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In-Depth Information
0.6
0.6
a1
a2
0.4
0.4
0.2
0.2
0
0
0
500
1000
1500
2000
0
0.2
0.4
0.6
t
c(t)
0.6
0.6
b2
0.4
0.4
b1
0.2
0.2
0
0
0
500
1000
1500
2000
0
0.2
0.4
0.6
t
c(t)
0.6
0.6
c1
c2
0.4
0.4
0.2
0.2
0
0
0
500
1000
1500
2000
0
0.2
0.4
0.6
t
c(t)
Fig.9.
The time series of the concentration and the iterative map for rule
φ
c
, for three
different lattice sizes: (a) N=3000, (b) N=5000 y (c) N=10000. The transient time of
5
· N
Rules
φ
a
,φ
b
and
φ
d
shows a P3 behavior that can be seen as three clouds
of points in the iterative map and in the time series of the concentration as
three branches that interact and mix among them. The attractor of rule
φ
c
(
Figure 8-c ) is more complex and consists of a fuzzy cycle. The time series of
the concentration shows three branches that can be distinguished clearly. The
triangular shape of the attractor is maintained as the lattice size increases up
to
L
=10
4
and the time series of the concentration shows a quasiperiod-3 or a
P3 collective behavior as can be seen in Figure 9. However the QP3 behavior
seems to be metastable because after a long time the final state is a P3. For
example Figure 10 shows the time series of the concentration of
φ
c
during 10
4
time steps where it can be observed a sharp transition from a QP3 to a P3
collective behavior.
Under the fitness function
F
(
φ
)in
d
= 1 the evolutionary process selects
rules that starting from a random initial condition synchronize the whole sys-
tem to a three-state cycle. To see how such a synchronization is obtained we
