The persistent, the anti-persistent and the Brownian: when does the Hurst exponent warn us of impending catastrophes?
Abstract
The analogy between self-similar time series with given Hurst exponent H and Markovian, Gaussian stochastic processes with multiplicative noise and entropic index q (Borland, PRE 57, 6, 6634-6642, 1998) allows us to explain the empirical results reported in (Pavithran et al., EPL, 129 2020 24004) and (Pavithran et al. Sci. Reports 10.1 (2020) 1-8) with the help of the properties of the nonextensive entropy Sq of index q: a dominant oscillating mode arises as H goes to zero in many different systems and its amplitude is proportional to 1/ H2 . Thus, a decrease of H acts as 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 event 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.