Sporadic randomness: The transition from the stationary to the nonstationary condition

We address the study of sporadic randomness by means of the Manneville map. We point out that the Manneville map is the generator of fluctuations yielding the Lévy processes, and that these processes are currently regarded by some authors as statistical manifestations of a nonextensive form of thermodynamics. For this reason we study the sensitivity to initial conditions with the help of a nonextensive form of the Lyapunov coefficient. The purpose of this research is twofold. The former is to assess whether a finite diffusion coefficient might emerge from the nonextensive approach. This property, at first sight, seems to be plausible in the nonstationary case, where conventional Kolmogorov-Sinai analysis predicts a vanishing Lyapunov coefficient. The latter purpose is to confirm or reject conjectures about the nonextensive nature of Lévy processes. We find that the adoption of a nonextensive approach does not serve any predictive purpose: It does not even signal a transition from a stationary to a nonstationary regime. These conclusions are reached by means of both numerical and analytical calculations that shed light on why the Lévy processes do not imply any need to depart from the adoption of traditional complexity measures.