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Numerical investigations of discrete scale invariance in fractals and multifractal measures

W. -X. Zhou·D. Sornette
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Abstract

Fractals and multifractals and their associated scaling laws provide a quantification of the complexity of a variety of scale invariant complex systems. Here, we focus on lattice multifractals which exhibit complex exponents associated with observable log-periodicity. We perform detailed numerical analyses of lattice multifractals and explain the origin of three different scaling regions found in the moments. A novel numerical approach is proposed to extract the log-frequencies. In the non-lattice case, there is no visible log-periodicity, {\em{i.e.}}, no preferred scaling ratio since the set of complex exponents spread irregularly within the complex plane. A non-lattice multifractal can be approximated by a sequence of lattice multifractals so that the sets of complex exponents of the lattice sequence converge to the set of complex exponents of the non-lattice one. An algorithm for the construction of the lattice sequence is proposed explicitly.

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W. -X. Zhou, D. Sornette. 2007-04-17. Numerical investigations of discrete scale invariance in fractals and multifractal measures. https://doi.org/10.1016/j.physa.2009.03.023

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