Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 296

generated from an initial condition in .0;1/,entersŒc 1 ;cafter a finite number of
iterations. Following the terminology introduced in Mira et al. (1996), the interval
Œc 1 ;cis called absorbing .
In general, for an n-dimensional map, an absorbing region

A R

(intervals in ,

3 ;:::) is defined as a bounded set whose boundary is
given by portions of the critical set CS and its images of increasing order CS k

R

2 , volumes in

R

areas in

D

T k .CS/, such that a neighborhoodU A exists whose points enter A after a finite

number of iterations and then never escape it, since T. A / A , which is to say that

A is trapping (see for example Mira et al. (1996) for more details).

Loosely speaking, we can say that the iterated application of a noninvertible map,
folding and folding again the space, defines trapping regions bounded by critical sets
of increasing order.
Sometimes, smaller absorbing regions are nested inside a bigger one. This can
be illustrated, again, for the logistic map (C.1), as shown in Fig. C.6a, where inside
the absorbing interval Œc 1 ;c a trapping subset is obtained by higher rank images
of the critical point, given by A D Œc 1 ;c 3 [ Œc 2 ;c.InFig.C.6bitisshownthat,

for the same parameter value D 3:61 as in Fig. C.6a, the numerical iteration of the
logistic map gives points that are trapped inside the two-cyclic interval

.
Inside an absorbing region one or more attractors may exist. However, if a chaotic
attractor exists which fills up a whole absorbing region then the boundary of the
chaotic attractor is formed by portions of critical sets. This is the situation shown
in Fig. C.6a, b, where the absorbing interval

A

A D Œc 1 ;c 3 [ Œc 2 ;cis invariant and
filled up by a chaotic trajectory,asshowninFig.C.6b.
To better illustrate the foregoing point, we also give a two-dimensional example,
obtained by using the map (C.5). In Fig. C.7a, a chaotic trajectory is shown, and in
Fig. C.7b its outer boundary is obtained by the union of a segment of LC and three
iterates LC i

D T i .LC/, i D 1;2;3.

1

x t

c c

c 2 = f ( c 1 )
c 3 = f ( c 2 )

.8

c 2

c 3

.6

.4

c 1 = f ( c )

c 1

.2

0 10 20 30 40 50 60 70 80 90

t

100

0 1

(a) (b)

Fig. C.6 Illustrating a trapping subset inside the absorbing set of Fig. C.5 for the quadratic map

with D 3:61.( a ) The delineation of the trapping subset Œc 1 ;c 3 [ Œc 2 ;c.( b ) The iterates of the

map remain trapped inside the two cyclic interval