Figure 9
From: Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications
Real world evidence of fractals and control indicators. (a) Sample autocorrelation function (ACF) ρn of four heartbeat series from a young man slowly decay as the lag (n) increases (the dependence rises) as the absolute slope indicator |η1| and the amplitude indicator η2 increase. (b) Decay of the sample ACF of the heartbeats from elderly persons (the top three lines) reduces (the dependence increases) as the values of |η1| and η2 increase, while the fluctuating decay of the sample ACF of the heartbeats (bottom) with the smallest |η1| and the second largest η2 in this group is slow. (c) Trace of the sample ACF of the U.S./U.K. foreign exchange rate for the period 1971–1980 decays more slowly (the higher dependence level) than that for the period 1981–2016; the former has a larger |η1| and η2 than the latter. (d) Decay of the sample ACF of the U.S. industrial production index for the period 1927–1971 decays more slowly (the higher dependence level) than that for the period 1972–2016, the former has larger |η1| than that of the latter and both have the same η2. (e) Fluctuating decay of the sample ACF, the yearly mean total sunspot number for the period 1858–2016 is much slower (the higher dependence level) than that for the period 1700–1857; the η2 for the latter is much larger than that of the former and the η1 is slightly greater than zero. (f) For the period 1749–2016, the trace of the sample ACF, which is the yearly mean total sunspot number, exhibits persistent oscillations, whereas the trace of the sample ACF, which is the monthly mean total sunspot number, rapidly declines, and the η2 in the yearly data is much larger than that in the monthly data and the η1 is slightly greater than zero. (g) Plotting the standard deviation similarity ratios sd(i,im) versus size (i) results in a roughly horizontal line when m = 2 and fluctuations occur along the horizontal lines when m = 3, 4, 5, 6 because the sample length is not long enough to exhibit the self-similarity of the system. (h) Average of the lag one similarity ratios (r1m) of r(i,im) over size (i) follows a power function with power −1.02.
