On multifractality and fractional derivatives

Details On multifractality and fractional derivatives On multifractality and fractional derivatives

2001-07-25

arxiv.org/abs/nlin/0107057v3

J. Stat. Phys. 108, 1181-1202 (2002)

Nonlinear Sciences

Chaotic Dynamics

LaTeX 11 pages, 11 figures, published version

Scientific paper

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.

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