Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 300

so they have only themselves a unique preimage (as for an invertible map). However,
the situation drastically changes if, for example, some parameter change causes the
minimum value c min to move downwards sufficiently that it goes below q (as in
Fig. C.9b). After the global bifurcation, which occurs when c min D q , the portion
.c min ;q / enters Z 3 , so new preimages f k .c min ;q / appear with k 1.These
preimages constitute an infinite (countable) set of disconnected portions of B . z /

nested inside .A/, represented by the thick portions of the diagonal in Fig. C.9b,

bounded by the infinitely many preimages of any rank, say q k , k 2 N

B

,ofq ,
that accumulate in a left neighborhood of the fixed point r . In fact, as r is a
repelling fixed point for the forward iteration of f , it is an attracting fixed point
for the backward iteration of the same map. So, the contact between the critical
point c min and the basin boundary q marks the transition from simple connected
to disconnected basins. Similar global bifurcations, due to contacts between critical
sets and basin boundaries, also occur in higher dimensional maps.
Also in higher dimensional cases, the global bifurcations which give rise to com-
plex topological structures of the basins, like those formed by disconnected sets,
can be explained in terms of contacts of basin boundaries and critical sets. In fact,
if a parameter variation causes a crossing between a basin boundary and a critical
set which separates different regions Z k so that a portion of a basin enters a region
where an higher number of inverses is defined, then new components of the basin
may suddenly appear at the contact. However, for maps of dimension greater than 1,
such kinds of bifurcations can be very rarely studied by analytical methods, since
the analytical equations of such singularities are not known in general. Hence such
studies are mainly performed by geometric and numerical methods.
Several examples of two-dimensional noninvertible maps that have disconnected
basins can be found in this topic. See also Agliari et al. (2000a, b), Agliari et al.
(2002b), Agliari et al. (2004), Bischi and Kopel (2001), Bischi and Kopel (2003 a ),
Bischi and Naimzada (1999), Bischi et al. (2000a), Bischi et al. (2003b), Puu (2003).
Examples in three dimensions are given in Agliari et al. (2000b) and Bischi et al.
(2001b).