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The persistent, anti-persistent and the Brownian: A toy model for investigating the connection between Hurst exponent and the emergence of oscillatory behaviour

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Abstract

The analogy between self-similar time series with the given Hurst exponent H and Markovian, Gaussian stochastic processes with multiplicative noise and entropic index q (Borland, PRE 57, 6634 (1998)) allows us to explain the empirical results reported in Pavithran et al, EPL 129, 24004 (2020) and Pavithran et al, Sci. Rep. 10, 1 (2020) with the help of the properties of the non-extensive entropy \( S_{q} \) of index q: a dominant oscillating mode arises as H goes to zero in many different systems and its amplitude is proportional to \({1}/{H^2} \). Thus, a decrease of H acts as the precursor of large oscillations of the state variable, which corresponds to catastrophic events in many problems of practical interest. In contrast, if H goes to 1 then the time series is strongly intermittent, fluctuations of the state variable follow a power law, whose exponent depends on H, and exceedingly large events are basically unpredictable. These predictions agree with observations in problems of aeroacoustics, aeroelasticity, electric engineering, hydrology, laser physics, meteorology, plasma physics, plasticity, polemology, seismology and thermoacoustics.

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Acknowledgements

Useful discussions with Prof. R I Sujith, Department of Aerospace Engineering, IIT Madras, Chennai and his warm encouragement are gratefully acknowledged.

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Di Vita, A. The persistent, anti-persistent and the Brownian: A toy model for investigating the connection between Hurst exponent and the emergence of oscillatory behaviour. Pramana - J Phys 97, 128 (2023). https://doi.org/10.1007/s12043-023-02606-0

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