Chemistry Reference
In-Depth Information
x
'
x
'
x
'
Z
0
Z
1
Z
c
M
1
c
c
Z
2
Z
3
Z
2
c
m
Z
1
2
c
Z
x
c
−1
c
−1
c
−
x
c
−1
(a)
x
1
(b)
(c)
Fig. C.2
The preimage regions of certain maps. (
a
) The tent-map. (
b
) A bimodal piecewise linear
map. (
c
) A discontinuous map. Notice that in (
a
)and(
b
) the number of preimages in adjacent
regions differ by 2, whereas in (
c
)theydifferby1
a typical Z
0
Z
2
tent map is shown, where the kink point behaves like the critical
point of the logistic map even if it is not obtained as the image of a point with
vanishing derivative. The same reasoning applies to the “bimodal” Z
1
Z
3
Z
1
piecewise linear function shown in Fig. C.2b.
Up to now we have considered continuous maps, but the properties of critical
points can easily be extended also to piecewise continuous maps T . In this case a
point of discontinuity may behave as a critical point of T , even if the definition in
terms of merging preimages cannot be applied. This happens when the ranges of
the map on the two sides of the discontinuity have an overlapping zone, so that at
least one of the two limiting values of the function at the discontinuity separates
regions having a different number of rank-1 preimages (see for example the map
shown in Fig. C.2c). The difference with respect to the case of a continuous map
is that now the number of distinct rank-1 preimages through a critical point differs
generally by one (instead of two), that is, a critical value c (in general the critical
set CS) separates regions Z
k
and Z
kC1
: A one-dimensional example is shown in
Fig. C.2c, where the point of discontinuity is a critical point c
1
, and both the two
limiting values of the function in c
1
are critical points, say c
1
and c
2
, associated
with c
1
; as both c
1
and c
2
separate regions Z
1
and Z
2
: Notice that now the critical
points have no merging rank-1 preimages. More on the properties and bifurcations
of discontinuous maps of the plane can be found in Mira et al. (1996).
In order to explain the geometric action of a critical point in a continuous map,
let us consider, again, the logistic map, and note that as x moves from 0 to 1 the
corresponding image f.x/spans the interval Œ0;c twice, the critical point c being
the turning point. In other words, if we consider how the segment
D
Œ0;1 is trans-
formed by the map f , we can say that it is
folded and pleated
to obtain the image
0
D
Œ0;c. Such folding gives a geometric reason why two distinct points of ,say
x
1
and x
2
, located symmetrically with respect to the point c
1
D
1=2, are mapped
into the same point x
0
2
0
due to the folding action of f (see Fig. C.1b). The same
conclusions can be obtained by looking at the two inverse mappings f
1
1
and f
1
2
Search WWH ::
Custom Search