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4
1.41
h
=0.1
3
0.91
2
0.52
2.4
2.6
2.8
3
3.2
log
n
Fig. 5.4. Estimation of temporal auto-correlations of the state of the system by the detrended
uctuation analysis (DFA) method [Peng et al. (1995)]. We show the log-log plot of the uctuations
F (n) in the state of the system, versus time scale n for the time series shown in Figs. 5.3a-c. In
such a plot a straight line indicates a power-law dependence F (n)/n
. The slope of the lines
yields the scaling exponent . For panel a in Fig. 5.3 the value of the exponents is 0.52, for panel
b is 0.91, and for panel c is 1.41.
Finally, in Fig. 5.4 we show the DFA of the time series of Fig. 5.3. As explained
in Sect. 5.3 a straight line in this plot corresponds to self-similar dynamics. In terms
of the noise correlations we can say in this case that the top set of points (left in
Fig. 5.1, bottom in Fig. 5.3) is very close to Brownian noise ( = 1:41), the bottom
set (right in Fig. 5.1, top in Fig. 5.3) to white noise ( = 0:52) whereas the middle
set (center in Fig. 5.1, middle in Fig. 5.3) corresponds to 1=f noise ( = 0:91).
5.4.4. Complex dynamics from simple models
The previous paragraphs show a way in which the system can be analyzed system-
atically. Actually, what we want to study is how a single characterization of the
system (in our case the exponent ) depends on the details, the topology, the noise
intensity, the Boolean rules, and so. To do so we systematically run simulations
for a xed time, and compute the exponent for a single run with a xed box size.
Afterwards, we perform averages over dierent initial conditions. Thus a given
rule, topology, and noise intensity give rise to and exponent value, and comparison
between dierent eects can be performed.
