In earlier parts of this course, fractal behavior and chaotic dynamics
originated in our consideration of collections of more than one linear
mapping. Linear mappings by themselves generally have fairly simple
dynamics; the exception to that rule has been rotations by an
irrational multiple of 
. The fragmentation of our dynamical
system into more than one linear mapping is the key to their
complicated dynamics. That fragmentation is also exhibited by a single
nonlinear function. In this part of the course, we will consider
iteration of a single function f(x) of a real variable, that is,
, 
, just as described before. (We also set
.) The novelty will be that f(x) can be almost any
function, for instance, 
or even
It is important here that the function be continuous and act as a mapping
of some interval (possibly 
) back into itself, so
that we can consider the dynamics.
Many of the concepts we have developed earlier are still valid. For
any starting point 
, the forward orbit of under
f is the set 
(we borrow a lot of notation from
[Dev89b]) of all iterates 
. On the other hand, the
backwards orbit 
is the set of all y such that
for some 
. In the case of a polynomial f(x) of
degree n, there could be as many as n solutions y to 
.
Thus, in general the backwards orbit gets very numerous very quickly
as 
. We are again interested in the long-term behavior of the
forward orbit.
There is an extremely nice graphical method for viewing the iteration
of a function. An example is shown here. In this picture, two graphs are drawn:
the graph of y=f(x)=3.7x(1-x) and the graph of y=x. Then a
sequence of of line segments is drawn starting from 
. The
first iterate is 
. It is the height of the graph of
f at 
. We draw a vertical line from 
to 
.
To calculate 
, we must move to the location of 
on the
horizontal axis. This is tantamount to drawing a horizontal line
segment from to 
on the line y=x (point
number 3). Again 
is the height of y=f(x) at
that location. Thus, a vertical line segment connects to
point number 4 which is 
. This continues, passing through
, 
, 
, 
, 
, etc.
The beautiful trail of segments resembles the craft of a spider
spinning its web from branch to branch. This picture is commonly
called a web diagram.
It is fairly easy to program web diagrams in MAPLE, and we include a worksheet that does just this. Before proceeding with our theoretical study, the reader might enjoy obtaining this worksheet
webpic.ms: MAPLE worksheet for drawing web diagrams for arbitrary functions, with lots of examples, some of which are discussed in this chapter.
