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Numerical investigations of discrete scale invariance in fractals and multifractal measures

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  • Zhou, Wei-Xing
  • Sornette, Didier

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, i.e., no preferred scaling ratio since the set of complex exponents spreads 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.

Suggested Citation

  • Zhou, Wei-Xing & Sornette, Didier, 2009. "Numerical investigations of discrete scale invariance in fractals and multifractal measures," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(13), pages 2623-2639.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:13:p:2623-2639
    DOI: 10.1016/j.physa.2009.03.023
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    References listed on IDEAS

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    1. Wei-Xing Zhou & Didier Sornette & Vladilen Pisarenko, 2003. "New Evidence Of Discrete Scale Invariance In The Energy Dissipation Of Three-Dimensional Turbulence: Correlation Approach And Direct Spectral Detection," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 14(04), pages 459-470.
    2. Zaslavsky, George M, 2000. "Multifractional kinetics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 288(1), pages 431-443.
    3. A. Johansen & H. Saleur & D. Sornette, 2000. "New evidence of earthquake precursory phenomena in the 17 January 1995 Kobe earthquake, Japan," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 15(3), pages 551-555, June.
    4. Mandelbrot, Benoit B., 1990. "New “anomalous” multiplicative multifractals: Left sided ƒ(α) and the modelling of DLA," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 168(1), pages 95-111.
    5. Wei-Xing Zhou & Didier Sornette, 2002. "Statistical Significance Of Periodicity And Log-Periodicity With Heavy-Tailed Correlated Noise," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 13(02), pages 137-169.
    6. A. Johansen & D. Sornette, 1998. "Evidence of Discrete Scale Invariance in DLA and Time-to-Failure by Canonical Averaging," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 9(03), pages 433-447.
    7. Zhou, Wei-Xing & Yu, Zun-Hong, 2001. "On the properties of random multiplicative measures with the multipliers exponentially distributed," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 294(3), pages 273-282.
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    Cited by:

    1. Jiang, Zhi-Qiang & Zhou, Wei-Xing & Sornette, Didier & Woodard, Ryan & Bastiaensen, Ken & Cauwels, Peter, 2010. "Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 Chinese stock market bubbles," Journal of Economic Behavior & Organization, Elsevier, vol. 74(3), pages 149-162, June.
    2. Jiang, Zhi-Qiang & Ren, Fei & Gu, Gao-Feng & Tan, Qun-Zhao & Zhou, Wei-Xing, 2010. "Statistical properties of online avatar numbers in a massive multiplayer online role-playing game," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(4), pages 807-814.

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