On multifractality and fractional derivatives

Abstract

It is shown phenomenologically that the fractional derivative $\xi =D^{\alpha }u$ of order $\alpha$ of a multifractal function has a power-law tail $\propto |\xi {|}^{-p_{\star }}$ in its cumulative probability, for a suitable range of $\alpha$'s. The exponent is determined by the condition ${\zeta }_{p_{\star }}=\alpha p_{\star }$, where ${\zeta }_p$ is the exponent of the structure function of order $p$. A detailed study is made for the case of random multiplicative processes (Benzi {\it et al.} 1993 Physica D {\bf 65}: 352) which are amenable to both theory and numerical simulations. Large deviations theory provides a concrete criterion, which involves the departure from straightness of the ${\zeta }_p$ graph, for the presence of power-law tails when there is only a limited range over which the data possess scaling properties (e.g. because of the presence of a viscous cutoff). The method is also applied to wind tunnel data and financial data.

LaTeX 11 pages, 11 figures, published version

References in corpus (2)

• Linear stochastic dynamics with nonlinear fractal properties
• Fine structure and complex exponents in power law distributions from random maps

Cited by in corpus (1)

• Multifractality of the Feigenbaum attractor and fractional derivatives