General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 131

this case the basins are no longer simply connected sets, and portions of each basin
are present both in the region above and below the diagonal . The basins are now
disconnected sets, and the adjustment process starting from initial conditions below
or above the diagonal may lead to convergence to either E 1 or E 2 .
The transition from simply connected basins to disconnected basins is caused
by a global bifurcation. We will now describe the mechanism which causes this
bifurcation in more detail. The argument begins by noticing that the map T defined

in (3.23) is noninvertible. Given a point x 0 1 ;x 0 2 2 S

, its preimages are computed
by solving with respect to x 1 and x 2 the algebraic system
8
<

.1 a/x 1 C ax 2 .1 x 2 / D x 0 1 ;

(3.26)

:

.1 a/x 2 C ax 1 .1 x 1 / D x 0 2 :

As noticed before, this is a fourth degree algebraic system, which may have four
or two real solutions, or no real solution at all. Hence, the strategy set

can be
subdivided into the regions Z 4 , Z 2 ,andZ 0 , separated by branches of the critical
curve LC. For the differentiable map (3.23) the curve LC 1 coincides with the set
of points at which the determinant of the Jacobian matrix vanishes (see Appendix C)
so that

S

x 1 x 2

.1 a/ 2
4a 2 2 :

1
2

1
2

D (3.27)

Equation (3.27) represents an equilateral hyperbola. The curve LC 1 is formed
by the union of two disjoint branches, say LC 1 D LC .a/ [ LC .b/

1 , which are
depicted in Fig. 3.13a. Also its image LC D T.LC 1 / is the union of two branches,
LC .a/

1

D T.LC .a/ D T.LC .b/

1 / and LC .b/ 1 /. This is shown in Fig. 3.13b. The

branch LC .a/ separates the region Z 0 , whose points have no preimages, from the
region Z 2 , whose points have two distinct rank-1 preimages. The other branch
LC .b/ separates the region Z 2 from the region Z 4 , whose points have four distinct
preimages. 4 In order to give a geometrical interpretation of the “unfolding action”
of the multivalued inverse T 1 , it is useful to consider a region Z k as the super-
position of k sheets, each associated with a different inverse. Such a representation
is known as Riemann foliation of the plane (see for example, Mira et al. (1996)).
Different sheets are connected by folds joining two sheets, and the projections of
such folds on the phase plane are arcs of LC. The foliation associated with the map
(3.23) is qualitatively represented in Fig. 3.13c. It can be noticed that the cusp point
of LC .b/ denoted by K is characterized by three merging preimages at the junction
of two folds.
This cusp point K of LC .b/ plays a crucial role in the analysis, since when K

enters the strategy set S (for a. C 1/ > 1, see below), suddenly points of S have

4 Following the terminology of Mira et al. (1996), we say that the map (3.23) is a noninvertible
map of Z 4 >Z 2 Z 0 type, where the symbol “> ” denotes the presence of a cusp point in the

branch LC .b/ .