is extended to a discrete optimal control model in order to determine optimal sustainable policies. The solutions are almost of the fastest approach type to the equilibrium type solution. Periodic or pulse solutions do not arise. Near optimal strategies are analyzed and other feasible strategies are also considered. In last part of this thesis the problem of extinction of population is investigated. The effect of various factors as well as natural factors on marine fish populations are studied, which are described by the model in order to get some idea on the size of various factors. Most fishing takes place near the break even intensity. It is defined as the fishing mortality at which cost equals profit. As a further aspect recovery scenarios are studied. Concrete numerical studies for optimal solutions are determined for cod and herring. Care is taken to take into account all possible realistic parameters but also to keep the model tractable for other species.

Quando um capital é alugado ou investido, uma parte age como o credor e o outro como o mutuário. O credor é o proprietário do capital e, como prêmio, o mutuário paga juros ao credor para o uso do capital do credor. Por exemplo, quando o dinheiro é depositado em uma conta poupança, o depositante é o credor e o banco é o mutuário. Este fenômeno gera uma dinâmica, este é o tema principal desta dissertação. Consideramos alguns modelos clássicos sob o ponto de vista das equações de diferenças, com ênfase no estudo da existência de um equilíbrio bem como as condições especiais para a sua estabilidade de soluções. / When capital is rented or invested, one part acts as the lender and the other one as the borrower. The lender is the owner of the capital, and, as prize, the borrower pays interest to the lender for the use of the lenders capital. For example, when money is deposited in a savings account, the depositor is the lender and the bank is the borrower. This phenomenon generates a dynamic, this is the main theme of this dissertation. We consider some classical models from the point of view of the difference equations, with emphasis on the study of the existence of an equilibrium as well the special conditions for its stability of solutions.

Quando um capital é alugado ou investido, uma parte age como o credor e o outro como o mutuário. O credor é o proprietário do capital e, como prêmio, o mutuário paga juros ao credor para o uso do capital do credor. Por exemplo, quando o dinheiro é depositado em uma conta poupança, o depositante é o credor e o banco é o mutuário. Este fenômeno gera uma dinâmica, este é o tema principal desta dissertação. Consideramos alguns modelos clássicos sob o ponto de vista das equações de diferenças, com ênfase no estudo da existência de um equilíbrio bem como as condições especiais para a sua estabilidade de soluções. / When capital is rented or invested, one part acts as the lender and the other one as the borrower. The lender is the owner of the capital, and, as prize, the borrower pays interest to the lender for the use of the lenders capital. For example, when money is deposited in a savings account, the depositor is the lender and the bank is the borrower. This phenomenon generates a dynamic, this is the main theme of this dissertation. We consider some classical models from the point of view of the difference equations, with emphasis on the study of the existence of an equilibrium as well the special conditions for its stability of solutions.

The diploma thesis deals with nonlinear dynamical systems with emphasis on typical phenomena like bifurcation or chaotic behavior. The basic theoretical knowledge is applied to analysis of selected (chaotic) models, namely, Lorenz, Rössler and Chen system. The practical part of the work is then focused on a numerical simulation to confirm the correctness of the theoretical results. In particular, an algorithm for calculating the largest Lyapunov exponent is created (under the MATLAB environment). It represents the main tool for indicating chaos in a system.

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