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T
1
v
1
v
2
v
3
v
1
v
2
v
3
1
2
3
T
2
v
2
0
I>T
I
0
1
−1
0
1
v
1
v
1
v
2
v
3
v
1
v
2
v
3
Fig. 2.
The
v
1
,v
2
system for
I>T
I
. The iterated maps operating in each region are
depicted near their corners.
dynamical laws are different depending on the values of the thresholds
T
1
,T
2
and
T
I
. The thresholds
T
1
,T
2
divide the square [
1
,
1] into four rectangular regions.
The equation of the maps changes as the coordinates of the pair
v
1
,v
2
lies on
one of these four regions. This is illustrated in figure 2.
The pair
v
1
(
t
)
,v
2
(
t
) tends to the fixed points corresponding to the maps as-
sociated to the region where the point is at a particular moment. Otherwise, the
orbit can leave the region on its way to the fixed point, leading to a different
dynamical law. The fixed points for each region are depicted in figure 3. In 1
the values of the xed points corresponding to each region are shown. Note that
W
21
−
0,
W
I
1
0and
W
I
1
+
W
21
1
−a
1
≤
W
I
1
1
−a
1
≤
1
−a
1
≥
, also from the normalization condition
1
−a
1
we obtain
W
12
1
−a
2
=1.
Examining figure 3 and considering all qualitatively different positions of the
thresholds one obtains a complete classification of all possible dynamical be-
haviours of the system, all the possible situations are listed in table 2.
Table 1.
Value of the fixed points
Region
0
1
2
3
(0
,
1
,
W
13
1
−
a
3
(
W
21
1
−
a
1
,
0
,
W
23
(
W
21
1
−
a
1
,
1
,
W
13
+
W
23
I<T
I
(0
,
0
,
0)
)
)
)
1
−
a
3
1
−
a
3
I>T
I
♦
(
W
I
1
1
−
a
1
,
0
,
0) (
W
I
1
1
−
a
1
,
1
,
W
13
)(
W
I
1
+
W
21
1
−
a
1
,
0
,
W
23
1
−
a
3
)(
W
I
1
+
W
21
1
−
a
1
,
1
,
W
13
+
W
23
1
−
a
3
)
1
−
a
3
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