Crossover in Scaling: Asymptotic Hyperbolic Behavior and Computation of Crossover Point

Abstract:

Scaling in many systems is often represented by a global Hurst exponent from the data. However, many cases show a crossover in scaling thus yielding two sub ranges of time windows with different scaling exponents. The crossover point is an important feature of the system but its computation is often complicated. A scale invariance in the two regimes with asymptotic behavior before and after the crossover suggests a hyperbolic fit for the fluctuation functions. Consequently a hyperbolic regression scheme is used to compute the crossover point (the center of the hyperbola) and the respective Hurst exponents (slopes of asymptotes). In the case of magnetospheric dynamics this method yields a measure of the time scales in the AL data before and after the crossover which are in agreement with previous studies that used power spectral density (PSD) or structure function (SF) to characterize the scaling break/crossover. Furthermore, a comparison of the crossover phenomenon in AL data to those from several theoretical models yields a Ornstein-Uhlenbeck Langevin (OUL) model which is then used for robust estimates of the crossover time by calculating its variance using multi-trajectory ensemble approach. The fluctuation analysis of AL data detrended using expected values from OUL model is shown to have no crossovers, unlike detrending by persistence or DFA of AL data, thus providing a new understanding of crossover phenomenon in fluctuation analysis.