Concave Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 67

to (1) the crossing of the equilibria (or points of a cycle) through sets where the
dynamical system is not differentiable, and (2) these sets of non-differentiability
separate regions where the maps that represent the dynamical system are differen-
tiable, but different. From the viewpoint of economics and oligopoly theory, these
bifurcations seem to be important since they may cause sudden stability switches
and/or the appearance or disappearance of equilibria, cycles, or chaotic attractors.
Although the study of border collision bifurcations is quite new even in the mathe-
matical literature, some studies report this phenomenon for economic models with
constraints (see for example Hommes (1991, 1995), Hommes and Nusse (1991),
Hommes et al. (1995), Puu and Sushko (2002), Puu et al. (2005), Puu and Sushko
(2006), Sushko et al. (2005, 2006). Some of the main results on this subject can
be found in Nusse and Yorke (1992, 1995), Maistrenko et al. (1993, 1995, 1998),
Di Bernardo et al. (1999), Banerjee et al. (2000a, b), Halse et al. (2003), Zhany-
bai and Mosekilde (2003), Zhusubaliyev et al. (2002, 2007). However, we should
stress that the study of the global dynamical properties of piecewise differentiable
dynamical systems is still at a pioneering stage. Many open problems still await a
systematic approach, even in the case of one-dimensional maps, see for example the
recent papers by Avrutin and Schanz (2006), Avrutin et al. (2006).
To show that after an equilibrium has lost its stability via a border collision bifur-
cation any kind of attractor may be created, in Fig. 2.6 we present a bifurcation
diagram of production x obtained for A D 16, B D 1, a D 0:5, c D 6 and N D 15
and increasing values of the capacity limit L in the range Œ0:2;1. In this case at the
bifurcation value L D

c
.N C1/B D 0:625 the stable equilibrium x D L becomes
unstable by crossing the kink of the map T , and a chaotic attractor is suddenly
created.
Dynamical systems generated by one-dimensional differentiable maps are among
the most frequently studied in the literature. It is well-known, for example, that
the critical point of the map, where the derivative vanishes, and its images play
an important role in deriving the bounds of the attractors in a bifurcation diagram
as well as the regions of higher density of points (see for instance Gumowski and
Mira (1980), Collet and Eckmann (1980), Mira et al. (1996)). In higher-dimensional
dynamical systems based on noninvertible maps the critical curves introduced ear-
lier assume this important role. For a piecewise differentiable map, like the one
encountered in the present example, the two kink points, the relative maximum
and minimum point, can be used to obtain the upper and lower boundaries of the
(chaotic) attractors. These points assume the role of critical (that is, folding) points
in our noninvertible map. Indeed, the piecewise linear map T is a noninvertible map
of Z 1 Z 3 Z 1 kind (see Appendix C). However, in contrast to the situations
studied before, these critical points are not found by looking for points of vanish-
ing derivative, as for differentiable maps, but they are the points where the map is
non-differentiable. In Fig. 2.7a, obtained with L D 0:8 and the other parameters as
in Fig. 2.6, the upper boundary of the attractor is the maximum value denoted by m,
and the lower boundary is its image m 1 D T.m/. In Fig. 2.7b, obtained with L D 0:9,
the chaotic interval is Œm 1 ;m. The property that the dynamics are trapped between
the critical points and their iterates is useful to bound the chaotic attractors in the

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