Figure 6 | Scientific Reports

Figure 6

From: Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications

Figure 6

Determining the smallest observation scale of fractals. (a) Plot of the absolute slope indicator (|η1(Xj)|) of the aggregation series (Xj) draws a declining line as size (j) increases in the stable fixed-point region (γ = 0.0264) and the stable period-cycle region (γ = 1.25). (b) Amplitude indicator (η2(Xj)) of the aggregation series (Xj) rapidly decays as size (j) increases in the stable fixed-point region (γ = 0.0264) and the stable period-cycle region (γ = 1.25). (c) Autocorrelation function (ACF) ρn of the aggregation series versus the lag n changes from rapid decay to almost unchanging in j = 5, 10, 15, 23, 30, 40, 50, 100, thus showing long-range dependence. (d) Similarity ratio (sd(i,2i)) of the aggregation series plotted versus size (i) tends to a horizontal line as size (j) increases from j = 5 to j = 100, thus showing self-similarity. (e,g) Plot of the sample ACF of the aggregation series versus the lag results in a horizontal line for even number (j) and period oscillations odd number (j), thus showing long-range dependence, although the dependence is unrelated to the aggregation size (j) or increased |η1(Xj)|. (f,h) Similarity ratio (sd(i,2i)) of the aggregation series versus size (i) generates damped oscillations, which are unrelated to size (j) or decreased η2(Xj).

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