Information Technology Reference
In-Depth Information
10
0
10
−1
10
−2
10
−3
10
100
1000
Log (L)
1
2
Fig.4.
Log-log plot of the Fitness function for rule
φ
a
, versus lattice size
L
=
N
.
generation 637 rule
φ
a
is found, its iterative map shows a triangular object
which is less dense than the previous one which results in a higher value of the
fitness function
F
(
φ
).
According to [2]
d
= 2 is a critical space dimension above which coherent
(quasiperiodic)oscillations can be observed. But surprisingly the GA has discov-
ered rules like
φ
a
and
φ
b
which shows a QP3. It might be that this quasiperiodic
behavior is only metastable or only seen on small lattices, so it is interesting to
see how the attractor scales with the lattice size, but a more complete study of
the existence of this behavior in the thermodynamic limit is out of the scope of
this work.
The return plot and the time series of the concentration for rule
φ
a
are
plotted in Figure 3 for three different lattices sizes, where it can be seen a fuzzy
limit cycle that looks like previous studied CAs in
d
= 3 with QP3 behavior.
The amplitude of the oscillations and the fitness function shrink as lattice size
increases, see Figure 4, however the attractor is found to become thinner and
sharper the larger the system. The largest runs were made on lattices of 10
6
cells
over 25000 time steps, the behavior is observed in Figure 5 where starting from
a random initial condition, the concentration oscillates among three different
branches that can be clearly distinguished initially, but as the system evolves
the three branches mix irregularly and the size of the attractor diminishes.
The stability of the attractor against external noise is shown in Figure 6 that
plots the fitness function
F
(
φ
) versus the amount of noise
η
in a log-log plot.
Noise is introduced in the system by flipping a fraction
η
of randomly selected
spins before each time the rule is applied. In d=2 the attractor is very sensitive
to very small amounts of noise which produces the shrinking of the attractor
from the triangular object initially obtained to a noisy cloud of points.
