Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 294

This means that if x 2 A then T.x/ 2 A,sothat Ais trapping, and every point of A
is an image of some point of A. A closed invariant set A is an attractor if (1) it is
Lyapunov stable, that is for every neighborhood W of A there exists a neighborhood
V of A such that T t .V/ W 8 t 0; (2) a neighborhood U of A exists such that
T t . x / ! A as t !C1 for each x 2 U .
The basin of an attractor A is the set of all points that generate trajectories
converging to A

.A/ D ˚ x j T t . x / ! A as t !C1 :

B (C.10)

Let U.A/be a neighborhood of an attractor A whose points converge to A. Of course
U.A/ B

.A/, and also the points that are mapped into U after a finite number of
iterations belong to B .A/. Hence, the basin of A is given by

[

T n .U.A//;

B .A/ D (C.11)

nD0

where T n .x/ represents the set of the rank-n preimages of x (the points mapped
into x after n applications of T ).
Let B be a basin of attraction and @ B its boundary. From the definition it follows

that is trapping with respect to the forward iteration of the map T and invariant

with respect to the backward iteration of all the inverses T 1 . Points belonging
to @

B

both under forward and backward iteration of T .This
implies that if an unstable fixed point or cycle belongs to @

B are mapped into @ B

must also
contain all of its preimages of any rank. In particular, if a saddle point, or a saddle
cycle, belongs to @

B then @ B

must also contain the whole stable set (see Gumowski
and Mira (1980), Mira et al. (1996)).
A problem that often arises in the study of nonlinear dynamical systems con-
cerns the existence of several attracting sets, each with its own basin of attraction.
In this case the dynamic process becomes path dependent, which means that the
kind of long-run dynamics that characterizes the system depends on the starting
condition. Another important problem in the study of applied dynamical systems is
the delineation of a bounded region of the state space in which the system dynamics
are ultimately trapped, despite the complexity of the long-run time patterns. This is
useful information, even more useful than a detailed description of the step-by-step
time evolution.
Both of these questions require an analysis of the global properties of the dynam-
ical system, that is, an analysis which is not based on the linear approximation of
the map. When the map T is noninvertible, its global dynamical properties can be
usefully characterized by using the formalism of critical sets, by which the folding
action associated with the application of the map, as well as the “unfolding” associ-
ated with the action of the inverses, can be described. Loosely speaking, the repeated
application of a noninvertible map repeatedly folds the state space along the critical
sets and their images, and often this allows one to define a bounded region in which

B ,then@ B