• Dynamical systems
• time series

In this thesis we present a method to analyze time series using the theory of nonlinear dynamical systems. Our main assumption throughout this work is that time series samples are generated by observations of a a nonlinear ergodic dynamical system. Given a time series, this method allows us to understand if the underlying dynamics is regular (periodic or quasi-periodic) or chaotic, computing a quantity K that takes the value 0 in the first case and 1 in the second.
First we focus on the hypotheses that a dynamical system and an observable have to satisfy to make the test work. In the case of periodic or quasi-periodic dynamics we give a proof of the fact that K=0, assuming some smoothness assumptions on the observable. Conversely, for chaotic dynamics, we show that the efficacy of the test is related to the speed of correlations decay of the system for a suitable class of observables. Indeed, if the correlations decay exponentially, the test identifies the dynamics as chaotic. To exemplify this situation, we consider smooth expanding maps. Furthermore, we explore the case where the correlations decay is subexponential, considering first when it is summable and then when it is not summable. The summability assumption alone is not sufficient for the successful application of the method. However, if further conditions on the dynamics are imposed, the test gives a value K equal to 1. Another condition guaranting the effectiveness of the method is slightly stronger than summability. Finally in the case where the correlations decay is not summable, a modified version of the test can be reformulated in terms of Cesàro averages.
We also apply the method to the Liverani-Saussol-Vaienti (LSV) maps, a family of intermittency maps depending on a parameter α. When α = 0 the map reduces to the doubling map with exponential decay of correlations, while for α in (0,1) we show that the decay of correlations is polynomial. Starting from this result, we determine the values of α for which the test can be formally proven to be effective.
At the end of this thesis we describe an implementation of the method, and we test it in numerical simulations on synthetic time series generated by LSV maps (including values of α not covered by the theoretical results) and by Logistic maps, in regular and chaotic regimes.
Finally, we consider series obtained by the maps with random perturbations. In this case, in our simulations, we utilize a slightly modified version of the test to extract information on the underlying deterministic dynamics. Our ultimate goal is the analysis of real physiological series that are assumed to be generated by an ergodic and randomly perturbed dynamical system.