General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 136

1
y

1
y

Z 0

( a ) ( a )

LC −1

Z 0

LC

LC ( a )

− 1

LC ( a )

(1)
H −2

E 2

E

Z 2

Z 2

E E S

LC ( b )

LC ( b )

Z 4 (1)

H

Z 4 − 1

(2)

H

E 1

( b ) − 1

E

LC −1

( b )

LC

H 0

− 1

(2)
H −2

x x

0 1 0 1

(a) (b)

Fig. 3.15 Linear inverse demand function and cost externalities. The case of duopoly with differ-
ent speeds of adjustment. ( a )Here D 3:6, a 1 D 0:55, a 2 D 0:7 - the basin of E 1 forms an island

inside the basin of E 2 .( b )Here 0:7 - a contact bifurcation has occurred

and the basin of E 2 becomes a set of disjoint islands inside the basin E 2

D 3:6, a 1 D 0:59, a 2 D

librium E i dominates E j in terms of the size of the basin if a i >a j . Figure 3.15a
shows that although the basin of E 1 is a simply connected set, the basin of E 2 is
now multiply connected. The basin of E 1 forms a big “hole” (or “island,” to use the
term of Mira et al. (1996)) inside the basin of E 2 . The stable set of E S , that is the
boundary which separates the two basins, is entirely included inside the regions Z 2
and Z 0 . Note, however, that the stable set of E S is close to the critical curve LC,
which is a signal for the occurrence of a global bifurcation. If a change in parame-
ters causes a contact between the stable set of E S (a basin boundary) and LC,then
this contact marks a bifurcation which normally causes a qualitative change in the
structure of the basins.
This is demonstrated in Fig. 3.15b, where D 3:6 and a 1 D 0:59, a 2 D 0:7.
Such a small change in the adjustment speed of player 2 causes a portion of the
basin of E 1 to enter the region Z 4 (denoted by H 0 in the figure). Consequently,
new rank-1 preimages of that portion will appear near LC .b/

1 , and such preimages
must belong to the basin of E 1 . These rank-1 preimages, denoted by H .1/

1 and H .2/ 1 ,

are located at opposite sides with respect to LC .b/

1 and merge onto it. Obviously,

D H .1/ [ H .2/ the set H 1

1 constitutes a disconnected portion of the basin of E 1 .
Moreover, since H 1 belongs to the region Z 4 , it also has four rank-1 preimages.
Two of them are located in the strategy space

1

and are denoted by H .j /

2 , j D 1;2.
Points belonging to these “islands” are mapped into H 0 in two iterations of the map
T . Indeed, infinitely many higher rank preimages of H 0 exist, even if only some of
them are inside the strategy space S D Œ0;1 Œ0;1, thus giving smaller disjoint
“islands” of the basin of E 1 . Hence, at the contact between the stable set of E S and

S