Figure 5
(a) Lyapunov dimension
versus network size
for five systems with periodic boundary conditions: Bär-Eiswirth model with
,
,
, and
; Gray-Scott model
with
(◻),
, and
(○); and Wacker-Schöll model with
,
,
,
, and
(△). The Lyapunov dimension for the Bär-Eiswirth model
and the Gray-Scott model
have very similar values, making them hard to distinguish in the graph, although their Lyapunov spectra are different. The full lines correspond to a least-squares linear fit of the corresponding data points. The error for each data point, not plotted in the figure, was estimated by computing the maximum difference between the final value of
and any intermediate value from the second half of the simulation; errors were smaller than 2.5% of the data point values and typically less than 0.3% for Gray-Scott and Bär-Eiswirth systems. (b) Deviation of
values from the linear fit in (a) for the Gray-Scott system with
. For this system
was calculated for different initial conditions for each network size
to estimate the convergence behavior. These data points were plotted in both figures, (a) and (b), but they are too close to be distinguished in (a). The horizontal line represents the
intercept of the linear fit, which was not subtracted when plotting deviation from the linear trend. (c) Sum of positive Lyapunov exponents
versus network size
for the same systems as in (a). The values for the Wacker-Schöll system have been multiplied by 20 to improve clarity.
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