Universality classes of fluctuation dynamics in hierarchical complex systems

A unified approach is proposed to describe the statistics of the short-time dynamics of multiscale complex systems. The probability density function of the relevant time series (signal) is represented as a statistical superposition of a large time-scale distribution weighted by the distribution of certain internal variables that characterize the slowly changing background. The dynamics of the background is formulated as a hierarchical stochastic model whose form is derived from simple physical constraints, which in turn restrict the dynamics to only two possible classes. The probability distributions of both the signal and the background have simple representations in terms of Meijer $G$ functions. The two universality classes for the background dynamics manifest themselves in the signal distribution as two types of tails: power law and stretched exponential, respectively. A detailed analysis of empirical data from classical turbulence and financial markets shows excellent agreement with the theory.

Figure 1

(a) Experimental distribution for velocity increments (gray circles) in a turbulent jet flow and model predictions (lines) for $N=1$ and $\beta =3.26$ [yellow (light gray) dashed], $N=2$ and $\beta =5.16$ [brown (gray) dashed], $N=3$ and $\beta =7,47$ [green (gray) dotted], $N=7$ and $\beta =15.5$ (black full); (b) histogram (gray circles) of the variance series $\epsilon \left(t\right)$ and model predictions (lines) with same parameters and color conventions as in (a). Inset shows the compounding [red (dark gray) line] of $\epsilon \left(t\right)$ with a Gaussian and the experimental distribution (black circles) of velocity increments.

Figure 2

(a) Empirical distribution of intraday returns of the Ibovespa index (gray squares) and model predictions (lines) for $N=1$ and $\beta =0.15$ [yellow (light gray) dashed], $N=2$ and $\beta =0.5$ [brown (gray) dashed], $N=3$ and $\beta =1.2$ (black full), $N=4$ and $\beta =1.32$ [green (gray) dotted]; (b) histogram (black dots) of the variance series $\epsilon \left(t\right)$ and model predictions (lines) with same parameters and color conventions as in (a). Inset shows the superposition [red (dark gray) line] of $\epsilon \left(t\right)$ with a Gaussian and the empirical distribution (black squares).