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See [Deva86] for a much more thorough discussion of these ideas. It turns out
that almost any nonlinear feedback process of this type leads to an interesting dynam-
ical system. In particular, let us look at rational maps of the complex plane C . These
provide a rich source of chaotic maps
Consider the map
f c
:
CC
Æ
defined by
() =+
2
f c zzc
.
(22.2)
This map was studied by Mandelbrot [Mand83] and looks innocent enough, but looks
are deceiving. We have the sequence
2
() =+
z
=
f
z
z
c
.
n
+
1
c
n
n
The trivial case where c = 0 is easy to analyze. If | z |<1, then the z n will converge to
0. If | z |>1, then the z n will converge to •. On the other hand, things are not so simple
for may other values of c . For example, Figure 22.2 shows the case where c = 0.31 +
0.04 i . Iterates of points in the interior of the black region converge to the point labeled
“attractive fixed point” (the terminology will be explained shortly) and those of points
outside the region converge to •. The boundary curve of the black region is a very
wild curve.
We need some more definitions. Assume f: R n Æ R n . (This includes the important
special case of complex maps since the complex plane can be identified with R 2 .)
Definition. A fixed point p of f is a repelling fixed point of f if f ¢( p ) has eigenvalues
larger than 1. The point p is an attractive fixed point of f if f ¢( p ) has eigenvalues smaller
than 1. If p is a periodic point of period k, then p is called an attractive periodic point
if it is an attractive fixed point of f k .
Definition.
If p is an attractive fixed point of f, then its basin of attraction , A(f, p ), is
defined by
{
}
k
A f p
(
,
) =
xxp
f
() Æ
as k
.
The map z 2
Figure 22.2.
+ (0.31 + 0.04 i ).
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