Chemistry Reference
In-Depth Information
D
T.LC
.b/
Then the coordinates of the cusp point of the curve LC
.b/
1
/ are given by
.a.
C
1/
1/.a
C
3.1
a//
4a
K
D
LC
.b/
\
D
.k;k/ with k
D
f.k
1
/
D
;
where the one-dimensional map f.x/
D
.1
C
a.
1//x
ax
2
is the restriction
of the map T to the diagonal. It now becomes obvious that at a.
C
1/
D
1 the cusp
point K enters the strategy set
and that after this bifurcation there are points in
the strategy set that have a higher number of preimages.
To elaborate a little further on the workings of the mechanism which transforms
the basins from simply connected sets to disconnected sets, consider the origin O
D
.0;0/.If0<a<1=.
C
1/,thenO
2
Z
2
and there are just two rank-1 preimages
of O. Both belong to the diagonal , with one preimage is O itself (since O is a
fixed point), and the other preimage is
S
1
C
a.
1/
a
:
;
1
C
a.
1/
a
O
.1/
D
1
This can be easily seen by using the restriction of the map T to the diagonal. The
situation is depicted in Fig. 3.14a, where for the sake of mathematical exposition we
show the whole extent of the basins of attraction of the locally stable equilibria E
1
and E
2
(and not just the region belonging to the strategy space
S
as in Fig. 3.12).
Observe that as long as the cusp point is outside the basins of attraction, the basins
are simple and connected sets. If however a>1=.
C
1/, then the origin O
2
Z
4
since the cusp point has entered
S
, and two more rank-1 preimages of O exist. These
2.3
1.4
x
2
x
2
(1)
O
−1
(1)
O
−1
LC
(
a
)
Z
0
Δ
LC
(
a
)
Δ
(3)
O
−1
Z
0
E
2
Z
2
K
E
2
E
1
Z
2
E
S
LC
(
b
)
Z
4
E
1
O
O
(2)
O
−1
K
LC
(
b
)
Z
4
2.3
x
1
x
1
1.4
(a)
(b)
Fig. 3.14
Linear inverse demand function and cost externalities. The case of duopoly with identi-
cal speeds of adjustment - basins of attraction of the two equilibria E
1
and E
2
.(
a
)Here D 3:4,
a D 0:2 < 1=.C1/, and the basins of attraction are simple and connected sets. (
b
)Here D 3:4,
a D 0:5 < 1=. 1/, and the basins of attraction become disconnected
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