Noninvertible Maps and Critical Sets - Nonlinear Oligopolies: Stability and Bifurcations - page 293

n

CS 1 J 0 D f p 2 R j det J .p/ D 0 g (C.7)

In fact, in any neighborhood of a point of CS 1 there are at least two distinct
points which are mapped by T in the same point. Accordingly, the map is not locally
invertible in points of CS 1 , and (C.7) follows from the implicit function theorem.
This property provides an easy method to compute the critical set for continuously
differentiable maps - from the expression of the Jacobian determinant one computes
the locus of points at which it vanishes, then the set obtained after an application of
the map to these points is the critical set CS.
Also the geometric properties illustrated above for the two-dimensional nonin-
vertible map (C.5) can be easily generalized to the case of the critical set of an
n-dimensional noninvertible map. It is worth noting that, in general, for piecewise
differentiable maps the set of points where the map is not differentiable may belong
to CS 1 , that is the images by T of such points may separate regions characterized
by a different number of rank-1 preimages (see for example Mira (1987)). Moreover,
piecewise continuous maps may have points of CS 1 at the discontinuities and, dif-
ferently from the case of continuous maps, the corresponding portions of CS may
separate regions that differ by an odd number of preimages (see Mira (1987)). In any
case, the importance of the set CS lies in the fact that its points separate regions Z k
characterized by a different number of preimages. This property may also be shared
by points where some inverses are not defined due to a vanishing denominator, as
shown in Bischi et al. (1999, 2001a, 2003a).

C.2 Discrete Time Dynamical Systems as Iterated Maps

A discrete-time dynamical system , defined by the difference equation

x .t C 1/ D T. x .t//; (C.8)

can be viewed as the result of the repeated application (or iteration )ofamapT .
Indeed, the point x represents the state of a system, and T represents the “unit time
advancement operator” T W x .t/ ! x .t C 1/. Starting from an initial condition
x 0 2 S, the iteration of T inductively defines a unique trajectory
. x 0 / D ˚ x .t/ D T t . x 0 /;t D 0;1;2;::: ; (C.9)

where T 0 D T.T t 1 /.Ast !C1 , a trajector y may

diverge , o r it m ay converge to a fixed point of the map T , which is a point x such
that T. x / D x . It may also asymptotically approach another kind of invariant set,
such as a periodic cycle , or a closed invariant curve or a more complex attrac-
tor, for example a so called chaotic attractor (see for example Devaney (1989),
Guckenheimer and Holmes (1983) and Medio and Lines (2001)). We recall that
asetA R

is the identity map and T t

n

is invariant for the map T if it is mapped onto itself, T.A/ D A.