Difference equations arise in many fields of mathematics, both as discrete analogs of continuous behavior (analysis, numerical approximations) and as independent models for discrete behavior (population dynamics, economics, biology, ecology, etc.). In recent years, many models - especially in mathematical biology - are based on higher order nonlinear difference equations. As a result, there has been much focus on the existence of periodic solutions of certain classes of these equations and the asymptotic behavior of these periodic solutions. In this dissertation, we study the existence and global attractivity of both periodic and quasiperiodic solutions of two different higher order nonlinear difference equations. Both equations arise in biological applications.

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.

[pt] Consideramos a equação de Boltzmann espacialmente não homogênea para esferas duras inelásticas, com coeficiente de restituição constante alfa pertence (0, 1), sob a termalização induzida por um meio hospedeiro com uma distribuição Maxwelliana fixa e fixando e pertence (0, 1) qualquer. Quando o coeficiente de restituição alfa é próximo de 1, comprovamos a existência de soluções globais considerando o regime próximo ao equilíbrio. Também estudamos o comportamento de longo prazo dessas soluções e comprovamos uma convergência para o equilíbrio com uma taxa exponencial. / [en] We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient alpha element of (0, 1), under the thermalization induced by a host medium with a fixed Maxwellian distribution and any fixed e element of (0, 1). When the restitution coefficient alpha is close to 1 we prove existence of global solutions considering the close-to-equilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.