Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 290

defined in . 1 ;a=4 according to (C.2). We can consider the range of the map f
formed by the superposition of two half-lines . 1 ;a=4, joined at the critical point
c D a=4 (Fig. C.1c), and on each of these half-lines a different inverse is defined. In
other words, instead of saying that two distinct maps are defined on the same half-
line we say that the range is formed by two distinct half lines on each of which a
unique inverse map is defined. This point of view gives a geometric visualization of
the critical point c as the point in which two distinct inverses merge. The action of
the inverses, say f 1 D f 1 [ f 2 , causes an unfolding of the range by mapping

c into c 1 and by opening the two half-lines one on the right and one on the left of
c 1 , so that the whole real line R is covered. So, the map f folds the real line, the

two inverses unfold it.
Another interpretation of the folding action of a critical point is the following.
Since f.x/is increasing for x 2 Œ0;1=2/ and decreasing for x 2 .1=2;1, its appli-
cation to a segment 1 Œ0;1=2/ is orientation preserving, whereas its application
to a segment 2 .1=2;1 is orientation reversing. This suggests that an application
of f to a segment 3 D Œa;b including the point c 1 D 1=2 preserves the orienta-
tion of the portion Œa;c 1 ,thatisf.Œa;c 1 / D Œf.a/;c, whereas it reverses the
portion Œc 1 ;b,sothatf.Œc 1 ;b/ D Œf.b/;c,sothat 3

D f. 3 / is folded, the

folding point being the critical point c.
Let us now consider the case of a continuous two-dimensional map T W S ! S,
S R

2 ,definedby

x 0 1

D T 1 .x 1 ;x 2 /;

T W (C.3)

x 0 2 D T 2 .x 1 ;x 2 /:

If we solve the system of the two (C.3) with respect to the unknowns x 1 and x 2 ,

then, for a given x 0 1 ;x 0 2 , we may have several solutions, representing rank-1 preim-

ages (or backward iterates) of x 0 1 ;x 0 2 ,say.x 1 ;x 2 / D T 1 x 0 1 ;x 0 2 ,whereT 1 is
in general a multivalued relation. In this case we say that T is noninvertible, and
the critical set (formed by critical curves, denoted by LC from the French “Ligne
Critique”) constitutes the set of boundaries that separate regions of the plane charac-
terized by a different number of rank-1 preimages. According to the definition, along
LC at least two inverses give merging preimages, located on LC 1 (following the
notations of Gumowski and Mira (1980), Mira et al. (1996)).
For a continuous and (at least piecewise) differentiable noninvertible map of the
plane, the set LC 1 is included in the set where det J .x 1 ;x 2 / changes sign, with
J being the Jacobian matrix of T ,sinceT is locally an orientation preserving
map near points .x 1 ;x 2 / such that det J .x 1 ;x 2 />0and orientation reversing if
det J .x 1 ;x 2 /<0. In order to explain this point, let us recall that when an affine
transformation x 0 D A x C b ,where A D ˚ a ij is a 2 2 matrix and b 2R

2 ,is
applied to a plane figure, then the area of the transformed figure grows, or shrinks, by
a factor D j det Aj ,andif det A >0then the orientation of the figure is preserved,
whereas if det A <0then the orientation is reversed. This property also holds for
the linear approximation of (C.3) in a neighborhood of a point p D .x 1 ;x 2 /,given
by an affine map with A D J , J being the Jacobian matrix evaluated at the point p