Herramientas para la no linealidad

Abstract

Abstract The interest of economists in chaos theory started in the 1980s. One of the reasons is the apparently random behavior of simple nonlinear dynamical systems with few degrees of freedom. The knowledge of this particular behavior from time series obtained from these dynamical systems has motivated us to study some of the tools used in the analysis of nonlinear dynamical systems. The thesis is divided into five chapters. The first is the introduction which includes an overview of discrete dynamical systems and outlines the contents of the following chapters. The second chapter is devoted to the embedding theory, where a Takens theorem is one of the theoretical justifications that allows us to construct an orbit of an unknown deterministic dynamical systems from a time series of measurements obtained from this system. The Whitney theorem is fundamental in the proof of the Takens theorem. In this chapter we provide two new proofs of the Whitney theorem and an extension of the Takens theorem (results published in Acta Mathematica Hungarica, 88 (4)). In chapter 3 we examine in depth the implications of the sign of the Lyapunov exponent of an orbit of a one-dimensional dynamical system and we prove a result that complements another given by Koçak and Palmer (2010) as well as new examples of one-dimensional and two-dimensional dynamical systems having the paradoxical behavior of dynamical systems shown by Demir and Koçak (2001) (results published in the International Journal of Bifurcation and Chaos, 23). The chapter 4 is devoted to Recurrence Plots. These graphics are a tool for visualizing the correlation integral of a vectorial time series. In recent decades recurrence plots and the measures defined on them have been used to analyze economic time series, where the main objective is to approach the shift points of the dynamic behavior of the process generator of data. In this chapter we use the recurrence quantitative analysis with a non-stationary time series obtained from Liu's dynamical system to approach Liu bifurcation points (results in Chaos, Solitons and Fractals, 36 (3)). A nonparametric test of independence is developed in Chapter 5. This test is the result of applying the main theorem that we show to the asymptotic distribution of a linear transformation of the statistic that estimates the symbolic correlation integral of an embedded independent and identically distributed series. The concept of symbolic correlation integral is also defined for any compact set of a m-dimensional vectorial space, and is based on its symbolization, that is, assign to each element a permutation of the set {1,2, ..., m}, which represents the order of its components. For this test the null hypothesis is that the time series is independent and identically distributed against any type of dependence in the time series. We studied the size and power of this test and the results show good statistical properties. The chapter ends defining symbolic recurrence plots and colored symbolic recurrence plots and we have calculate measures on them. Each chapter ends with a conclusions section on the results proven and questions that remain open, which we will seek to close in futur works.