Nonautonomous difference equations with applications

This work is divided in two parts. In the first part we develop the theory of discrete nonautonomous dynamical systems. In particular, we investigate skew-product dynamical system, periodicity, stability, center manifold, and bifurcation. In the second part we present some concrete models that are used in ecology/biology and economics. In addition to developing the mathematical theory of these models, we use simulations to construct graphs that illustrate and describe the dynamics of the models. One of the main contributions of this dissertation is the study of the stability of some concrete nonlinear maps using the center manifold theory. Moreover, the second contribution is the study of bifurcation, and in particular the construction of bifurcation diagrams in the parameter space of the autonomous Ricker competition model. Since the dynamics of the Ricker competition model is similar to the logistic competition model, we believe that there exists a certain class of two-dimensional maps with which we can generalize our results. Finally, using the Brouwer’s fixed point theorem and the construction of a compact invariant and convex subset of the space, we present a proof of the existence of a positive periodic solution of the nonautonomous Ricker competition model.

An essay on economic growth: classical determinants, convergence and financial development

This dissertation surveys the literature on economic growth. I review a substantial number of articles published by some of the most renowned researchers engaged in the study of economic growth. The literature is so vast that before undertaking new studies it is very important to know what has been done in the field. The dissertation has six chapters. In Chapter 1, I introduce the reader to the topic of economic growth. In Chapter 2, I present the Solow model and other contributions to the exogenous growth theory proposed in the literature. I also briefly discuss the endogenous approach to growth. In Chapter 3, I summarize the variety of econometric problems that affect the cross-country regressions. The factors that contribute to economic growth are highlighted and the validity of the empirical results is discussed. In Chapter 4, the existence of convergence, whether conditional or not, is analyzed. The literature using both cross-sectional and panel data is reviewed. An analysis on the topic of convergence using a quantile-regression framework is also provided. In Chapter 5, the controversial relationship between financial development and economic growth is analyzed. Particularly, I discuss the arguments in favour and against the Schumpeterian view that considers financial development as an important determinant of innovation and economic growth. Chapter 6 concludes the dissertation. Summing up, the literature appears to be not fully conclusive about the main determinants of economic growth, the existence of convergence and the impact of finance on growth.