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Correlation between the Hurst exponent and the maximal Lyapunov exponent: Examining some low-dimensional conservative maps

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  • Tarnopolski, Mariusz

Abstract

The Chirikov standard map and the 2D Froeschlé map are investigated. A few thousand values of the Hurst exponent (HE) and the maximal Lyapunov exponent (mLE) are plotted in a mixed space of the nonlinear parameter versus the initial condition. Both characteristic exponents reveal remarkably similar structures in this space. A tight correlation between the HEs and mLEs is found, with the Spearman rank ρ=0.83 and ρ=0.75 for the Chirikov and 2D Froeschlé maps, respectively. Based on this relation, a machine learning (ML) procedure, using the nearest neighbor algorithm, is performed to reproduce the HE distribution based on the mLE distribution alone. A few thousand HE and mLE values from the mixed spaces were used for training, and then using 2−2.4×105 mLEs, the HEs were retrieved. The ML procedure allowed to reproduce the structure of the mixed spaces in great detail.

Suggested Citation

  • Tarnopolski, Mariusz, 2018. "Correlation between the Hurst exponent and the maximal Lyapunov exponent: Examining some low-dimensional conservative maps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 834-844.
    • Handle: RePEc:eee:phsmap:v:490:y:2018:i:c:p:834-844
    DOI: 10.1016/j.physa.2017.08.159
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    References listed on IDEAS

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    1. Carbone, A. & Castelli, G. & Stanley, H.E., 2004. "Time-dependent Hurst exponent in financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 267-271.
    2. Arianos, Sergio & Carbone, Anna, 2007. "Detrending moving average algorithm: A closed-form approximation of the scaling law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 9-15.
    3. Tarnopolski, Mariusz, 2016. "On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 662-673.
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