Istituto di Scienza e Tecnologie dell'Informazione     
Paradisi P., Allegrini P. Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency. In: Chaos Solitons & Fractals, vol. 81 (B) pp. 451 - 462. Special Issue: The emergence of self-organization in complex systems. Elsevier, 2015.
In many complex systems the non-linear cooperative dynamics determine the emergence of self-organized, metastable, structures that are associated with a birth-death process of coop- eration. This is found to be described by a renewal point process, i.e., a sequence of crucial birth-death events corresponding to transitions among states that are faster than the typical long-life time of the metastable states. Metastable states are highly correlated, but the occur- rence of crucial events is typically associated with a fast memory drop, which is the reason for the renewal condition. Consequently, these complex systems display a power-law decay and, thus, a long-range or scale-free behavior, in both time correlations and distribution of inter-event times, i.e., fractal intermittency. The emergence of fractal intermittency is then a signature of complexity. However, the scaling features of complex systems are, in general, affected by the presence of added white or short-term noise. This has been found also for fractal intermittency. In this work, after a brief review on metastability and noise in complex systems, we discuss the emerging paradigm of Temporal Complexity. Then, we propose a model of noisy fractal in- termittency, where noise is interpreted as a renewal Poisson process with event rate r p . We show that the presence of Poisson noise causes the emergence of a normal diffusion scaling in the long-time range of diffusion generated by a telegraph signal driven by noisy fractal intermittency. We analytically derive the scaling law of the long-time normal diffusivity coef- ficient. We find the surprising result that this long-time normal diffusivity depends not only on the Poisson event rate, but also on the parameters of the complex component of the signal: the power exponent μ of the inter-event time distribution, denoted as complexity index, and the time scale T needed to reach the asymptotic power-law behavior marking the emergence of complexity. In particular, in the range μ < 3, we find the counter-intuitive result that normal diffusivity increases as the Poisson rate decreases. Starting from the diffusivity scaling law here derived, we propose a novel scaling analysis of complex signals being able to estimate both the complexity index μ and the Poisson noise rate r_p .
URL: http://www.sciencedirect.com/science/article/pii/S0960077915001952
DOI: 10.1016/j.chaos.2015.07.003
Subject Scaling
Time series analysis
Signal processing
Fractal intermittency
Complex systems
62M10 Time series, auto-correlation, regression, etc.
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
62-07 Data analysis
93E03 Stochastic systems, general
60-XX Probability theory and stochastic processes
60H30 Applications of stochastic analysis (to PDE, etc.)
60K05 Renewal theory

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