General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 120

change in the qualitative dynamics. At a 2 ' 0:2462 the 2-cycle undergoes a flip
bifurcation and a stable cycle of period 4 appears. As before, just after the bifurca-
tion the four periodic points are close to the 2-cycle saddle, and far from the lines
of non-differentiability. However, as the parameter a 2 is further increased, one of
the periodic points moves towards the folding line F ,andata 2

' 0:2466,aperi-

odic point intersects the boundary of region

.1/ , that is the “folding line” F (see
Fig. 3.8a). This marks the occurrence of a true border collision bifurcation, with the
effect of a transition to a 4-piece chaotic attractor (see Fig. 3.8b with a 2 D 0:26).
As can be seen, the chaotic attractor crosses the folding line F . Hence, it is
bounded by the images of this line, denoted as T .i / .F/, i D 1:::;8,inFig.3.8b.
This suggests that when a chaotic attractor intersects a folding line F , the boundary
of the chaotic area includes points belonging to images of increasing rank of F .
This is a well-known property of the critical lines of smooth noninvertible maps
(see Appendix C), which is here extended to the lines of non-differentiability of a
piecewise differentiable map (see Mira et al. (1996)). As a 2 is further increased,
the 4-cyclic chaotic attractor becomes wider (see Fig. 3.9a) until the merging of the
pieces occurs. This merging leads to a 2-cyclic chaotic attractor (this occurs at a 2 '
0:2765/ and then a unique large chaotic attractor emerges (see Fig. 3.9b), obtained
for a 2 ' 0:2965). Also in this case, the boundary of the chaotic area is given by the
images of a suitable portion of the folding line F . Finally, we once again point out
that in the two cases shown in Fig. 3.9, the upper portion of the chaotic attractors
is included in the region with negative profits, that is above the lines representing
the equation x 1 C .N 1/x 2 D A=c k , k D 1;2. This means that along the chaotic
trajectories that describe the long run time evolution of the production decisions of
the firms, some periods with negative profits are involved.

D

3.1.3 Continuous Time Models and Local Stability

In this section model (1.31), describing the continuous time dynamics of par-
tial adjustment towards the best response with naive expectations, is examined in
the isoelastic case. The Jacobian of the system again has the form (2.46), and
its characteristic equation has the special form of (2.47). We assume again that
a k D ˛ 0 k .0/>0 for all k. Here either all r k values are in the interval . 1;0,or
exactly one r k value is positive. If none of the r k values is positive, then the local
asymptotic behavior of the equilibrium is the same as in the concave case. By adding
up the terms with identical denominators in the bracketed factor of (2.47) we obtain
(2.48), where at most one j >0.Ifall j 0, then the problem is the same as in the
concave case, so the equilibrium is always locally asymptotically stable. Therefore
we may assume that j 0 >0for some j 0 .If j ¤ 0 and m j D 1,then a j .1 C r j /
is not an eigenvalue of the Jacobian. Otherwise it is, and the other eigenvalues are
the roots of the equation