The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 46

(see also Appendix C) can be used to obtain the critical curves (for a given set of
parameters):
1. The map (1.46) is continuously differentiable, so the (folding) set LC 1 can
be obtained numerically as the locus of points .x 1 ;x 2 / for which the Jacobian
determinant of T g vanishes.
2. The critical curves LC, which separate the regions Z k , are obtained by comput-
ing the images of the points belonging to LC 1 ,thatisLC D T g .LC 1 /.
In Fig. 1.15a the set of points at which the Jacobian vanishes gives the curves
denoted by LC .a/

1 and LC .b/ 1 . It is formed by the union of the two branches of a

hyperbola. Also the critical curve LC D T g .LC 1 / is formed by two branches,
denoted by LC .a/ D T g .LC .a/ D T g .LC .b/

1 /.ThecurveLC .b/ sep-
arates the region Z 0 , whose points have no preimages, from the region Z 2 , whose
points have two distinct rank-1 preimages. The curve LC .a/ separates the region Z 2
from Z 4 , whose points have four distinct preimages.
Our analysis based on the critical curves of the map now reveals why the set
of initial conditions that lead to convergence to the Nash equilibrium, bounded by
! 1 , ! 2 and its preimages .! 1 / 1 and .! 2 / 1 , is a rather simple set. It is due to the
fact that only preimages of rank-1 of ! 1 and ! 2 exist. Note that .! 1 / 1 and .! 2 / 1
are entirely included in Z 0 , that is a region of the feasible set whose points have no
preimages. Therefore, the preimages .! i / 1 .i D 1;2/ of the invariant axes, have
no preimages of higher rank. Consequently, the whole boundary that separates the
basin B.E/ and the infeasible set B. 1 / is
F D [ nD0 T g .! 1 / [ [ nD0 T g .! 2 / ;

1 / and LC .b/

(1.56)

that is, the union of all the preimages of the segments ! 1 and ! 2 (see Appendix C),
which is a rather simple set.
To conclude this subsection, we would like to stress the fact that the properties of
the basin boundaries are related to the global dynamics of our duopoly model. Such
a simple structure of the basin may be also maintained when the Nash equilibrium
loses stability due to local (period-doubling) bifurcations. In Fig. 1.15b, obtained
with the same parameters as before except that a 1 D 0:015 and a 2 D 0:0165,we
depict a situation where (after the usual period-doubling sequence) a chaotic attrac-
tor describes the long run evolution of the production decisions of the duopolists.
Despite the fact that the dynamic behavior can be considered as complex, the basin
boundaries are still given by the same quadrilateral.
The reader should notice, however, that basins are not always as simple as in the
examples presented so far in this topic. Indeed, a closer look at Fig. 1.15b reveals
that the critical curve LC .b/ is rather close to a basin boundary. This indicates that
a small shift of this curve due to a parameter variation may cause a contact, after
which a portion of the set of infeasible points B. 1 / crosses the critical curve and,
consequently, enters the region Z 2 . In the next subsection we will show that such
contact bifurcations may have a considerable impact on the topological structure of
the feasible set.