Study of multifractal characteristics and correlation behavior in complex systems

Abstract

All natural systems would not be so complex, nonlinear without presence of fluctuations or random processes which makes the observations interesting and worthwhile. Using statistical mechanics and its macroscopic counterpart thermodynamics,which forms the mathematical theory; one can understand the magnitude and time scale of these natural or spontaneous fluctuations to analyze system behavior. It is also worth specifying that all natural complex systems could be represented in form of a time series. Non-stationary time series have been investigated through several approaches. In particular, the characterization of fluctuations and their scaling behavior have been the focus for many studies, since they reveal the nature of the dynamics. The availability of large amount of data in various fields such as finance, Physical, Biological, and Physiological process has attracted considerable attention for such study. These non-stationary time series are well known to exhibit multiple characteristics depending on the scale of observation through its self-similar features. Analysis of multifractal behavior requires isolation of local fluctuations at both lower and higher scales which motivates to study their statistical characteristics more carefully. Apart from the study of their non-Gaussian properties at different scales, this analysis will also help in throwing more light on the non-statistical nature of the fluctuations. The present thesis work is primarily devoted to the study of multifractal characteristics, correlation and cross-correlation behavior in a variety of complex systems and their analysis through methods developed using statistical mechanics principle. We have made use of the recently developed methods such as wavelet based fluctuation analysis (WBFA) and multifractal detrended cross-correlation analysis (MF-X-DFA) for our study.