Chemistry Reference
In-Depth Information
7
6.5
(2)
O
−1
x
2
x
2
Z
E
2
E
2
(
a
)
LC
−1
Z
2
(
a
)
LC
Z
2
−
1
(3)
O
−
1
E
(
b
)
LC
−
(
b
)
LC
−1
1
LC
(
b
)
LC
(
a
)
LC
(
b
)
Z
4
LC
(
a
)
Z
4
x
1
0
E
1
7.5
(1)
O
−1
x
1
0
E
1
6.5
(a)
(b)
Fig. 1.15
The Cournot duopoly with a gradient type adjustment process and linear
demand/quadratic cost. Illustrating the regions of preimages of different ranks, the sets of points
where the Jacobian vanishes (LC
.a/
1
and LC
.b/
1
) and the critical curves LC
a
and LC
b
.(
a
)The
parameters are the same as in Fig. 1.13. (
b
) The speeds of adjustment are slightly higher, E
becomes unstable and a strange attractor emerges, but the basic structure of the basin remains
the same as in (
a
). Note however that the critical curve LC
.b/
in now quite close to the boundary
of the
white
and
grey
regions
rank-1 preimages, given by O itself (since T
g
.0;0/
D
.0;0/)andO
.i /
1
, i
D
1;2;3
(since T
g
.O
.i /
1
/
D
.0;0/ as well). The regions Z
k
are separated by segments of
critical curves denoted as LC
.a/
and LC
.b/
in Fig. 1.15a.
An intuitive understanding of the importance of critical curves can be obtained
by referring to the folding or unfolding mechanism of a map. The map (1.46) is
noninvertible, which means that distinct points in the action set can be mapped into
the same point by T
g
. This can be geometrically envisioned by imagining a process
which folds the action space onto itself (so that points which are in different loca-
tions are folded onto each other). A result from algebraic geometry tells us that the
folding process can be characterized by a change of sign of the determinant of the
Jacobian of the map: if the sign is positive, then the map is orientation preserving,
whereas it is orientation reversing otherwise.
4
The folding curves where the sign
change occurs is the locus of points where the determinant of the Jacobian of the
map vanishes. Its
image
gives the so-called critical curve, which separates zones
or regions with different numbers of
preimages
(this indicates the importance of
the unfolding action of the map). To sum up, the following numerical procedure
4
Consider a one-dimensional, continuously differentiable map g.y/.Ifg
0
.y/ > 0, then for x<y,
it follows that g.x/< g.y/. If, on the other hand, g
0
.y/ < 0, the orientation is reversed. Obviously,
the change of signs occurs exactly at the point where the derivative vanishes.
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