Chemistry Reference
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is followed by other period doublings and, in general, by the well-known period
doubling cascade, that constitutes the typical route to chaotic behavior for smooth
unimodal maps. So, complex dynamics, that include periodic cycles of any period
and chaotic motion, can be obtained if the map is unimodal and the fixed point is
unstable, that is if
2B.N
C
1/
Nc
C
A
<a<
B.N
C
1/
A
:
(5.105)
This range is non-empty provided that A<Nc, that is if the reservation price is
less than the firms' aggregated marginal costs. For example, if we consider the set
of parameters N
D
3, A
D
3, c
D
2, B
D
2, the range given by (5.105) is
16=9
'
1:78<a<8=3
'
2:67. This is confirmed by a numerical computation of
the bifurcation diagram shown in Fig. 5.5b. The asymptotic dynamics are trapped
inside the interval Œm;g.m/,wherem
D
g."
min
/>0is the minimum value of the
map (see Fig. 5.5a). For increasing values of the adjustment coefficient a the mini-
mum value m decreases until it reaches the value m
D
0 (for the set of parameters
used to obtain the bifurcation diagram of Fig. 5.5b, this occurs at a=B
'
1:1858).
This is the
final bifurcation
, after which the generic trajectory involves negative val-
ues. It is worth stressing that the same kind of bifurcation diagram, as the one shown
in Fig. 5.5b can be obtained by increasing the reservation price A or by increasing
the marginal costs c. In cases where the sequence of scaling factors ".t/ does not
converge learning does not occur in the long run.
Example 5.13.
We now consider the case of a duopoly with heterogeneous play-
ers and we give a detailed study of the region of stability in the space of the
parameters. For N
D
2, the dynamic model (5.91) assumes the form of an iterated
two-dimensional map T
W
."
1
.t/;"
2
.t//
!
."
1
.t
C
1/;"
2
.t
C
1// defined by
A
1
C
B
2c
1
c
2
.c
1
C
c
2
/
;
a
1
3
"
1
.t/
B
2c
2
c
1
"
2
.t/
"
1
.t
C
1/
D
"
1
.t/
C
"
1
.t/
C
(5.106)
A
1
C
B
2c
1
c
2
.c
1
C
c
2
/
:
a
2
3
"
2
.t/
B
2c
2
c
1
"
2
.t/
"
2
.t
C
1/
D
"
2
.t/
C
"
1
.t/
C
In order to study the stability of the unique positive steady state "
D
."
1
; "
2
/
D
.B;B/, we consider the Jacobian matrix computed at the equilibrium
1
;
a
1
a
1
2c
2
c
1
3B
3B
.A
C
2c
1
c
2
/
(5.107)
a
2
2c
1
c
2
3B
a
2
1
3B
.A
C
2c
2
c
1
/
from which the standard stability conditions are obtained (see Appendix F), namely
1
Tr
C
Det > 0; 1
C
Tr
C
Det > 0; Det < 1
(5.108)
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