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.

Color measurement using a smartphone applied to Madeira wines

Computer vision is a field that uses techniques to acquire, process, analyze and understand images from the real world in order to produce numeric or symbolic information in the form of decisions [1]. This project aims to use computer vision to prepare an app to analyze a Madeira Wine and characterize it (identify its variety) by its color. Dry or sweet wines, young or old wines have a specific color. It uses techniques to compare histograms in order to analyze the images taken from a test sample inside a special container designed for this purpose. The color analysis from a wine sample using an image captured by a smartphone can be difficult. Many factors affect the captured image such as, light conditions, the background of the sample container due to the many positions the photo can be taken (different to capture facing a white wall or facing the floor for example). Using new technologies such as 3D printing it was possible to create a prototype that aims to control the effect of those external factors on the captured image. The results for this experiment are good indicators for future works. Although it’s necessary to do more tests, the first tests had a success rate of 80% to 90% of correct results. This report documents the development of this project and all the techniques and steps required to execute the tests.