General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 135

the knowledge of the critical curves, we can still analyze the structure of the basins
of the two coexisting stable Nash equilibria and we can characterize the bifurcations
that cause their qualitative changes using numerical and graphical procedures.
As in the case of identical speeds of adjustment, there exists a rather large set of
parameter values for ;a 1 ,anda 2 for which two stable equilibria exist. Moreover,
it is easy to realize that small differences between the two adjustment coefficients do
not cause significant changes in the local stability properties, that is in the modulus
of the eigenvalues. On the other hand, as will be demonstrated below, such small
differences may cause drastic effects with regard to the structure of the basins. Many
of the arguments given in the previous section for the study of the boundaries of the
basins and their global bifurcations continue to hold for non-identical adjustment
speeds. However, there are some important differences.
The main difference is that the diagonal is no longer invariant. Even if
the fixed points remain the same, the basins are no longer symmetric with respect
to .
The preimages of the unstable fixed point O belong to the boundary of the set
of points which generate bounded trajectories, but a simple analytical expression
of the preimages of O cannot be obtained. Since they are solutions of a fourth
degree algebraic equation, they can be computed by standard numerical routines.
For increasing values of or a i the point O enters the region Z 4 .However
the exact values of the parameters at which this occurs cannot be computed
analytically.
Although the boundary which separates the basins of E 1 and E 2 is still formed
by the whole stable set of E S , in the case of a 1 ¤ a 2 the local stable set of E S
is not along the diagonal . The contact between the stable set of E S and the
critical curve LC .b/ , which causes the transition from simple to complex basins,
does not occur at the fixed point O (since now the origin O does not belong to
the stable set of E S ) and no longer involves the cusp point of LC .b/ . Again, the
parameter values at which such contact bifurcations occur cannot be computed
analytically. However, the bifurcation is always caused by a contact between LC
and a basin boundary.
We will finally demonstrate that the occurrence of these bifurcations can be
detected by computer-assisted proofs, based on the knowledge of the properties
of the critical curves and their graphical representation. As mentioned before, this
“modus operandi” is typical in the study of the global bifurcations of nonlinear
two-dimensional maps. Figure 3.15a shows the situation obtained for D 3:6 and
a 1 D 0:55, a 2 D 0:7. The stable set of E S forms the boundary of the basin of E 1 .
On the one hand, the effect of such a small asymmetry in the adjustment speeds on
the local stability properties is negligible. The eigenvalues of the two fixed points are
exactly the same and are very close to the eigenvalues obtained for identical adjust-
ment speeds with the same value of and with, for example, a D .a 1 C a 2 /=2.On
the other hand, as far as the global dynamics is concerned, non-identical adjustment
speeds have a strong effect on the structure of the basins of attraction of the Nash
equilibria E 1 and E 2 . Our numerical simulations show that in general the Nash equi-