The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 48

6

Z 0

x 2

E 2

Z 2

LC ( b )

Z 4

LC ( a )

x 1

0 E 1 6

Fig. 1.17 The Cournot duopoly with a gradient type adjustment process and linear
demand/quadratic cost. The same situation as in Fig. 1.16, but with slightly higher speeds of
adjustment. Note how the holes have become larger and connected along the vertical axis

H .2/

2 , contain initial conditions which are mapped into the main hole and then into
the infeasible set. The sets H .1/

2 and H .2/

2 are bounded by preimages of rank 3
of ! 1 . Since these smaller holes are again both inside Z 2 , each of them has again
two further preimages inside Z 2 , and so on. Summarizing, we can conclude that
the global bifurcation which we have just described transforms a simply connected
basin into a multiply connected basin. The latter set has a countably infinite number
of holes, called an arborescent sequence of holes , which belong to the infeasible
set B. 1 /. As the speeds of adjustment are further increased, the holes become
more pronounced and they become connected along the vertical axis as shown in
Fig. 1.17.
Our numerical results show that the structure of the basins may become consider-
ably more complex as the adjustment speeds are increased. The transition between
qualitatively different structures of the boundary occur through so called contact
bifurcations (see for example Mira et al. (1996)) and these bifurcations can be
described in terms of contacts between the basin boundaries and arcs of the crit-
ical curves . To conclude this chapter, we would like to stress that in general there
is no relation between the bifurcations which change the qualitative properties of
the basins (global bifurcations) and the bifurcations which change the qualitative
properties of the attractor (sequences of local bifurcations). The former is related to
the global dynamics, whereas the latter focuses on the local (stability) properties.
In later chapters we will encounter situations where the attractor is a rather sim-
ple set (that is, an equilibrium), but the structure of its basin is quite complex. As
demonstrated above, in other situations exactly the opposite might be the case.