Dependent Variables in Broad Band Time Series

The traditional tools of data analysis; correlation functions, Fourier transforms, and linear regression have great difficulty distinguishing deterministic chaos from randomness. In this dissertation, we present the development of a class of new techniques of data analysis based on the ideas of chaotic dynamics and fractal geometry, and then study their application to a variety of mathematical models, as well as an experiment in turbulent fluid flow. Throughout the dissertation, we establish the theme that these methods are sensitive to quite different characteristics of the data than are traditional statistical methods, and moreover are sensitive to deterministic structures which the traditional methods often miss completely. In the first chapter, we develop new methods for the analysis of discrete time series. We show that our statistics are able to indicate dependence among values of a time series at different times even in cases where the data is uncorrelated. In the second chapter, we generalize the methods of chapter I to the case of continuous time series. We demonstrate that even if the system has an underlying simplicity, the best description of the system for a given purpose may not involve those variables that generated the complex behavior in the first place. In chapter III, we use the statistical indicators developed in chapter I to define and study new time scales in series generated by some one-dimensional dynamical maps. In chapter IV, we apply our statistics to the analysis of a Couette-Taylor experiment in turbulent fluid flow. An analysis of this data using time scales as defined in chapter III reveals structure in the flow which time scales based on torque measurements, correlation functions, or the average time between zero crossings fail to show. And an analysis using the continuous time series methods of chapter II reveals that certain characteristics of the flow depend on the velocity, u, separately from their dependence on the velocity through the Reynold's number. This result is surprising and clearly demonstrates the utility of our statistical methods.