The importance of the privilege for appearance of inverse-power solutions in Ito equations

Three-level description of the domino cellular automaton

“…7. Clouds of points which illustrate functions a(ρ) and b(ρ) are found by using the histogram method of constructing the Ito equation from time series ( [10], [11], [12]). The fit is satisfactory, particularly in the case of function a(ρ).…”

The construction of an Ito model for geoelectrical signals

“…On the other hand, to thi s aim, there were derived Ito equations for cases of toy 8 models, in which the whole microscopic evolution is controlled [5,26]. …”

“…Therefore, in the paper we assume that the time series under 9 investigation fulfills these properties; the verification will be conducted by the 10 comparison of the input time series with that generated by the reconstructed Ito embedding method [25]) because it is both stochastic and nonlinear. Moreover, Ito 22 models give various types of distribution functions: from short-tail to long-tail 23 distributions (see [26]). It should be underlined that unlike neural network models, 1 which onl y reproduce data, the Ito model has the clear ph ysical interpretation.…”

“…Both terms can be the result of the averaged interactive dynamics on the 10 microscopic level. These effects were investi gated for cases of toy models in [26] and 11 [5]. 12 In order to appl y the Ito-based equations the signal was firstl y reduced to a zero-21 mean stationary signal.…”

“…3 Stationary marginal distribution functions p(y) are shown in Fig.5a (in lin-log 16 scales) and in Fi g. 5b (in log-log scales). Two different behaviors can be viewed for 17 the whole series (red symbols): exponential for y < 0.5 and inverse-power for 1.8 > y 18 > 0.5 (see Table 1 in [26]). The excess of large fluctuations is evidenced by a hump 19 in Fig.…”