Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 71

computed at the two periodic points : .C 2 / D .1 a/ 1

2 .Fortheset
of parameters used in Fig. 2.9(b) we obtain .C 2 / D 0:72, hence it is stable,
however .C 2 / decreases for increasing values of N and it reaches the bifurcation
value .C 2 / D 1 at N D

a.N C 1/

1a 1 D 12:33 for a D 0:4 (see Fig. 2.5).
When the cycle C 2 becomes unstable, a stable cycle of period 4,sayC 4 , appears
(see Fig. 2.9(c), obtained for N D 13) and a further increase of N leads to a period
halving bifurcation, typical of bimodal maps (one-dimensional maps with a local
maximum and a local minimum), at which C 4 is replaced by another stable cycle of
period 2, with periodic points located on the first and third branches (as in Fig. 2.9(d)
for N D 15). This means that the multiplier associated with this cycle is .1 a/ 2 ,
independent of N, from which we deduce that this 2-cycle will remain stable for
each value of N.
To sum up, this example has illustrated that in a piecewise linear map, even if the
loss of stability of an equilibrium point or of a periodic cycle is related to the local
values of their multipliers, the effects of such bifurcations, as well as the location
of the emerging attractors in the phase space, is related to the global shape of the
iterated map. In particular, the periodic cycles emerging from such bifurcations have
periodic points belonging to branches of the map that are far from the bifurcating
equilibrium. This property also holds for piecewise linear dynamical systems of
dimension greater than one, as we shall see in the next example.

a 2 a

2

Example 2.4. In this example we consider the semi-symmetric oligopoly, which is
obtained by assuming c 2 D ::: D c N , a 2 D ::: D a N ,andL 2 D ::: D L N : Further
let x 1 .0/ and x 2 .0/ D ::: D x N .0/ denote the initial production quantities of the
firms. If the firms partially adjust their production quantities towards the best replies
with linear adjustment functions, then the decisions made by firm 1 and the identical
firms 2;:::;N are captured by the two-dimensional dynamical system

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 R 1 ..N 1/x 2 .t//;
x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 R 2 .x 1 .t/ C .N 2/x 2 .t//;

T W (2.34)

where 8

<

Ac 1
B.N

0 if x 2

1/ ;

A c 1 2L 1

N

R 1 ..N 1/x 2 / D

L 1 if x 2

1 ;

:

B.N 1/

A c 1

2B

1

2 .N 1/x 2

otherwise;

and

8
<

A c B ;

0 if x 1 C .N 2/x 2

A c 2
B

R 2 .x 1 C .N 2/x 2 / D

L 2 if x 1 C .N 2/x 2

2L 2 ;

:

A c 2

2B

1

2 Œx 1 C .N 2/x 2 otherwise: